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# Coupled maps: local bifurcations of fixed points initiate global phase portrait changes.

Abstract

We present two map examples such that bifurcations of their fixed point which is embedded Inserted into. See embedded system.  in a topologically to·pol·o·gy
n. pl. to·pol·o·gies
1. Topographic study of a given place, especially the history of a region as indicated by its topography.

2.
transitive transitive - A relation R is transitive if x R y & y R z => x R z. Equivalence relations, pre-, partial and total orders are all transitive.  invariant (programming) invariant - A rule, such as the ordering of an ordered list or heap, that applies throughout the life of a data structure or procedure. Each change to the data structure must maintain the correctness of the invariant.  chaotic chaotic /cha·ot·ic/ (ka-ot´ik) completely confused, disorganized, or irregular.  set can generate global map phase portrait A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is representated by a different curve, or point. Examples
Simple pendulum see picture (right).
changes. To be more precise, we consider two coupled map families such that the family maps all have the same fixed point which is nested within the same topologically transitive invariant set which is nested in turn within the same invariant subspace In mathematics, an invariant subspace of a linear mapping

T : VV

from some vector space V to itself is a subspace W of V such that T(W) is contained in W.
. We prove in such a case that these point bifurcations which are transversal to the invariant subspace generate two periodic of period 2 points in a neighbourhood of the given point and besides can simultaneously give rise to orbits The following is a list of types of orbits: By orbital characteristics
• Box orbit
• Circular orbit
• Ecliptic orbit
• Elliptic orbit
• Highly Elliptical Orbit
• Graveyard orbit
• Hohmann transfer orbit
that are homoclinic to the periodic points. These orbits appear suddenly and consist of points of transversal intersections of stable manifolds In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor.  and unstable unstable,
adj 1. not firm or fixed in one place; likely to move.
2. capable of undergoing spontaneous change. A nuclide in an unstable state is called
radioactive. An atom in an unstable state is called
excited.
ones built up at the periodic points. Therefore, at a moment immediately just after the bifurcation Bifurcation

A term used in finance that refers to a splitting of something into two separate pieces.

Notes:
Generally, this term is used to refer to the splitting of a security into two separate pieces for the purpose of complex taxation advantages.
, a countable set “Countable” redirects here. For the linguistic concept, see Count noun.

In mathematics, a countable set is a set with the same cardinality (i.e., number of elements) as some subset of the set of natural numbers.
of periodic points and, moreover, a whole large invariant topologically transitive set In set theory, a set (or class) A is transitive, if
• whenever xA, and yx, then yA, or, equivalently,
• whenever xA, and x is not an urelement, then
appear in a neighbourhood of the invariant set. Thus, in the case under study, a local bifurcation of fixed point initiates a global one of phase portrait of map. AMS AMS - Andrew Message System  subject classification: 37G35, 37D45, 37E99.

Keywords Keywords are the words that are used to reveal the internal structure of an author's reasoning. While they are used primarily for rhetoric, they are also used in a strictly grammatical sense for structural composition, reasoning, and comprehension. : Bifurcation, dynamical systems Dynamical Systems

A system of equations where the output of one equation is part of the input for another. A simple version of a dynamical system is linear simultaneous equations. Non-linear simultaneous equations are nonlinear dynamical systems.
, fixed points, coupled maps.

1. Introduction and Main Results

Although there are very many results concerning dynamical system dynamical system
n.
Mathematics A space together with a transformation of that space, such as the solar system transforming over time according to the equations of celestial mechanics.

Noun 1.
properties, one can determine very seldom a priori a priori

In epistemology, knowledge that is independent of all particular experiences, as opposed to a posteriori (or empirical) knowledge, which derives from experience.
whether a system has a certain property or not. A bright example is orbits homoclinic (biasymptotic) to fixed and periodic points. It is well known how much of dynamical system phase pattern complexity arises in a vicinity of these orbits. However we know very little on causes and conditions favourable for an appearance of such orbits. The present paper is devoted to the mechanics mechanics, branch of physics concerned with motion and the forces that tend to cause it; it includes study of the mechanical properties of matter, such as density, elasticity, and viscosity.  of appearance of new homoclinic orbits In mathematics, a homoclinic orbit is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium.  at maps used to study a synchronization (1) See synchronous and synchronous transmission.

(2) Ensuring that two sets of data are always the same. See data synchronization.

(3) Keeping time-of-day clocks in two devices set to the same time. See NTP.
phenomenon. These are so-called so-called
1. Commonly called: "new buildings ... in so-called modern style" Graham Greene.

2.
coupled maps. We shall examine further the coupled maps which have a fixed point embedded into the map "diagonal". The latter is their invariant subspace. For such kind of maps, we shall show that these point bifurcations which are transversal to the "diagonal" can initiate INITIATE. A right which is incomplete. By the birth of a child, the husband becomes tenant by the curtesy initiate, but his estate is not consummate until the death of the wife. 2 Bouv. Inst. n. 1725.  a new transversal homoclinic orbit appearance and, as a corollary corollary: see theorem. , generate global changes of the phase map portrait An orientation in which the data is printed across the narrow side of the form. Contrast with landscape.

. Thus bifurcations of the fixed point embedded in the invariant subspace (to be more precise, in the invariant topologically transitive set of the given subspace Noun 1. subspace - a space that is contained within another space
mathematical space, topological space - (mathematics) any set of points that satisfy a set of postulates of some kind; "assume that the topological space is finite dimensional"
) can differ very strongly from those for isolated fixed points.

So, let [??] = ([??], [??]). [R.sup.2]. Consider

[MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression.  NOT REPRODUCIBLE re·pro·duce
v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es

v.tr.
1. To produce a counterpart, image, or copy of.

2. Biology To generate (offspring) by sexual or asexual means.
IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (1.1)

Here [[PHI phi
n.
Symbol The 21st letter of the Greek alphabet.

PHI,
n See health information, protected.
].sub.a](t) = at (1-t) and a [member of] (2, 4), [epsilon] [member of] (0, 1/2) are parameters. Usually a is called the coupling parameter The coupling parameter of the resonator, specifies the part of the energy of the laser field, which is output at each round-trip.

The coupling parameter should not be confused with the round-trip loss, which refers to the part of the energy of the ...?...
. This map family is known as one of coupled logistic maps The logistic map is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time  and is used very often as a sample for researching of coupled map properties. It is easy to show that there are parameter (1) Any value passed to a program by the user or by another program in order to customize the program for a particular purpose. A parameter may be anything; for example, a file name, a coordinate, a range of values, a money amount or a code of some kind.  values such that the map [??] has a nontrivial nontrivial - Requiring real thought or significant computing power. Often used as an understated way of saying that a problem is quite difficult or impractical, or even entirely unsolvable ("Proving P=NP is nontrivial"). The preferred emphatic form is "decidedly nontrivial".  topologically transitive attractor on the "diagonal"[??] = [??]. Indeed, denote de·note
tr.v. de·not·ed, de·not·ing, de·notes
1. To mark; indicate: a frown that denoted increasing impatience.

2.
by [??] [subset A group of commands or functions that do not include all the capabilities of the original specification. Software or hardware components designed for the subset will also work with the original. ] (0,4) a set of values of a such that [[??].sub.a] : t [right arrow] [[??].sub.a](t) has the aforesaid Before, already said, referred to, or recited.

This term is used frequently in deeds, leases, and contracts of sale of real property to refer to the property without describing it in detail each time it is mentioned; for example,"the aforesaid premises.
attractor provided a [member of] [??]. As is well known [6], [??] [not equal to] = [??]. Moreover, mes([??]) > 0, where mes(.) is the Lebesgue measure In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. . Therefore, if a [member of] [??] and the "diagonal" is an asymptotically stable subspace, then [??] has the aforementioned a·fore·men·tioned
Mentioned previously.

n.
The one or ones mentioned previously.

aforementioned

mentioned before

attractor.

Further we shall deal with the following map family of [R.sup.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

where z [equivalent to] (x, y) [member of] [R.sup.2], and a, a are the same as in (1.1). The family (1.2) can be obtained from (1.1) by a change of variables (see [4]). At that the "diagonal"[??] = [??] transforms to the y-axis See X-Y matrix. .

Denote O = (0, 0), [LAMBDA The Greek letter "L," which is used as a symbol for "wavelength." A lambda is a particular frequency of light, and the term is widely used in optical networking. Sending "multiple lambdas" down a fiber is the same as sending "multiple frequencies" or "multiple colors. ] = [[F.sup.2](O), F(O)], [P.sub.+] = (0, a - 2) and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be neighbourhoods of the point [P.sub.+] and the set [LAMBDA] respectively.

Theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance.  1.1. There exists A [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that, for any a [member of] (A, 4) and arbitrary Irrational; capricious.

The term arbitrary describes a course of action or a decision that is not based on reason or judgment but on personal will or discretion without regard to rules or standards.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], there is [[epsilon].sub.a] [member of] (0, a - 3/2(a - 2)) such that if [epsilon] [member of] ([[epsilon].sub.a], a - 3/2(a - 2)), then the following is satisfied:

1) F has a couple of periodic of period 2 points belonging to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] \{(x, y) : x = 0};

2) F has a nontrivial, invariant, topologically transitive set belonging to [U.sub.[LAMBDA]]\{(x, y) : x = 0}, and periodic points of F are dense on it.

Corollary 1.2. There exists A [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that, for any a [member of] (A, 4) [intersection intersection /in·ter·sec·tion/ (-sek´shun) a site at which one structure crosses another.

intersection

a site at which one structure crosses another.
] [??] and arbitrary [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [U.sub.[LAMBDA]], there is [[epsilon].sub.a] [member of] (0, a - 3/2(a - 2)) such that if [epsilon] [member of] ([[epsilon].sub.a], a - 3/2(a - 2)), then the following is satisfied:

1) [LAMBDA] is a simply connected topologically transitive attractor for the restriction restriction - A bug or design error that limits a program's capabilities, and which is sufficiently egregious that nobody can quite work up enough nerve to describe it as a feature.  F[|.sub.{(x,y):x=0};

2) F has a couple of periodic of period 2 points belonging to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] \{(x, y) : x = 0};

3) F has a nontrivial, invariant, topologically transitive set belonging to [U.sub.[LAMBDA]]\{(x, y) : x = 0}, and periodic points of F are dense on it.

To show that the same results take place for diffeomorphisms also, let us consider the following maps of [R.sup.4]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here 0 < [absolute value of b] < 1, c [eta] [not equal to] 1/2 and a, a are the same as above. Let O = (0, 0, 0, 0), [P.sub.+]([[gamma].sub[[gamma].sub.b]]) be the fixed point of [[gamma].sub.b] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = (0, 0, a-2, a-2) = [P.sub.+]([[gamma].sub.0]). Denote by [[LAMBDA].sub[[gamma].sub.b]], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a closure of the unstable manifold manifold

In mathematics, a topological space (see topology) with a family of local coordinate systems related to each other by certain classes of coordinate transformations. Manifolds occur in algebraic geometry, differential equations, and classical dynamics.
of [[LAMBDA].sub[[gamma].sub.b]] at [P.sub.+]([[LAMBDA].sub[[gamma].sub.b]]), a neighbourhood of [P.sb.+]([[LAMBDA].sub[[gamma].sub.b]]) and a neighbourhood of [LAMBDA].sub[[gamma].sub.b]]([LAMBDA].sub[[gamma].sub.b]]), respectively.

Theorem 1.3. There exist A [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and B > 0 so that, for any a [member of] (A, 4), [absolute value of b] [member of] (0,B) and arbitrary [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], there is [[epsilon].sub.a] [member of] (0, a - 3/2(a - 2)) such that if [epsilon] [member of] ([[epsilon].sub.a], a - 3/2(a - 2)), then the following is satisfied:

1) [[gamma].sub.b] has a couple of periodic of period 2 points belonging to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]\{(x, z, y, t) : x = z = 0};

2) [[gamma].sub.b] has a nontrivial, invariant, topologically transitive set belonging to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]\{(x, z, y, t) : x = z = 0}, and periodic points of [[gamma].sub.b] are dense on it.

Considering [[gamma].sub.b] one can easily see that the plane {(x, z, y, t) : x = z = 0} is invariant with respect to [[gamma].sub.b] and the restriction of [[gamma].sub.b] to this plane [[gamma].sub.b][|.sub.{(x,z,y,t):x=z=0}] is (within the accuracy of change of variables and parameters) the Henon Henon is a type of temperate bamboo. It is usually a grayish green in color.  diffeomorphism In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth. .

2. Proof of Theorem 1.1

An idea is to show that, at the moment of bifurcation of fixed point embedded in an invariant subspace, the following takes place: 1) two "saddle" hyperbolic hy·per·bol·ic   also hy·per·bol·i·cal
1. Of, relating to, or employing hyperbole.

2. Mathematics
a. Of, relating to, or having the form of a hyperbola.

b.
periodic of period 2 points appear outside this subspace in a vicinity of the given fixed point; 2) there are nontrivial points of transversal intersection of the stable manifolds and unstable ones of F which are built up at the periodic points. The latter implies (logic) implies - (=> or a thin right arrow) A binary Boolean function and logical connective. A => B is true unless A is true and B is false. The truth table is

A B | A => B ----+------- F F | T F T | T T F | F T T | T

It is surprising at first that A =>
an existence of transversal orbits homoclinic to these points. As is well known [5, 9] an arbitrary small neighbourhood of such kind of orbit contains a nontrivial, invariant, topologically transitive set. Periodic points of F are everywhere dense on this set. Since the stable manifolds of periodic points (due to their definition) are located outside the invariant subspace, the aforesaid homoclinic orbits as well as the invariant sets associated with ones are located outside this subspace too. The phase map portrait bifurcation enlarges upon a whole neighbourhood of the chaotic topologically transitive set because the above mentioned unstable manifolds This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics. For categorical listings see and its subcategories. Generic families of manifolds
• Euclidean space, Rn
• n-sphere, S
expand along the whole chaotic set at once as soon as these manifolds appear.

From now on we assume that a > 3. Designate des·ig·nate
tr.v. des·ig·nat·ed, des·ig·nat·ing, des·ig·nates
1. To indicate or specify; point out.

2. To give a name or title to; characterize.

3.
a closure by Cl(.), an interior by Int A programming statement that specifies an interrupt or that declares an integer variable. See interrupt and integer.

1. (programming) int - A common name for the integer data type. In C for example, it means a (signed) integer of the computer's native word length.
(.), a neighbourhood by U(.), an a-neighbourhood by [U.sub.[epsilon]](.) and an origin of coordinates coordinates

of a point on a graph or grid map, the points on the horizontal and vertical axes which identify the location of the point on the graph/map.
by O = (0, 0). Let [P.sub.+] = (0, a - 2) be the fixed point of F, [[lambda].sub.1]([P.sub.+]) = 2 - a, [[lambda].sub.2]([P.sub.+]) = (2 - a)(1 - 2[epsilon]) be eigenvalues eigenvalues

statistical term meaning latent root.
of DF at [P.sub.+], [S.sub.[+ or -]] = ([+ or -] [square rot rot (rot)
1. decay.

2. a disease of sheep, and sometimes of humans, due to Fasciola hepatica.

rot

decay.
of a(a - 2) - (3 - 4[epsilon])[(1 - 2[epsilon]).sup.-2], 1/(1 - 2[epsilon])) be two periodic of period 2 points of F and [[micro].sub.1]([S.sb.[+ or -]]), l [member of] {1, 2} be eigenvalues of D[F.sup.2] at [S.sub.[+ or -]]. One can easily see that [S.sub.[+ or -]] exists for any [epsilon] > 3 provided [epsilon] > 0 such that 1/(1 - 2[epsilon]) < a - 2. If 1/(1 - 2[epsilon]) [right arrow] a - 2, then [S.sub.[+ or -]] [right arrow] [P.sub.+] and coincides with the latter provided [epsilon] = [[epsilon].sub.0], where [[epsilon].sub.0] = a - 3/2(a - 2)

Lemma lemma (lĕm`ə): see theorem.

(logic) lemma - A result already proved, which is needed in the proof of some further result.
2.1. There exists [??] [member of] (0, [[epsilon].sub.0]) such that 0 < [[micro].sub.1]([S.sub.[+ or -]]) < 1 < [micro]2([S.sub.[+ or -]]) for all [epsilon] [member of] ([??], [[epsilon].sub.0]).

Proof. It is easy to see that [[micro].sub.1]([S.sub.[+ or -]]), l [member of] {1, 2} are solutions of the equation

0 = [[micro] + a(a - 2)(1 - 2[epsilon]) - 3 - [(1 - 2[epsilon]).sup.-1]] x [micro] + a(a - 2)(1 - 2[epsilon]) - 2 - [(1 - 2[epsilon]).sup.-1] - [(1 - 2[epsilon]).sup.-2]] + 4[[epsilon].sup.2]/(1 - 2[epsilon]) a(a - 2) - 2/(1 - 2[epsilon]) - [(1 - 2[epsilon]).sup.-2]].

[[micro].sub.1]([epsilon]) [right arrow] [[micro].sub.1]([[epsilon]0]) as [epsilon] [??] [[epsilon].sub.0]. Here [[micro].sub.1]([[epsilon].sub.0]) = 1, [[micro].sub.2]([[epsilon].sub.0]) = [(a - 2).sup.2] > 1. It is clear too that [[micro].sub.2]([epsilon]) > 1 for [epsilon] < [[epsilon].sub.0] close enough to [[epsilon].sub.0]. Let us show that [[micro].sub.1](a) < 1 for the same [epsilon]. Choose [epsilon] so that [gamma] [equivalent of] a - 2 - 1/(1 - 2[epsilon]) > 0 is small enough. Accurate to O([[gamma].sup.2]), eigenvalues of D[F.sup.2] at [S.sub.[+ or -]] coincide with solutions of the equation

0 = [[micro].sub.2] - 1 + [(a - 2).sup.2] - 2[gamma] ([a.sup.2] - 2a + 2)/(a - 2)][micro] + [(a - 2).sup.2] - [gamma](3[a.sup.2] - 10a - 8)/(a - 2). (2.1)

Solving (2.1) one can find accurate to o([gamma]) that [[micro].sub.1](a) =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence 0 < [[micro].sub.1](a) < 1 for [[epsilon].sub.0] - [epsilon] > 0 small enough. Thus [S.sub.[+ or -]] are the hyperbolic "saddles".

Denote by [W.sup.s.sub.loc] ([S.sub.[+ or -]]), [W.sup.u.sub.loc] ([S.sub.[+ or -]]) locally stable and unstable manifolds of F at [S.sub.[+ or -]]. Due to analyticity Noun 1. analyticity - the property of being analytic
property - a basic or essential attribute shared by all members of a class; "a study of the physical properties of atomic particles"
of F, these manifolds are analytic an·a·lyt·ic or an·a·lyt·i·cal
1. Of or relating to analysis or analytics.

2. Expert in or using analysis, especially one who thinks in a logical manner.

3. Psychoanalytic.
[1]. Consider [W.sup.u.sub.loc]([S.sub.+]), [W.sup.u.sub.loc]([S.sub.-]). It is easy to see that [W.sup.u.sub.loc[([S.sub.+]), [W.sup.u.sub.loc]([S.sub.-]) are curves symmetric No difference in opposing modes. It typically refers to speed. For example, in symmetric operations, it takes the same time to compress and encrypt data as it does to decompress and decrypt it. Contrast with asymmetric.

(mathematics) symmetric - 1.
with respect to the y-axis. Let x = [omega](y) be a function whose graph graph, figure that shows relationships between quantities. The graph of a function y=f (x) is the set of points with coordinates [x, f (x)] in the xy-plane, when x and y are numbers.  coincides with [W.sup.u.sub.loc]([S.sub.+]). The analyticity of [W.sup.u.sub.loc]([S.sub.+]) implies the one of x = [omega](y). Substituting coordinates of [S.sub.+] one can find that [omega] (1/(1 - 2[epsilon])) = [square root of a(a - 2) - (3 - 4[epsilon])][(1 - 2[epsilon]).sup.-2]. Let us designate the graph of x = [omega](y) by graph([omega]) and Graph([omega]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (graph([omega])). Since F ([W.sup.u]([S.sub.[+ or -]])) = [W.sup.u]([S.sub.[??]]),

we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus the equation

[omega] (a(a - 2)/2 - [[[omega](y)].sup.2] + [y.sup.2]/2) = (1 - 2[epsilon])y[omega](y) (2.2)

holds. The latter means an invariance in·var·i·ant
1. Not varying; constant.

2. Mathematics Unaffected by a designated operation, as a transformation of coordinates.

n.
An invariant quantity, function, configuration, or system.
of Graph(U) with respect to the map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Denoting

Y = a(a - 2)/2 - [[[omega](y)].sup.2] + [y.sup.2]/2 [delta] [square root of a(a - 2) - (3 - 4[epsilon])[(1 - 2[epsilon]).sup.-2],

we obtain

[omega] (1/1 - 2[epsilon]) = [delta], [omega](Y) = (1 - 2[epsilon])y[omega](y). (2.3)

Differentiating the latter we find

[omega]'(Y) = -(1 - 2[epsilon]) y[omega]'(y) + [omega](y)/[omega](y)[omega]'(y) + y. (2.4)

Changing y, [omega](y) here to 1/(1 - 2[epsilon]), [delta] respectively and taking into account that Y = 1/(1 - 2[epsilon]) when y = 1/(1 - 2[epsilon]), we obtain an equation with respect to [omega]' (1/[1 - 2[epsilon]]).

Solving it we find

[omega]' (1/[1 - 2[epsilon]]) = (1 - [epsilon])[(1 - 2[epsilon]).sup.-1]/[delta [+ or -] [square root of [[(1 - [epsilon]).sup.2][[delta].sup.-2][(1 - 2[epsilon]).sup.-2] - (1 - 2[epsilon])].

Since the sign "+" corresponds to a direction tangent tangent, in mathematics.

1 In geometry, the tangent to a circle or sphere is a straight line that intersects the circle or sphere in one and only one point.
to [W.sup.u.sub.loc]([S.sub.+]) at [S.sub.+], accurate to O([[delta].sup.3]), we have

[omega]'(1/1 - 2[epsilon]) = -[delta][(1 - 2[epsilon]).sup.2]/2(1 - [epsilon]).

Formulas for [omega]'' (1/[1 - 2[epsilon]]), [omega]''' (1/[1 - 2[epsilon]]) can be found in the same way. Keeping in mind that 1/(1-2[[epsilon].sub.0]) = a - 2, let us take limits for [omega] (1/[1 - 2[epsilon]]), [omega]'' (1/[1 - 2[epsilon]]), [omega]''' (1/[1 - 2[epsilon]]) as [epsilon] [??] [[epsilon].sub.0]. We find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, for [epsilon] close enough to [[epsilon].sub.0], there is a neighbourhood of y = 1/(1-2[epsilon]) within which x = [omega](y) is a convex function In mathematics, a real-valued function f defined on an interval (or on any convex subset of some vector space) is called convex, or concave up, if for any two points x and y in its domain C and any t in [0,1], we have
. That is, for the same [epsilon], there exists a neighbourhood of [S.sub.+] in which [W.sup.u.sub.loc]([S.sub.+]) is a convex Convex

Curved, as in the shape of the outside of a circle. Usually referring to the price/required yield relationship for option-free bonds.
curve. Here and further we call a curve convex (concave Concave

Property that a curve is below a straight line connecting two end points. If the curve falls above the straight line, it is called convex.
) when it is a graph of a convex (concave) function.

Denote [kappa Kappa

Used in regression analysis, Kappa represents the ratio of the dollar price change in the price of an option to a 1% change in the expected price volatility.

Notes:
Remember, the price of the option increases simultaneously with the volatility.
] = [omega]' (1/[1 - 2[epsilon]]). In view of (2.3) and (2.4), [delta][[kappa].sup.2] + 2[kappa](1 - [epsilon])/(1 - 2[epsilon]) = -(1 - 2[epsilon])[delta]. Since x = [omega](y) is a function convex in a vicinity of y = 1/(1 - 2[epsilon]), there exist [[??].sub.-] < 0 < [[??].sub.+] such that graph(U) is located under and to the left of x = [delta] + [kappa] [y 1/(1 - 2[epsilon])] provided y [member of] (1/[1 - 2[epsilon]]+[[??].sub.-], 1/[1-2[epsilon]]+[[??].sub.+]). We show now that the whole graph graph([omega]) is located under and to the left of x = [delta] + [kappa] [y - 1/(1 - 2[epsilon])] too. Let us make a change of variables [??] = x - [delta], [??] = y - 1/(1 - 2[epsilon]). Taking into account that a(a - 2) - [(1 - 2[epsilon]).sup.-2] - [[delta].sup.2] = 2/(1 - 2[epsilon]), one can easily observe that the map (1.2) takes the following form in coordinates [??], [??]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.5)

[FIGURE 1 OMITTED]

In doing so the tangent equation simplifies to [??] = [kappa][??]. Consider [??](([??] = [[kappa].sup.2])[t.sup.2]/2. This is a curve [??] = q([??]), where

[??] = [kappa]t + (1 - 2[epsilon])([delta] + [kappa]t)t, [??] = -[1/(1 - 2[epsilon]) + [kappa][delta]] t - (1 + [[kappa].sup.2])[t.sup.2]/2.

We prove that the curve locates under and to the left of [??] = [kappa] [??]. Computing computing - computer  its derivative derivative: see calculus.
derivative

In mathematics, a fundamental concept of differential calculus representing the instantaneous rate of change of a function.
we find q'([??]) = [kappa] + (1 - 2[epsilon])[delta] + 2(1 - 2[epsilon])[kappa]t/1/(1 - 2[epsilon]) + [kappa][delta] + (1 + [[kappa].sup.2)t. Let us show that q'([??]) > [kappa] provided [??] < 0 and q'([??]) < [kappa] when [??] > 0. In view of (2.5) it is clear that

[kappa]e 1 + 1/(1 - 2[epsilon])(3 - 4[epsilon] + [[kappa].sup.2])t + [kappa][delta]] < -(1 - 2[epsilon])[delta], for t > 0, (2.6)

[kappa] [1 + 1/(1 - 2[epsilon])(3 - 4[epsilon] + [[kappa].sup.2])t + [kappa][delta]] > -(1 - 2[epsilon])[delta], for t < 0. (2.7)

Besides, (2.6) implies - [kappa] -(1 - 2[epsilon])[delta] -2(1 - 2[epsilon])[kappa]t > [kappa]/(1 - 2[epsilon]) + [[kappa].sup.2] [delta] + [kappa](1 + [[kappa].sup.2])t . Since [delta] [right arrow] 0, [kappa] [right arrow] 0 as [epsilon] [??] [[epsilon].sub.0], there exists [??] < [[epsilon].sub.0] such that 1/(1 - 2[epsilon]) > [delta][absolute value of [kappa]] when [epsilon] [member of] ([??], [[epsilon].sub.0]) and a [member of] (1 + [square root of 5], 4). It implies in turn that if a and [epsilon] are the same as above, then the inequality inequality, in mathematics, statement that a mathematical expression is less than or greater than some other expression; an inequality is not as specific as an equation, but it does contain information about the expressions involved.  1/(1 - 2[epsilon])+ [kappa][delta] +(1 + [[kappa].sup.2])t > 0 is satisfied for any t > 0. Keeping in mind that [??] < 0 for t > 0, we see that q'([??]) > [kappa] for [??] < 0. On the other hand, in the same way (2.7) implies -[kappa] -(1 - 2[epsilon])[delta] - 2(1 - 2[epsilon])[kappa]t < [kappa]/(1 - 2[epsilon]) + [[kappa].sup.2][delta] + [kappa](1 + [[kappa].sup.2])t .

If [1/(1 - 2[epsilon]) + [kappa][delta]/1 + [[kappa].sup.2] < t < 0, then q'([??]) < [kappa] and simultaneously [??] > 0. However if t < [1/(1 - 2[epsilon]) + [kappa][delta]]/1 + [[kappa].sup.2], then q'([??]) > [kappa] and simultaneously [??] < 0. Hence the graph [??] = q([??]) is indeed located under and to the left of [??] = [kappa][??].

Let us study [??] = q([??]) in detail. One can easily see that its graph consists of two arcs which merge See mail merge and concatenate.  in one curve at the point ([[??].sub.0], [[??].sub.0]), where [[??].sub.0] = [??]([t.sub.0]), [[??].sub.0] = [??]([t.sub.0]), [t.sub.0] = [1/(1 - 2[epsilon]) + [kappa][delta]]/1 + [[kappa].sup.2]. One of the arcs is convex and another is concave. Indeed, computing q''([??]), we find q''([??]) = (1 - [[kappa].sup.2])[(1 - 2[epsilon])[delta] - [[kappa]]/[1/(1 - 2[epsilon]) + [kappa][delta] + (1 + [[kappa].sup.2])t].sup.3]. Keeping this in mind, one can easily infer that q''([??]) < 0 for t > [t.sub.0] and q''([??]) > 0 for t < [t.sub.0]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exits.

Let us locate graph([omega]) when 0 [less than or equal to] y [less than or equal to] [square root of a(a - 2)]. Consider the quadrangle quadrangle

Rectangular open space completely or partially enclosed by buildings of an academic or civic character. The grounds of a quadrangle are often grassy or landscaped.
[DELTA] bounded by segments of the lines x = 0, y = 0, y = [square root of (a - 2)] and x = [delta] + [kappa] [y - 1/(1 - 2[epsilon])]. Computing [bar.F]([DELTA]), we find that [bar.F]([DELTA}) is a "curvilinear curvilinear

a line appearing as a curve; nonlinear.

curvilinear regression
see curvilinear regression.
" quadrangle bounded by arcs of the curves y = 0, y = a(a - 2)/2 - x/(1 - 2[epsilon]), [x.sup.2] = -2a(a - 2)[(1 - 2[epsilon]).sup.2]y and [??] = q([??]). Taking into account mutual positions of [??] = [kappa][??] and [??] = q([??]), we see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By construction, [{[[DELTA].sub.n]}.sup.[infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ].sub.n=0] is a monotone mon·o·tone
n.
1. A succession of sounds or words uttered in a single tone of voice.

2. Music
a. A single tone repeated with different words or time values, especially in a rendering of a liturgical text.
decreasing sequence of nested "curvilinear" quadrangles and [bar.F]([[DELTA].sub.*]) [intersection] {(x, y) : 0 [less than or equal to] y [less than or equal to] [square root of a(a - 2)}] = [[DELTA].sub.*]. Due to the invariance of [[DELTA].sub.*] with respect to [bar.F], it is clear that graph([omega]) [subset] [[DELTA].sub.*]. This implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us consider the fraction of boundary BOUNDARY, estates. By this term is understood in general, every separation, natural or artificial, which marks the confines or line of division of two contiguous estates. 3 Toull. n. 171.
2.
of [[DELTA].sub.*] which is a limit of sequence of images of the line x = [delta] + [kappa] [y - 1/(1 - 2[epsilon])]. We denote it by [L.sub.[omega]] and show that [L.sub.[omega]] is a smooth function graph.

Let [XI] > 0 be a constant, [rho] = _[square root of a(a - 2)], I [equivalent to] = [0.99[square root of [[rho].sup.2] - 2[rho], [rho]] be a line segment, X = [chi](y) be a [C.sup.2]-smooth function such that 0 [less than or equal to] [chi] (y) < 2[delta] < [??] < 0.4[square root of [[rho].sup.2] - 2[rho], [absolute value of [chi]'(y)] < [XI], [absolute value of [chi]''(y)] < 100 [XI] on I. Set x = [[chi].sub.F] (y) by the formulas x = (1 - 2[epsilon])t[chi](t), y = a(a - 2) - [t.sup.2] - [chi square chi square (kī),
n a nonparametric statistic used with discrete data in the form of frequency count (nominal data) or percentages or proportions that can be reduced to frequencies.
](t)]/2.

Lemma 2.2. Let a > 1 + [square root of 4 + 2 [square root of 2]] and x = [chi](y) as above. There exists [[epsilon].sub.F] < [[epsilon].sub.0] so close to [[epsilon].sub.0] that a domain of the function x = [[chi].sub.F] (y) includes I and [absolute value of [[chi]'.sub.F](y)] < [XI], [absolute value of [[chi]'.sub.F](y)] < 100 [XI] for [epsilon] [member of] ([[epsilon].sub.F], [[epsilon].sub.0]).

Proof. Computations give y(0.99 [square root of [[rho].sup.2] - 2[rho])] > [rho], y([rho]) < 0. Thus, the domain of

x = [[chi].sub.F] (y) includes I . Computing [absolute value of [[chi]'.sub.F]], we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where t [member of] [0.99[square root of [[rho].sup.2] - 2[rho], [rho]]. If a > 1 + [square root of 5], then 1/(a - 2) < (1 + [square root of 5])/4 < 0.9. So, 1 - 2[epsilon] < 0.9 for [epsilon] sufficiently close to [[epsilon].sub.0]. Fix [[epsilon].sub.F] so that [delta[XI] < 0.001 [square root of [[rho].sup.2] - 2[rho]], 1 - 2[epsilon] < 0.9, [[delta].sup.2] < [XI] for all [epsilon] [member of] ([[epsilon].sub.F], [[epsilon].sub.0]). In doing so we get [absolute value of [[chi]'.sub.F](y)] < [XI] for any y [member of] I . As for [absolute value of [[chi]''.sub.F](y)] we can estimate it by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Keeping in mind that [square root of [[rho].sup.2] - 2[rho]] = [square root of a(a - 2) -2 [square root of a(a - 2)]] > 1 for a > 1 + [square root of 4 + 2[square root of 2]], we find [absolute value of [chi]''(t)] 100 [XI]. < 100. _

Let us choose [XI] = [10.sup.-6]. As is shown above, values of the x-coordinate of points belonging to images of the line x = [delta] + [kappa] [y - 1/(1 - 2[epsilon])] are less than 2[delta]. Since [absolute value of [kappa]] < [[XI].sup.2] for [[epsilon].sub.F] sufficiently close to [[epsilon].sub.0], it means due to Lemma2.2 that the absolute values of derivatives derivatives

In finance, contracts whose value is derived from another asset, which can include stocks, bonds, currencies, interest rates, commodities, and related indexes. Purchasers of derivatives are essentially wagering on the future performance of that asset.
of these images are uniformly bounded on I and do not exceed [10.sup.-6]. By construction, the images constitute a monotone sequence. Therefore the sequence limit is a one-valued [C.sup.1]-smooth function. Obviously its domain includes I and the absolute value of its derivative is less than [10.sup.-6]. But this is a fraction of [L.sub.[omega]] only. However if we consider the [bar.F]-image of the given fraction, one can prove then the same for whole [L.sub.[omega]]. Let us show now that [L.sub.[omega]] = graph([omega]). Indeed, by definition, the arc [L.sub.[omega]] is a graph of the one-valued [C.sup.1]-smooth function having uniformly bounded derivative. It is invariant with respect to [bar.F] and includes [S.sub.+]. However there is a unique curve which is situated in a vicinity of [S.sub.+], belongs to [[DELTA].sub.*], is invariant with respect to [bar.F] and has the absolute value of the derivative less than [10.sup.-6]. It is [W.sup.u.sub.loc] ([S.sub.+]). Hence [W.sup.u.sub.loc]([S.sub.+]) [subset] [L.sub.[omega]]. The contrary is true: [W.sup.u]([S.sub.+]) [contains] [L.sub.[omega]]. Actually if not, then Cl([W.sup.u]([S.sub.+])) \[W.sup.u]([S.sub.+]) is a couple of periodic of period 2 points which belong to [L.sub.[omega]] by construction. Straightforward computations show that these points exist when a [epsilon] [member of] (0, [??]), where [??] = (1 - [square root of 3/[a(a - 2)]])/2, and disappear or, to be more precise, merge with [S.sub.+] at [epsilon] = [??]. Moreover, the same computations show that if a [epsilon] [member of] ([??], [[epsilon].sub.0]), then, excluding [S.sub.[+ or -]], F has no other periodic points in [R.sup.2] \ {(x, y) : x = 0}. Therefore [L.sub.[omega]] = graph([omega]).

Remark 2.3. The same arguments show Graph([omega]) = [W.sup.u]([S.sub.+]) provided [epsilon] [member of] ([??], [[epsilon].sub.0]).

Keeping in mind that Graph([omega]) consists of the [bar.F]-images of graph([omega]), and using (2.3), one can estimate the absolute x-coordinate value of points of Graph([omega]) by [absolute value of [omega](y)] < 2[delta] [[a(a - 2)/2].sup.n]. Here n is the number of iterations of graph([omega]). We shall see later that it is enough to consider not more than 5 iterations of [L.sub.[omega]]. As for values of (jargon) for values of - A common rhetorical maneuver at MIT is to use any of the canonical random numbers as placeholders for variables. "The max function takes 42 arguments, for arbitrary values of 42". "There are 69 ways to leave your lover, for 69 = 50".  derivatives of Graph([omega]), those can be estimated in an appropriate way on suitable subsets of Graph([omega]). To determine these subsets, one should use a critical set of F [2, 3]. In the case under study, the critical set is K = {(x, y) : x = y} [union] {(x, y) : x = -y}.

[FIGURE 2 OMITTED]

Consider [[bar.F].sup.2]([L.sub.[omega]]). Lett Lett
n.
A member of a Baltic people constituting the main population of Latvia.

[German Lette, from Latvian Latvi.]
> 0 be small enough. Denote [Z.sub.+] = ([X.sub.+], [Y.sub.+]) = [[bar.F].sup.2]([L.sub.[omega]])[intersection] {(x, y) : x > 0, y > 0}, Z- [equivalent to] ([X.sub.-], [Y.sub.-]) = [[bar.F].sup.2]([L.sub.[omega]]) [intersection] {(x, y) : x > 0, y < 0}, [Z.sub.t] = [L.sub.[omega]] n {(x, y) : y = [Y.sub.+] + t} and by [L.sub.[omega]t] the arc of Graph([omega]) stretching of [[bar.F].sup.2]([Z.sub.t]) to [bar.F]([Z.sub.t]).

Lemma 2.4. Let a [member of] (1 + [square root of 5], 4) and t > 0 small enough. There exists [[epsilon].sub.F] > 0 such

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Choose [??] > 0 so close to [[epsilon].sub.0] that [delta] < [t.sup.4], [omega](y) < 0.01[delta] for all (x, y) [member of] [L.sub.[omega]], [epsilon] [member of] ([??], [[epsilon].sub.0]). In this case [omega](y) < [delta]/3 for all (x, y) [member of] [L.sub.[omega]t] Fix [??] > 0 so that 100[XI] < [t.sup.4] [much less than] t , [absolute value of [omega]'(y)] < [XI] for all (x, y) [member of] [L.sub.[omega]], [epsilon] [member of] ([??], [[epsilon].sub.0]) and consider [bar.F]([Z.sub.t]). Using (2.4) we find accurate to O([[delta].sup.2]) that

[omega]'(a[a - 2]/2 - [[omega]([[Y.sub.+] + t)].sup.2]/2 - [[[Y.sub.+] + t].sup.2]/2) = -(1 -2[epsilon])[[omega]'([Y.sub.+] + t) + [omega]([Y.sub.+] + t)/[Y.sub.+] + t].

Taking into account [omega]([Y.sub.+] + t) < 0.01[t.sup.3], [absolute value of [omega]'([Y.sub.+] + t)] < 0.01[t.sup.2], we can see that

[absolute value of 1 - 2[epsilon] [absolute value of [omega]'([Y.sub.+] + t) + [omega]([Y.sub.+] + t)/[Y.sub.+] + t] < t/30

of course, provided [epsilon] [member of] ([[epsilon].sub.F], [[epsilon].sub.0]), where [[epsilon].sub.F] = min([??], [??]). Computing and estimating [absolute value of [omega]'(y)] on [[bar.F].sup.2]([Z.sub.t]), we find that it is much less than t . It is clear that analogous analogous /anal·o·gous/ (ah-nal´ah-gus) resembling or similar in some respects, as in function or appearance, but not in origin or development.

a·nal·o·gous
reasons can be used for an arbitrary point of (x, y) [member of] [L.sub.[omega]] with y > [Y.sub.+] + t.

Remark 2.5. To prove that the other fractions of Graph([omega]) possess similar properties, one should make arguments similar to those aforesaid. For example, let us consider an arc of Graph([omega]) between points [[bar.Z].sub.t] = [[bar.F].sup.2]([L.sub.[omega]]) [intersection] {(x, y) : y = [Y.sub.-] - t} and [[bar.F].sup.2]([Z.sub.t]). If we try to apply the reasoning presented above to this arc, we can see then that these arguments all remain valid. A crucial circumstance Circumstance or circumstances can refer to:
• Legal terms:
• Aggravating circumstances
• Attendant circumstance
determining the possibility to use the aforesaid arguments to any arc is that the arc points as well as its images all do not belong to the critical set K. Moreover, the arc closure should be separated from the set [bar.K] [equivalent to] {(x, y) : - [XI] [less than or equal to] x [less than or equal to] 0, x [less than or equal to] y [less than or equal to] -x} [union] {(x, y) : 0 [less than or equal to] x [less than or equal to] [XI], - x [less than or equal to] y = x} as well as from its images.

We now discuss the behaviour of [[bar.F].sup.3]([L.sub.[omega]]) inside [bar.F]([bar.K]). Analyzing formulas (2.3) and (2.4), we can see the following. First, [bar.F]([Z.sub.x]) [member of] {(x, y) : x = 0}, where [Z.sub.x] [equivalent to] [L.sub.[omega]] [intersection] {(x, y): y = 0}. Second, since [absolute value of [omega](y)[omega]'(y)] is O([[delta].sup.2]) and [Y.sub.[+ or -]] = [+ or -][delta], there exists [??] [member of] (-[delta], [delta]) such that [omega]([??])[omega]([??]) + [??] = 0. The latter means in turn that a tangent to Graph([omega]) is parallel to the x-axis See x-y matrix. . Third, [bar.F]([Z.sub.+]) and [bar.F]([Z.sub.-]) are situated on different sides of the y-axis, i.e., [bar.F]([Z.sub.+]) [member of] {(x, y) : x > 0}, [bar.F]([Z.sub.-]) [member of] {(x, y) : x < 0}.

We have studied properties of [[bar.F].sup.3]([L.sub.[omega]]) on [[bar.F].sup.2]([L.sub.[omega]]) and within [bar.F]([bar.K]). It is clear that the same approach can be used to research a behaviour of the "remainder" [[bar.F].sup.3]([L.sub.[omega]]) [[bar.F].sup.2]([L.sub.[omega]]) [unin] [bar.F]([bar.K]). Outside a neighbourhood of [bar.F]([bar.K]) [union] [[bar.F].sup.2]([bar.K]), the given "remainder" has the same properties as [L.sub.[omega]t]. However, inside [[bar.F].sup.2]([bar.K]), its behaviour is similar to that of [[bar.F].sup.3]([L.sub.[omega]t]) inside [bar.F]([bar.K]).

An analogous analysis can be made with respect to properties of [[bar.F].sup.4]([L.sub.[omega]]), [[bar.F].sup.5]([L.sub.[omega]]) and [W.sup.u]([S.sub.+]) in whole. The only thing that should be kept in mind is the following. When a bit of the arc [[bar.F].sup.3]([L.sub.[omega]]) falls inside [bar.F]([ba.K]), this gives rise to a necessity to study its image properties inside [[bar.F].sup.2]([bar.K]). Similarly the most essential and principal piece of researching of behaviour of [[bar.F].sup.4]([L.sub.[omega]]) consists in a study of properties of all fractions of [[bar.F].sup.4]([bar.K]) \ [[bar.F].sup.3]([bar.K]) which have a nonempty intersection with [[bar.F].sup.2]([bar.K]). In order to do this, it is necessary to locate where the given intersection image is. In order to locate the latter, it is necessary in turn to consider [[bar.F].sup.3]([bar.K]), and so on. However if we analyze an·a·lyze
v.
1. To examine methodically by separating into parts and studying their interrelations.

2. To separate a chemical substance into its constituent elements to determine their nature or proportions.

3.
reasons which guarantee a validity of the arguments presented above in detail, it becomes clear then that it is enough to study the behaviour of [[bar.F].sup.2]([L.sub.[omega]t]) only. Among the reasons which should be taken into account, the principal ones are following: 1) a point [P.sub.a] [??] (0, a(a - 2)[4 + 2a - [a.sup.2]]/8) is a fixed one of F and [bar.F] at a = 4; 2) there is a continuous dependence of [[bar.F].sup.j] ([bar.K]), j = 1, 2, ... on parameters; 3) for any j = 1, 2, ... , [[bar.F].sup.j+1]([bar.K]) tends to [P.sub.a] as [epsilon][??] [[epsilon].sub.0], a [??] 4 in the sense that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here dist(.,.) is the distance.

Consider the local stable manifolds of F at [S.sub.[+ or -]] now. As well as the unstable manifolds, [W.sup.s.sub.loc]([S.sub.+]), [W.sup.s.sub.loc]([S.sub.-]) are curves symmetric with respect to the y-axis. Let y = [OMEGA](x) be the function whose graph coincides with [W.sup.s.sub.loc]([S.sub.+]). Since [W.sup.s.sub.loc]([S.sub.+]) is an analytic manifold In mathematics, an analytic manifold is a topological manifold with analytic transition maps. Every complex manifold is an analytic manifold. , y = [OMEGA](x) is an analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. Analytic functions can be thought of as a bridge between polynomials and general functions. . Taking into account coordinates of [S.sub.+], we find [OMEGA]([delta]) = 1/(1-2[epsilon]). Denote the graph of y = [OMEGA](x) by graph([OMEGA]). Because F([W.sup.s]([S.sub.[+ or -]])) = [W.sup.s]([S.sub.[+ or -]]), we infer that F : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [??] = -(1 - 2[epsilon]) x [OMEGA](x), [OMEGA]([??]) = a(a - 2) - [([OMEGA](x)).sup.2] - [x.sup.2]/2. Keeping in mind that [OMEGA]([??]) = [OMEGA](-[??]), we obtain the functional equation a(a - 2)/2 - {[[[OMEGA](x)].sup.2] + [x.sup.2]}/2 = [OMEGA]((1 - 2[epsilon])x[OMEGA[(x)). Therefore graph([OMEGA]) is invariant with respect to [bar.F]. Setting X = (1 - 2[epsilon])x[OMEGA](x), we arrive at the system

[OMEGA]([delta]]) = 1/1 - 2[epsilon], [OMEGA](X) = a(a - 2)/2 - [[OMEGA](x)].sup.2] + [x.sup.2]/2. (2.8)

Differentiation differentiation, in biology, series of changes that occur in cells and tissues during development, resulting in their specialization. This, in turn, permits a greater variety of organisms.  of (2.8) gives - [OMEGA](x)[OMEGA]'(x) + x] = (1 - 2[epsilon])[x[OMEGA]'(x) + [OMEGA](x)][OMEGA]'(X). Changing x, [OMEGA](x) to a, [delta], 1/1 - 2[epsilon] respectively and taking into account that X = [delta] when x = [delta], we obtain an equation with respect to [OMEGA]([delta]). Its solution is [OMEGA]'([delta]) = -1 - [epsilon]/[delta][(1 - 2[epsilon]).sup.2] [+ or -] [square root of [[1 - [epsilon]/[delta][(1 - 2[epsilon]).sup.2].sup.2] - 1/1 - 2[epsilon]]. The sign "+" corresponds to the direction tangent to [W.sup.s.sub.loc]([S.sub.+]) at [S.sub.+]. Therefore, accurate to O([[delta].sup.3]),

[OMEGA]'([delta] = 1 - [epsilon]/[delta][(1 - 2[epsilon]).sup.2] + [square root of [[1 - [epsilon]/[delta][(1 - 2[epsilon]).sup.2]].sup.2] - 1/1 - 2[epsilon]] = (1 - 2[epsilon])[delta]/2(1 - [epsilon]).

Formulas of [OMEGA]'([delta]), [OMEGA]'''([delta]) can be found in a similar way. Taking limit as [epsilon] [??] [[epsilon].sub.0] in these formulas and keeping in mind that 1/(1 - 2[[epsilon].sub.0]) = a - 2, one can easily check that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, for [epsilon] close enough to [[epsilon].sub.0], there is a neighbourhood of x = [delta] within which y = [OMEGA](x) is a convex function.

Therefore, for the same [epsilon], there exists a neighbourhood of [S.sub.+] such that [W.sup.s.sub.loc]([S.sub.+]) is a convex curve inside this neighbourhood.

Consider a curvilinear quadrangle AH [bar.F](H) [bar.F](A) [subset] {(x, y) : x [greater than or equal to] 0, y [greater than or equal to] 0} which is constituted in such a manner. AH, [bar.F](A) [bar.F](H) are arcs of curves [(1 - 2[epsilon]).sup.2] [x.sup.2] [y.sup.2] + [a(a - 2) - [x.sup.2] - [y.sup.2]].sup.2]/4 = a(a - 2), [x.sup.2] + [y.sup.2] = a(a - 2) which are adjacent to A and [bar.F](A), respectively. H, [bar.F](H) are points of intersection of these arcs with [L.sub.[omega]. In doing so, H [bar.F](H) is a fraction of [L.sub.[omega]] between H and [bar.F](H), and A [bar.F](A) is a segment of the y-axis. Obviously, there exist a and [epsilon] such that [bar.F] is a diffeomorphism of A H [bar.F](H) [bar.F](A) on [bar.F](A) [bar.F](H) [[bar.F].sup.2](H) [[bar.F].sup.2](A). Indeed, since [bar.F](O) = (0, a(a - 2)/2), [bar.F]([P.sub.+]) = [P.sub.+] and a - 2 < [square root of a(a - 2)] < a(a - 2)/2 provided a > 1 + [square root of 5], there exists a pre-image of (0, [square root of a(a - 2))] [equivalent to] [bar.F](A) inside the interval interval, in music, the difference in pitch between two tones. Intervals may be measured acoustically in terms of their vibration numbers. They are more generally named according to the number of steps they contain in the diatonic scale of the piano; e.g.  (O, [P.sub.+]). This is A. Let [??] [member of] (1 + [square root of 5], 4), [eta] > 0 such that [U.sub.[eta]](O) [??] A for all a [member of] ([??], 4). Since [L.sub.[omega]] converges to a segment of the y-axis uniformly as [epsilon] [??] [[epsilon].sub.0], there is [??] > 0 so that [bar.F](A) [bar.F](H)[subset] [bar.F]([U.sub.[eta]](A)) for [epsilon] [member of] ([??], [[epsilon].sub.0]). Therefore AH [subset] {(x, y) : x [greater than or equal to] 0, y > 0}. Observing that [bar.F](AH) = [bar.F](A)[bar.F](H), H [bar.F](H) [subset] [bar.F](H [bar.F](H)), A [bar.F](A) [subset] [bar.F](A [bar.F](A)), [[bar.F].sup.2](A) [[bar.F].sup.2](H) [subst] {(x, y) : x [greater than or equal to] 0, y = 0} and taking into account that a restriction of [bar.F] to {(x, y) : x [greater than or equal to] 0, y [greater than or equal to] 0} is a diffeomorphism, one can easily find that [bar.F] is a diffeomorphism of A H [bar.F](H) [bar.F](A) to [bar.F](A) [bar.F](H) [[bar.F].sup.2](H) [[bar.F].sup.2](A).

Lemma 2.6. Let a > 1 + [square root of 5]. Then there exists [??] < [[epsilon].sub.0] such that y > x for all points (x, y) [member of] AH provided that [epsilon] [member of] ([??], [[epsilon].sub.0]).

Proof. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then y > x for any (x, y) [member of] [U.sub.[zeta]] (A), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Choosing [??] so close to [[epsilon].sub.0] that [delta] = [zeta] for all [epsilon] [member of] ([??], [[epsilon].sub.0]) concludes the proof.

Denote the map inverse (mathematics) inverse - Given a function, f : D -> C, a function g : C -> D is called a left inverse for f if for all d in D, g (f d) = d and a right inverse if, for all c in C, f (g c) = c and an inverse if both conditions hold.  of [bar.F] on AH [bar.F](H) [bar.F](A) by [[bar.F].sup.-1]. Taking into account that y [greater than or equal to] x [greater than or equal to] 0 for all (x, y) [member of] AH [bar.F](H) [bar.F](A), we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. while [[??].sup.s.sub.+] [intersection] {x, y) : x = 0} = [phi], it is clear that [[??].sup.s.sub.+] is situated between AH and [bar.F](A)[bar.F](H).

Let us show that Cl([[??].sup.s.sub.+] leans against {(x, y) : x = 0}. The map [[bar.F].sup.-1] induces a smooth function map in such a manner. Let 0 < [sigma] [much less than] [cube root cube root
n.
A number whose cube is equal to a given number.

cube root
Noun

the number or quantity whose cube is a given number or quantity: 2 is the cube root of 8
of [sigma]] < [10.sup.-3] be small enough. Denote by [S.sub.[psi PSI - Portable Scheme Interpreter ]] the space of [C.sup.2]-smooth functions y = [psi](x) such that

[absolute value of [psi](x) - 1/(1 - 2[epsilon]] < [sigma], [absolute value of [psi]'(x)] < 0.01, < 5 [absolute value of [psi]''(x)] 5 on [??] [equivalent to] [0, [sigma]]. Define [PHI] : [S.sub.[psi]] [??] [S.sub.[psi]] by means of the formula [PHI](y = [psi](x)) = [y.sub.F] = [[psi].sub.F] ([x.sub.F])), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.9)

Lemma 2.7. Let a > 1 + [square root of 4 + 2 [square root of 2]] and y = [psi](x) as above. There exists [[epsilon].sub.F] < [[epsilon].sub.0] so that if [epsilon] [member of] ([[epsilon].sub.F], [[epsilon].sub.0]), then [absolute value of [[psi].sub.F] ([x.sub.F]) - 1/(1 - 2[epsilon])] < [sigma], [absolute value of [psi]'.sub.F]([x.sub.F])] < 0.01, [absolute value of [[psi]''.sub.F]([x.sub.F])] < 5 for all x [member of] [??].

Proof. Denote [psi](x) = a - 2 + r(x). Substituting this expression in the formulas (2.9) instead of [psi](x), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Taking into account that [square root of b [+ or -] c] = [square root of b] [1 [+ or -] c/(2b) + O ([c.sup.2][(2b).sup.-2])], [absolute value of r(x)] < [sigma], [absolute value of x/(1 - 2[epsilon])] < 2[sigma], we find [y.sub.F] = a-2+r(x)/(a-2)+O([[sigma].sup.2], [x.sub.F] = x/(a - 2)(1 - 2[epsilon])+ O([[sigma].sup.2]) for [epsilon] close enough to [[epsilon].sub.0]. Since 1 - 2[[epsilon].sub.0] = 1/(a - 2), the latter implies: j) that [absolute value of [y.sub.F] - (a - 2)] < [sigma][square root of 2/2] when x [member of] [??] I and jj) that the largest value of [x.sub.F] [approximately ap·prox·i·mate
1. Almost exact or correct: the approximate time of the accident.

2.
equal to] [sigma] when x = [sigma]. As for derivatives, those can be found by differentiating of the identities x = (1 - 2[epsilon])[x.sub.F][[psi].sub.F] ([x.sub.F]), [psi](x) = a(a - 2) - [x.sup.2.sub.F] - [[psi].sup.2.sub.F] ([x.sub.F])_/2. We finally find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because [x.sub.F] < 2[sigma] [much less than] [[psi].sub.F] ([x.sub.F]) [approximately equal to] a - 2, we obtain that [absolute value of [[psi]'.sub.F]([x.sub.F])] = [absolute value of (1 - 2[epsilon])[psi]'(x) - O([sigma])] < 0.01 when [epsilon] is close enough to [[epsilon].sub.0]. Since [[psi]'.sup.2.sub.F] ([x.sub.F]) < 0.0001, [absolute value of [x.sub.F][[psi]'.sub.F]([x.sub.F])] < 0.02, [[psi].sub.F] ([x.sub.F]) = a - 2 + O([sigma]), we find further that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Choosing [[epsilon].sub.F] so that 1 - 2[epsilon] = 1/(a - 2) + O([[sigma].sup.2]) for any [epsilon] [member of] ([[epsilon].sub.F], [[epsilon].sub.0]), we infer that [absolute value of [[psi]''.sub.F]] < 5 for all a > 1 + [square root of 4 + 2 [square root of 2]] and [epsilon] [member of] ([[epsilon].sub.F], [[epsilon].sub.0]).

Given a > 1 + [square root of 4 + 2 [square root of 2]], let us consider the function y = [square root of a(a - 2) - [x.sup.2]] on [0, [??]], where [??] < [sigma]. Its graph coincides with the arc [bar.F](A) [bar.F](H). Denote this function by y = [[psi].sub.0](x). The functions y = [[psi].sub.j] (x), j = 1, 2, ... are defined by means of their graphs This partial list of graphs contains definitions of graphs and graph families which are known by particular names, but do not have a Wikipedia article of their own.

For collected definitions of graph theory terms that do not refer to individual graph types, such as
in such a manner: The arc [[bar.F].sup.1]-j(AH) is the graph of y = [[psi].sub.j] (x). It is easy to see that [W.sup.u]([P.sub.+]) [contains] [bar.F](A) [[bar.F].sup.2](A) provided a > 1 + [square root of 5]. On the other hand, [W.sup.u]([S.sub.+]) [contains] [L.usb.[omega]t] for [epsilon] [member of] ([??], [[epsilon].sub.0]). Because of this, [[bar.F].sup.-j] (A) [right arrow] [P.sub.+], [[bar.F].sup.-j](H) [right arrow] [S.sub.+] as j [right arrow] [infinity]. Hence there is J that []bar.F].sup.-j] (A) [member of] [U.sub.[sigma]] ([P.sub.+]) for all j [greater than or equal to] J. Due to the orthogonality orthogonality

In mathematics, a property synonymous with perpendicularity when applied to vectors but applicable more generally to functions. Two elements of an inner product space are orthogonal when their inner product—for vectors, the dot product (see
of the coordinate Belonging to a system of indexing by two or more terms. For example, points on a plane, cells in a spreadsheet and bits in dynamic RAM chips are identified by a pair of coordinates. Points in space are identified by sets of three coordinates.  axes axes

[L., Gr.] plural of axis. The straight lines which intersect at right angles and on which graphs are drawn. Usually the horizontal axis is the x-axis and the vertical one the y-axis. Called also axes of reference.
as well as that the matrix ofDF is diagonal at points belonging to the y-axis, pre-images of [[bar.F].sup.-j] ({(x, y) : y = 0}) are orthogonal At right angles. The term is used to describe electronic signals that appear at 90 degree angles to each other. It is also widely used to describe conditions that are contradictory, or opposite, rather than in parallel or in sync with each other.  to the y-axis at their intersection points with the y-axis. Since {(x, y) : x2 + y2 = a(a - 2)} = [[bar.F].sup.-1] ({(x, y) : y = 0}), pre-images of y = [square root of a(a - 2) - [x.sup.2]] have the same property. Due to their smoothness and smooth dependence on parameters, it is clear that the following statement holds: There is [??] < [[epsilon].sub.0] such that, for any [epsilon] [member of] ([??], [[epsilon].sub.0]), there is a sub-arc of [[bar.F].sup.-J+1] ({(x, y) : y = [square root of a(a - 2) - [x.sup.2]], 0 [less than or equal to] x [less than or equal to] [sigma]}) which belongs to [??] = {(x, y) : 1/(1-2[[epsilon].sub.0])-[sigma] < y <1/(1-2[[epsilon].sub.0]) + [sigma], 0 [less than or equal to] x [less than or equal to] [??]} and stretches throughout the whole [??] from one end to another. Due to Lemma 2.4, we know that [L.sub.[omega]] erects and converges to the y-axis when a tends to [[epsilon].sub.0]. To be more precise, for any t > 0 (including t [right arrow] 0), there is [[epsilon].sub.t] > 0 such that if [epsilon] [member of] ([[epsilon].sub.t], [[epsilon].sub.0]), then x = [omega](y) < [t.sup.2] [much less than] t , [absolute value of [omega]'(y)] < [t.sup.2] [much less than] t for any (x, y) [member of] [L.sub.[omega]t] . Therefore there is [??] = max([??], [[epsilon].sub.t]) so that [[bar.F].sup.-J+1] (AH) [subset] [??]. In doing so one can fix [??] so close to [[epsilon].sub.0] that [absolute value of [[psi]'.sub.J]J (x)] < 0.01, [absolute value of [[psi]''.sub.J]J (x)] < 5 at those points (x, y) [member of] [[bar.F]-J+1](AH) whose x [member of] [0, [??]]. Thus y = [[psi].sub.J] (x) is the same as y = [psi](x) of Lemma 2.7.

Let [??] be a family of [C.sup.2]-smooth functions of the kind y = [psi](x) and y = [[psi].sub.m](x) be restrictions of y = [[psi].sub.J+m](x), m = 1, 2, ... to [0, [??]]. We show that y = [[psi].sub.m](x) belongs to [??].

Lemma 2.8. Let a, [epsilon], y = [[psi].sub.m](x) be such that Lemmas This following is a list of lemmas (or, "lemmata", i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures.  2.1-2.7 are valid. Then

1) [{y = [[psi].sub.m](x)}.sup.[infinity].sub.(m=1)] converges to its limit y = [[psi].sub.*](x) as m [right arrow] [infinity] uniformly with respect to x and

2) graph([[psi].sub.*]) = Cl([[??].sup.s.sub.+].

Proof. The maps [[bar.F].sup.-j] shrink shrink Vox populi noun A psychiatrist  AH [bar.F](H) [bar.F](A) to the y-axis. Therefore [bar.F-j] (AH [bar.F](H) [bar.F](A)) is a sequence of nested curvilinear quadrangles. Their sides [bar.F.sup.2j](AH) and [[bar.F].sup.-2j+1](AH) form two monotone sequences, one of which is increasing while the other is decreasing. As is shown above [[bar.F].sup.-J+1](AH) [subset] [??]. It is clear too that we can without loss of generality Without loss of generality (abbreviated to WLOG or WOLOG and less commonly stated as without any loss of generality) is a frequently used expression in mathematics.  assume that [[bar.F].sup.-J](AH) [subset] [??]. It follows from the latter inclusion that [[bar.F].sup.-j](AH) [subset] [??] for all j [greater than or equal to] J - 1. Due to Lemma 2.7, this means that y = [[psi].sub.m](x) belongs to [??] In view of the compactness of [??] as a subset of the [C.sup.1]-smooth function set, there are [C.sup.1]-smooth (invariant with respect to [PHI]) functions which are limits of the sequences [{[[psi].sub.J+2m]}.sup.[infinity].sub.(m=1)] and [{[[psi].sub.J+1m]}.sup.[infinity].sub.(m=1)]. Denote these functions by [[psi].sub.*] and [[psi].sup.*], respectively. Obviously [[psi].sub.*] and [[psi].sup.*] are one-valued functions a quantity that has one, and only one, value for each value of the variable.

. By definition, their graphs contain [P.sub.+] and [S.sub.+]. It is easy to see that y = [[psi].sub.*](x), y = [[psi].sup.*](x) should be solutions of (2.8). However if a neighbourhood of [S.sub.+] is small enough, then, as we know, there exists only one curve invariant with respect to [bar.F] whose slope is less than 0.01. This is [W.sup.s.sub.loc] ([S.sub.+]). Therefore, within the given neighbourhood, the graphs of y = .*(x), y = [[psi].sub.*](x) coincide with [W.sup.s.sub.loc]([S.sub.+]).

Let us show that graph([[psi.sub.*]) = graph([[psi.sub.*]) = Cl([[??].sup.s.sub.+]). Denote det(D [bar.F]) = (1 - 2[epsilon])([y.sup.2] - [x.sup.2]). Estimating [absolute value of det(D [bar.F])], we observe [absolute value of det(D [bar.F])] > 1 in [??] provided [epsilon] and [sigma] are small enough. This implies [absolute value of det(D [bar.F-1])] < 1 in [??]. Due to the smoothness and the invariance of graph ([[psi.sub.*]), graph([[psi.sub.*]) with respect to [[bar.F].sup.-1], the latter means graph([[psi.sub.*]) = graph([[psi].sup.*]). Obviously Cl([[??].sup.s.sub.+]) [subset] graph([[psi].sub.*]). Since y = [[psi].sub.*](x) is a one-valued function, Cl([[??].sup.s.sub.+]+ is a fixed point. But inside AH [bar.F](H) [bar.F](A) except [S.sub.+] there is one fixed point only. It is [P.sub.+]. Since [P.sub.+] belongs to the y-axis, we infer that graph([[psi].sub.*]) = Cl([[??].sup.s.sub.+]). As is well known [1], analytic maps have analytic stable and unstable manifolds at their hyperbolic fixed points. Therefore, y = [[psi].sub.*](x) is an analytic function.

The method presented above can be used in a neighbourhood to the right of [S.sub.+], i.e., forx > [delta]. Indeed let J be fixed in such a manner that the arc [[bar.F].sup.-J](AH) coincides with the graph of the function y = [[psi].sub.J] (x), where [[psi].sub.J] [member of] [??]. Choose [zeta] so that [delta] < [[zeta].sup.3] [much less than] < [sigma] and [[bar.F].sup.-J](AH) can be prolonged pro·long
tr.v. pro·longed, pro·long·ing, pro·longs
1. To lengthen in duration; protract.

2. To lengthen in extent.
inside the domain U [equivalent to] {(x, y) : 1/(1-2[[epsilon].sub.0])- [sigma] < y < 1/(1-2[[epsilon].sub.0]) + [sigma], 0 [less than or equal to] x [less than or equal to] [zeta]}. Let [[bar.F].sup.-J] (B) be the first intersection point of prolongation PROLONGATION. Time added to the duration of something.
2. When the time is lengthened during which a party is to perform a contract, the sureties of such a party are in general discharged, unless the sureties consent to such prolongation. See Giving time.
of [[bar.F].sup.-J](AH) with x = [zeta]. Thus [[bar.F].sup.-J] (BH) is a simply connected arc in U. This arc contains [[bar.F].sup.-J](AH), intersects x = [zeta], and is the graph of a function For the more general concept of the graph of a relation, see relation. For another use of the term "graph" in mathematics, see graph theory. For a graph-theoretic representation of a function from a set to the same set, see functional graph.  which belongs to [??] at [0, [zeta]]. Denote this function by y = [[PSI].sub.0](x). Without loss of generality, we can assume that the domain of [[bar.F].sup.-1] includes U. Then we can, with the aid of the map [PHI], define functions y = [[PSI].sub.m](x) which are restrictions of y = [[PSI].sub.J+m](x) to [0, [zeta]]. For the sake of simplicity Simplicity is the property, condition, or quality of being simple or un-combined. It often denotes beauty, purity or clarity. Simple things are usually easier to explain and understand than complicated ones. Simplicity can mean freedom from hardship, effort or confusion. , the restriction of [[bar.F].sup.-1] to U will be denoted by the same symbol [[bar.F].sup.-1] in the sequel.

Consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and let [[??].sup.s.sub.+] [subset] U be its simply connected component such that [S.sub.+] [member of] [[??].sup.s.sub.+].

Lemma 2.9. Let a, [epsilon], y = [[PSI].sub.m](x) be such that Lemmas 2.1-2.7 are valid. Then

1) [{y = [[PSI].sub.m](x)}.sup.[infinity].sub.(m=1)] converges to its limit y = [[PSI].sub.*](x) as m [right arrow] uniformly with respect to x and

2) graph([[PSI].sub.*]) = Cl([[??].sup.s.sub.+].

Proof. Let [[PSI].sub.0] [member of] [??] Repeating the computations produced in the proof of Lemma 2.8 it is not difficult to check that [PHI] transforms y = [[PSI].sub.0](x) into another function, say, y = [[PSI].sub.1](x) such that [[PSI].sub.1] [member of] [??] too. That is, [PHI] : [??] [right arrow] [??] Repeating the computations again and again, one can as above construct two function sequences [{[[PSI].sub.2m]}.sup.[infinity].sub.(m=1)] and [{[[PSI].sub.2m+1]}.sup.[infinity].sub.(m=1)]. The same reasons as above show that these sequences both are convergent con·ver·gence
n.
1. The act, condition, quality, or fact of converging.

2. Mathematics The property or manner of approaching a limit, such as a point, line, function, or value.

3.
. Their limits are continuous functions. Let us denote these functions by y = [[PSI].sub.*](x) and y = [[PSI].sup.*](x), respectively. We show that the limits coincide and determine a finite finite - compact  size fraction of [[??].sup.s.sub.+]. In order to do this, it is necessary to prove that the domain of y = [[PSI].sub.m](x), m = 1, 2, ... contains the interval [0, -]. We shall do it for a = [[epsilon].sub.0]. A general case ensues from it by continuity.

So, let [epsilon] = [[epsilon].sub.0]. Then [S.sub.+] = [P.sub.+] and 1/(1-2[[epsilon].sub.0]) = a - 2. Observe first of all that [[bar.F].sup.-1] transforms a line x = [zeta] into a hyperbola hyperbola (hīpûr`bələ), plane curve consisting of all points such that the difference between the distances from any point on the curve to two fixed points (foci) is the same for all points.  x = [zeta]/(a - 2)y. The hyperbola intersects the line at a point T = ([zeta], a - 2). It is clear that there is no loss of generality gen·er·al·i·ty
n. pl. gen·er·al·i·ties
1. The state or quality of being general.

2. An observation or principle having general application; a generalization.

3.
in assuming y = [[PSI].sub.0](x) such that [[PSI].sub.0](x) > a-2 for all x [member of] [0, [zeta]]. Denote [T.sub.f] [equivalent to] graph([[PSI].sub.0])[intersection]{(x, y) : x = [zeta]} and let p [member of] (0, [sigma]) be such that a - 2 + p is the y-coordinate of [T.sub.f]. Assume additionally that [sigma] is chosen so that, in Cl(U), the action of [[bar.F].sup.-1] along the y-axis is a linear (accurate to the second order quantity of smallness) contraction contraction, in physics
contraction, in physics: see expansion.
contraction, in grammar
contraction, in writing: see abbreviation.

contraction - reduction
towards the line y = a - 2. Obviously this can be done. Then the shrinkage Shrinkage

The amount by which inventory on hand is shorter than the amount of inventory recorded.

Notes:
The missing inventory could be due to theft, damage, or book keeping errors.
coefficient coefficient /co·ef·fi·cient/ (ko?ah-fish´int)
1. an expression of the change or effect produced by variation in certain factors, or of the ratio between two different quantities.

2.
is equal to 1/(a - 2). Therefore, if [y.sub.f] denotes the value of the y-coordinate of [[bar.F].sup.-1]([T.sub.f]), then [y.sub.f] can be estimated by [y.sub.f] = a-2-p/(a-2) + O([p.sup.2]). Let us compute To perform mathematical operations or general computer processing. For an explanation of "The 3 C's," or how the computer processes data, see computer.  now [x.sub.f], the value of the x-coordinate of [[bar.F].sup.-1]([T.sub.f]). Denote the angle between x = [zeta] and x = [zeta]/(a - 2)y at T by h. Then tg(h) = -[zeta] /(a-2). Since o is small enough, the hyperbola arc which lies in Cl(U) coincides (accurate to the second order quantity of smallness) with its tangent at T . Therefore, accurate to O([p.sup.2]), [x.sub.f] = [zeta] ptg(h)/a - 2 = [zeta] [[1 + p(a - 2).sup.-2] = [zeta]'. So, we find that the domain of y = [[PSI].sub.1](x) is approximately equal to [0, [zeta]'] which is larger than [0, [zeta]].

In order to treat y = [[PSI].sub.2](x), it is enough to apply the aforesaid reasons to [T'.sub.f] = [[bar.F].sup.-1]([T.sub.f]). In doing so, one should keep in mind that [zeta]' > [zeta]. Let us denote the intersection point of x = [zeta]'/(a - 2)y with x = [zeta] by T', the angle between x = [zeta]'/(a - 2)y and x = [zeta] at T' by h', the coordinates of [[bar.F].sup.-1]([T'.sub.f]) by [x'.sub.f], [y'.sub.f]. Making computations similar to those presented above, we find accurate to O([p.sup.3]) that [y'.sub.f] = a - 2 + p[(a - 2).sup.-2], tg(h') = -[zeta]/(a-2) 1 + p[(a - 2).sup.-2]], [equivalent to] [zeta]''. Thus, we see that the domain of y = [[PSI].sub.2](x) is approximately equal to the interval [0, [zeta]] which is smaller than [0, [zeta]'] while larger than [0, [zeta]]. All this takes place provided a is close enough to 4, and so on. As a result we find that the domains of y = [[PSI].sub.m](x), m = 1, 2, ... all are larger than [0, [zeta]]. By continuity, the same is true for [epsilon], a sufficiently close to [[epsilon].sub.0], 4, respectively. Since [sigma] was fixed independently of values of the parameters a and [epsilon], there exist limits of the sequences of [{[[PSI].sub.2m]}.sup.[infinity].sub.(m=1)], [{[[PSI].sub.2m+1]}.sup.[infinity].sub.(m=1)], and these limits coincide with the arc [[??].sup.s.sub.+]. Actually it is easy to see that y = [[PSI].sub.*](x) and y = [[PSI].sup.*](x) should be the solutions of (2.8). However as we know, the only curve which is invariant with respect to [bar.F] and has the slope less than 0.01 is [[??].sup.s.sub.loc]([S.sub.+]). Hence, in the given neighbourhood, the graphs graph([[PSI].sub.*]), graph([PSI], *) coincide with the arc of [[??].sup.s.sub.+]. We show that this arc stretches throughout the whole interval [0, [zeta]]. Indeed, it is evident that [[??].sup.s.sub.+] [subset] graph([[PSI].sub.*]) [intersection] graph([[PSI].sup.*]). If [[??].sup.s.sub.+] does not intersect In a relational database, to match two files and produce a third file with records that are common in both. For example, intersecting an American file and a programmer file would yield American programmers.  x = [zeta] , then Cl([[??].sup.s.sub.+] is inside Cl(U) [intersection] {(x, y) : [delta] [less than or equal to] x [less than or equal to] [zeta]}. Since y = [[PSI].sub.*](x), y = [[PSI].sup.*](x) are one-valued functions, it is clear that Cl([[??].sup.s.sub.+] is a one point set. Due to its invariance with respect to [[bar.F].sup.-1], it should be a fixed point. But as we know, if [epsilon] [member of] ([??], [[epsilon].sub.0]), then except [S.sub.+] there are no other fixed points of F in Cl(U [intersection] {(x, y) : [delta] [less than or equal to] x [less than or equal to] [zeta]}). Therefore, our assumption is false.

[FIGURE 3 OMITTED]

Lemma 2.10. There is [??] [member of] (1 + [square root of 4 + 2 [square root of 2, 4])] such that, for any a [member of] ([??], 4) and arbitrary U([P.sub.+]), U([[LAMBDA].sub.a]), one can find [[epsilon].sub.a] [member of] (0, [[epsilon].sub.0]) and then J = J(a, U ([P.sub.+]),U([[LAMBDA].sub.a]), [[epsilon].sub.a]) such that, for all [epsilon] [member of] ([[epsilon].sub.a], [[epsilon].sub.0]), the following is fulfilled ful·fill also ful·fil
tr.v. ful·filled, ful·fill·ing, ful·fills also ful·fils
1. To bring into actuality; effect: fulfilled their promises.

2.
:

[iota]) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

[iota][iota]) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

[iota][iota][iota]) the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Contains points of transverse To cross from side to side.  intersection of the manifolds [W.sup.s.sub.loc](S.sub.+]) and [W.sup.u]([S.sub.+]).

Proof. Let [zeta] [member of] (0, [sigma]) be small enough. Choose [[epsilon].sub.F] > 0 so that Lemmas 2.8 and 2.9 are satisfied for any [epsilon] [member of] ([[epsilon].sub.F], [[epsilon].sub.0]). The latter permits us to consider [zeta] as a size of [W.sup.s.sub.loc]([S.sub.+]). Lemmas 2.8 and 2.9 imply that [iota])[W.sup.s.sub.loc]([S.sub.+]) [subset] U, [iota][iota]) Cl[W.sup.s.sub.loc] ([S.sub.+])[intersection]{(x, y) : x = 0} [not equal to] [phi], Cl([W.sup.s.sub.loc]([S.sub.+])[intersection] {(x, y) : x = [zeta]} [not equal to] [phi], [iota][iota][iota]) the slope of [W.sup.s.sub.loc]([S.sub.+]) to the x-axis does not exceed 0.01 at all points of [W.sup.s.sub.loc]([S.sub.+]). Here, U is the same as in Lemma 2.9.

Since [[bar.F].sup.2] [equivalent to] [F.sup.2] and [[bar.F].sup.j+1] ([W.sup.u.sub.loc]([S.sub.+]) [contains] [[bar.F].sup.j] ([W.sup.u.sub.loc]([S.sub.+])) for all natural j , the unstable manifold of F at [S.sub.+] coincides with that of [bar.F] at [S.sub.+]. Therefore, one can study the [bar.F]-images of [W.sup.u.sub.loc]([S.sub.+]) instead of the F-images of [W.sup.u.sub.loc]([S.sub.+]). We shall do this in the sequel.

Fix t > 0 so that [quadro root of t] < [zeta]. Without loss of generality, one can assume that [[epsilon].sub.F] is chosen so that [quadro root of [delta]] < t and Lemma 2.4 is fulfilled. This means that there exists the arc [L.sub.[omega]t] of [W.sup.u]([S.sub.+]) which stretches out of [[bar.F].sup.2]([Z.sub.t]) to [bar.F]([Z.sub.t]) (see the designations preceding Lemma 2.4) and such that: [iota]) the distance between [L.sub.[omega]t] and [[LAMBDA].sub.a] is less than t ; [iota][iota][R]) the slope of [W.sup.u]([S.sub.+]) to the x-axis exceeds 1/t [much greater than] 1 at all points of [L.sub.[omega]t]. Repeating the reasons and the computations presented after the proof of Lemma 2.4, we can find once more a fraction of [W.sup.u]([S.sub.+]). This fraction is situated on the other side of [[LAMBDA].sub.a], possesses the same properties as [L.sub.[omega]t] and stretches out of [F.sup.3](K) to F(K). Continuing computations and repeating reasons similar to those aforesaid, at the next step, we can obtain once more a fraction of Wu(S+) which is located on the same side of [[LAMBDA].sub.a] as [L.sub.[omega]t], possesses the same properties as [L.sub.[omega]t] and stretches out approximately of F(K) to [F.sup.3](K). It is easy to verify (1) To prove the correctness of data.

(2) In data entry operations, to compare the keystrokes of a second operator with the data entered by the first operator to ensure that the data were typed in accurately. See validate.
the validity of the last statements. To do this, it is necessary to localize lo·cal·ize
v. lo·cal·ized, lo·cal·iz·ing, lo·cal·iz·es

v.tr.
1. To make local: decentralize and localize political authority.

2.
[bar.F]-images of a few remarkable points. For example, let us consider [Z.sub.[+ or -]] and [Z.sub.x], the points of intersection of [L.sub.[omega]t] with the critical set K and with the x-axis, respectively. It is not difficult to see that if it is known where [bar.F]-images of the points of [Z.sub.[+ or -]] and [Z.sub.x] are, then [bar.F]([L.sub.[omega]t]) can be localized Translated into the spoken language of the country. See localization.  good enough. In particular because [iota]) we have continuous dependence of [L.sub.[omega]t] on a and [epsilon]; [iota][iota]) [Z.sub.t] [approximately equal to] [Z.sub.+] when a, [epsilon] close to 4, [[epsilon].sub.0], respectively; [??] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 0 for a close to 4; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Analogously a·nal·o·gous
1. Similar or alike in such a way as to permit the drawing of an analogy.

2. Biology Similar in function but not in structure and evolutionary origin.
in order to localize [[bar.F].sup.2]([L.sub.[omega]t]), it is necessary to find points of intersection of [bar.F]([L.sub.[omega]t]) with the critical set, with its [bar.F]-image and with the coordinate axes, respectively, and then to compute and to localize [bar.F]-images of these points, and so on. In doing so, the parameter values can be fixed such that the end points of [[bar].F.sup.j]([L.sub.[omega]t]), j = [??] both belong to a neighbourhood of [F.sup.2](O). Obviously the neighbourhood size (resp. natural J ) can be chosen as small (resp. large) as desirable. Indeed, since [F.sup.2](O) = [F.sup.j](O), j [greater than or equal to] 2 when a = 4 and [L.sub.[omega]t] [intersection] K [right arrow] O as [epsilon] [??] [[epsilon].sub.0], due to the continuous dependence of F on parameters, there exist [??] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[epsilon].sub.a] [member of] ([[epsilon].sub.F], [[epsilon].sub.0]) such that the aforesaid assertion (programming) assertion - 1. An expression which, if false, indicates an error. Assertions are used for debugging by catching can't happen errors.

2. In logic programming, a new fact or rule added to the database by the program at run time.
is valid for all j = [??] provided [epsilon] [member of] ([[epsilon].sub.a], [[epsilon].sub.0]). As for slopes of the fractions of [[bar.F].sup.j]([L.sub.[omega]t]) under consideration, one can show that these slopes exceed 1/t [much greater than] 1 at the points of [[bar.F].sup.j]([L.sub.[omega]t]) which lie between F(K) and [F.sup.j+2](K), j = 1, 2,.... This can be done in the same way as it is done above where the slopes of [bar.F]([L.sub.[omega]t]) are studied.

It is much simpler to collect and to pick up successive steps of the algorithm algorithm (ăl`gərĭth'əm) or algorism (–rĭz'əm) [for Al-Khowarizmi], a clearly defined procedure for obtaining the solution to a general type of problem, often numerical.  that is presented above in figures than to describe those steps by words. That is why we depict de·pict
tr.v. de·pict·ed, de·pict·ing, de·picts
1. To represent in a picture or sculpture.

2. To represent in words; describe. See Synonyms at represent.
several first steps of the given algorithm in Figs. 4-5.

Studying [[bar.F].sup.j]([L.sub.[omega]t]), j = 2, 3, ..., it is easy to observe that [[bar.F].sup.2]([L.sub.[omega]t]) is such that [[bar.F].sup.2]([L.sub.[omega]t]\[W.sup.u.sub.loc]([S.sub.+])) contains an arc which has the necessary properties and intersects [W.sup.s.sub.loc]([S.sub.+]) transversely. Denote by [[bar].F.sup.2] ([L.sub.[omega]t]\[W.sup.u.sub.loc]([S.sub.+])[intersection][W.sup.s.sub.loc] ([S.sub.+]) [??] [THETA] the intersection point. There is an alternative: either a) [THETA] = [S.sub.+] or b) [THETA} [not equal to] [S.sub.+]. Let us discuss the alternative.

If (a) is satisfied, then there is a point [??] [member of] [W.sup.u.sub.loc]([S.sub.+]) \ {[S.sub.+]} such that: 1) [[bar.F].sup.-m]([??]) [member of] [W.sup.u.sub.loc]([S.sub.+]) for all natural m; 2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] there is natural M such that [[bar.F].sup.M] (([bar.[THETA]]) = [[bar.F].sup.m] ([THETA]) [member of] [W.sup.s.sub.loc]([S.sub.+]) for all natural m and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the orbit of [??], by Orb([??]). Here [??] is

[??] or [bar.[THETA]], and N is a natural or countable (mathematics) countable - A term describing a set which is isomorphic to a subet of the natural numbers. A countable set has "countably many" elements. If the isomorphism is stated explicitly then the set is called "a counted set" or "an enumeration".  number. Considering the orbits Orb([??]) and Orb([bar.[THETA]]), we can see that these orbits both are homoclinic to [S.sub.+]. Taking into account that [[bar.F].sup.2] ([L.sub.[omega]t] \ [W.sup.u.sub.loc]([S.sub.+])) intersects [W.sup.s.sub.loc] ([S.sub.+]) transversely, we infer that the given orbits both are transverse homoclinic too. (1) (We notice that if (a) takes place, then there must be a neighbourhood [??] of [??] in [W.sup.u]([S.sub.+]) such that [[bar.F].sup.M]([??]) = [W.sup.u.sub.loc]([S.sub.+]) because [S.sub.+] is a hyperbolic saddle and there is only one 1-dimensional expansion direction of [bar.F] at [S.sub.+].) []

[FIGURE 4 OMITTED]

Proof of Theorem 1.1. Theorem 1.1 is a corollary of Lemma 2.10. Indeed, according to according to
prep.
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

3.
[5,9], any neighbourhood of an orbit which is transverse homoclinic to a hyperbolic fixed point contains a nontrivial topologically transitive set chaotic in the sense of Li and Yorke [7]. This set is invariant with regard to [F.sub.2], and periodical periodical, a publication that is issued regularly. It is distinguished from the newspaper in format in that its pages are smaller and are usually bound, and it is published at weekly, monthly, quarterly, or other intervals, rather than daily.  points are everywhere dense on the given set. Because F([S.sub.+]) = [S.sub.-], F([W.sup.u.sub.loc]([S.sub.+])) = [W.sup.u.sub.loc]([S.sub.-]), F([W.sup.s.sub.loc]([S.sub.+])) = [W.sup.s.sub.loc]([S.sub.-]), there exists another set which is symmetric with respect to the y-axis to the aforesaid one and has the same properties. Their union gives us a set possessing all the aforesaid one has the same properties. Their union gives us a set possessing all the necessary properties which is invariant with respect to F. []

[FIGURE 5 OMITTED]

Concluding the study of F, let us show that, at the bifurcation value [epsilon] = [[epsilon].sub.0], the phase pattern of F in a vicinity of U([P.sub.+]) does not differ from that for [epsilon] [member of] ([[epsilon].sub.0], 1/2).

Lemma 2.11. Given [epsilon] = [[epsilon].sub.0] and a > 3. Then F has the 1-dimensional local stable manifold [W.sup.s.sub.loc]([P.sub.+]) at the point [P.sub.+].

Proof. Let us transfer the origin of coordinates at the point [P.sub.+] and introduce coordinates

[??] = x, [??] = y - a + 2. Then, for [epsilon] = [[epsilon].sub.0], the map F takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In what follows, we will omit o·mit
tr.v. o·mit·ted, o·mit·ting, o·mits
1. To fail to include or mention; leave out: omit a word.

2.
a. To pass over; neglect.

b.
"hats" above the coordinates [??], [??]. The eigenvalues of D F at (0, 0) are equal to -1 and 2 - a. We show that there is a curve y = [psi](x) which is invariant with respect to [??] and that this curve is a locally stable manifold of [??] at (0, 0).

Assume the curve exists. Because of its invariance with respect to F, the inclusion F (graph([psi])) [subset] graph ([psi]) holds. In other words Adv. 1. in other words - otherwise stated; "in other words, we are broke"
put differently
, we have the functional identity

-(a - 2)[psi](x) - {[x.sup.2] + [[[psi](x)].sup.2]}/2 = [psi] (-x - x[psi](x)/(a - 2)), [psi](0) = 0.

Differentiating this identity three times, we find [psi]'(0) = 0, [psi]"(0) = -1/(a - 1), [psi]"'(0) = 0. Thus, if the function-solution exists, then y = -[x.sup.2]/[2(a - 1)] + O([x.sup.4]). We show that this indeed holds. Denote y = -[x.sup.2]/[2(a - 1)] [??] [[psi].sub.0](x) and consider its [??]-image. Due to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the image of y = [[psi].sub.0](x) is a curve [bar.y] = [[psi].sub.1]([bar.x]), where [bar.x] = -x + [x.sup.3] / 2(a - 1)(a - 2), [bar.y] = [x.sup.2]/2(a - 1) [1 + [x.sup.2] / 4(a - 1)]. Since [absolute value of [bar.x]] < [absolute value of x] and y([bar.x]) > y(x) > [bar.y]([bar.x]) for (x, y) [not equal to] (0, 0) [not equal to] ([bar.x], [bar.y]), it is clear that this curve is located under y = [psi]0(x). Denote [[product].sub.0] = {(x, y) : [[psi].sub.1](x) [less than or equal to] y [less than or equal to] [[psi].sub.0](x), -[gamma] [less than or equal to] x [less than or equal to] [gamma]}, where [less than or equal to] > 0 is a small constant. Consider [??]([[product].sub.0]). Since [??] is a diffeomorphism inside a neighbourhood of (0, 0), [??]-images of the curves y = [[psi].sub.j] (x), j = 0, 1 localize [??]([[product].sub.0]) completely. Computing, we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence the image of [bar.y] = [[psi].sub.1]([bar.x]) is a curve [??] = [[psi].sub.2]([??]), where

[??] = x - [x.sup.3][(a - 1).sup.-1][(a - 2).sup.-1] + O([x.sup.5]),

[??] = -[x.sup.2]/[2(a - 1)] 1 - (a - 3)[x.sup.2]/[4(a - 1)] - [x.sup.2]/(a - 2)}+ O([x.sup.6]).

We show that [??] = [psi]2([??]) is located above y = [[psi].sub.0](x). Computing [[psi].syb.0]([??]) we obtain [[psi].sub.0]([??]) = -[x.sup.2]/[2(a - 1)] + [x.sup.4][(a - 2).sup.-1][(a - 1).sup.-2] + O([x.sup.6]). Since

1 / (a - 2)[(a - 1).sup.2] < a - 3 / 8[(a - 1).sup.2] + 1 / 2(a - 2)(a - 1) when a > 3,

for x small enough, we obtain 1) the curve y = [[psi].sub.2](x) lies under y = 0 but above y = [[psi].sub.0](x); 2) the cone cone, in botany
cone or strobilus (strŏb`ələs), in botany, reproductive organ of the gymnosperms (the conifers, cycads, and ginkgoes).
[[product].sub.0] is located inside the cone [??]([[product].sub.0]). Notice that [Dom Dom (dōm), peak, 14,942 ft (4,554 m) high, Valais canton, S Switzerland, in the Mischabelhörner group. It is the highest peak entirely in Switzerland. .sub.j], the domains of y = [[psi].sub.j] (x), shrink slightly under the action of [??]. That is [Dom.sub.0] [contains] [Dom.sub.1] [contains] [Dom.sub.2] [contains] ... [contains] [Dom.sub.j]....

Choose [gamma] > 0 so that

[[product].sub.0] [intersection] {(x, y) : [gamma]/2 [less than or equal to] x = [less than or equal to]/2} [subset] [??]([[product].sub.0]) [subset] {(x, y) : [gamma]/2 [less than or equal to] x [less than or equal to] [gamma]/2}

and consider a sequence of [{[[product].sub.-m]}.sup.[infinity].sub.m=1], where [[product].sub.-m] = [[??].sup.- 1]([[product].sub.-m+1])[intersection][[product].sub.0],m = 1, 2,.... What is said above means that [{[[product].sub.- m]}.sup.[infinity].sub.m=1] constitute a set of the nested cones Cones
Receptor cells that allow the perception of colors.

Mentioned in: Color Blindness
. Denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This is an invariant set. Because [absolute value of det(D [[??].sup.-1]([P.sub.+]))] = 1/(a - 2) < 1, there is [gamma] > 0 so that [absolute value of det(D [[??].sup.-1]((x, y)))] < 1 for any (x, y) [member of] [U.sub.[gamma]] ((0, 0)). In view of the invariance of [product], the latter means that mes([product]) = 0. Therefore, the sides of [product] which are limits of sequences of graphs of the functions y = [[psi].sub.-2m](x) and y = [[psi].sub.-2m+1](x) coincide. Thus there is only one curve y = [psi](x) which is invariant with respect to [[??].sup.-1]. It is easy to check that this is a smooth arc. In fact, the graphs of y = [[psi].sub.0](x), y = [[psi].sub.1](x) touch each other at x = 0 and [[psi]'.sub.0](0) = [[psi]'.sub.1](0) = 0, [[psi]".sub.0](0) = [[psi]".sub.1](0) = -1/(a 2). The graphs of y = [[psi].sub.-j] (x), j = 2, 3, ... ,by construction, all are situated between them and [[psi]'.sub.-j] (0) = 0, [[psi]".sub.-j] (0) = -1/(a - 2) for all natural j. Therefore the function y = [psi](x) is at least twice differentiable dif·fer·en·tia·ble
1. That can be differentiated: differentiable species.

2. Mathematics Possessing a derivative.
at x = 0 and has a finite continuous first derivative Noun 1. first derivative - the result of mathematical differentiation; the instantaneous change of one quantity relative to another; df(x)/dx
derivative, derived function, differential, differential coefficient
in a neighbourhood of x = 0. So, there is the neighbourhood of [P.sub.+] in which the graph of the function y = [psi](x) is a [C.sup.1]-smooth arc. Thus, the [C.sup.1]-smooth locally stable manifold of F at [P.sub.+] prolongs to exist even when [epsilon] = [[epsilon].sub.0]. This means that the phase space pattern bifurcation does still not take place although the exponential 1. (mathematics) exponential - A function which raises some given constant (the "base") to the power of its argument. I.e.

f x = b^x

If no base is specified, e, the base of natural logarthims, is assumed.
2.
rate of convergence In numerical analysis (a branch of mathematics), the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical  to [P.sub.+] along [W.sup.s.sub.loc]([P.sub.+]) is already lost. []

3. Proof of Theorem 1.3

Proof of Theorem 1.3. The main idea is to find a suitable family of diffeomorphisms so that the results which are just presented for the endomorphisms can be applied to these diffeomorphisms. Since our approach consists in a successive use of continuity and continuous dependence, we restrict In the C programming language, the data pointed to by a pointer declared with the restrict qualifier may not be pointed to by any other pointer. This allows for more effective optimization.  ourselves to indicate the proof scheme in considerable detail.

Consider the families [Y.sub.0] and [Y.sub.b]. Obviously, the plane x = z = 0 is invariant with respect to [Y.sub.0], the arc [DELTA]([Y.sub.0]) [equivalent to] {(0, 0, y, t) : y = [a(a - 2) - [t.sup.2]]/2, [f.sup.2.sub.a] (0) [less than or equal to] t [less than or equal to] [f.sub.a](0)}, where [f.sub.a](0) = a(a - 2)/2, [F.sup.2.sub.a] (0) [equivalent to] a(a - 2)[a(a - 2) - 4]/8 belongs to this plane, [P.sub.+]([Y.sub.0]) = (0, 0, a - 2, a - 2) [member of] [LAMBDA]([Y.sub.0]) is a fixed point. It is evident too that [LAMBDA]([Y.sub.0]) is a topologically transitive set for a [member of] ([??], 4) [intersection] [??]. Indeed, [Y.sub.0]((0, 0, y, t)) [member of] [LAMBDA]([Y.sub.0]) for y [member of] [[f.sup.2.sub.a](0), [f.sub.a](0)] as [Y.sub.0]((0, 0, y, t)) = (0, 0, [a(a - 2) - [y.sup.2]]/2, y). Since there exists [??] [member of] [[f.sup.2.sub.a] (0), [f.sub.a](0)] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], projections of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are dense on the segment [[f.sup.2.sub.a] (0), [f.sub.a](0)] of the y-axis and on the same segment of the t-axis. The latter means that [LAMBDA]([Y.sub.0]) is a topologically transitive set. In a vicinity of [P.sub.+]([Y.sub.0]), there are two periodic of period 2 points [S.sub.[+ or -]]([Y.sub.0]) [equivalent to] ([+ or -][x.sub.S] (0),[+ or -][z.sub.S](0), [y.sub.S](0), [t.sub.S](0)) when [epsilon] [member of] ([[epsilon].sub.a], [[epsilon].sub.0]). Here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Denote by x O y the factor space [R.sup.4]/~ which is constituted by identification of the points of [R.sup.4] whose coordinates x, y are identical. That is x O y [equivalent to] [R.sup.4]/ ~= {(x, ., y, .)} or simply {(x, y)}. Obviously, the action of [Y.sub.0] restricted to x O y coincides with that of F.

Consider

{(x, z, y, t) : 0 [less than or equal to] x [less than or equal to] [zeta], -[infinity] < z < [infinity], y = [OMEGA](x),-[infinity] < t < [infinity]} [subset] [R.sup.4],

"a suspension" over a fraction of the stable manifold [W.sup.s]([P.sub.+]) [subset] x O y of F at [S.sub.+]. Denote it by [W.sup.s.sub.loc]([S.sub.+]([Y.sub.0])). It is not difficult to check that [W.sup.s.sub.loc]([S.sub.+]([Y.sub.0])) is a fraction of the stable manifold of [Y.sub.0] at the point [S.sub.+]([Y.sub.0]) adjoining to [S.sub.+]([Y.sub.0]). Indeed, since [Y.sub.j.sub.0] (([x.sub.0], z, [OMEGA]([x.sub.0]), t)) = ([x.sub.j], [x.sub.j-1], [OMEGA]([x.sub.j]), [OMEGA]([x.sub.j-1]), where

[x.sub.j] = -(1 - 2[epsilon])[x.sub.j-1][OMEGA]([x.sub.j]), [OMEGA](x.sub.j) = a(a-2) - [[OMEGA].sup.2]([x.sub.j-1]) - [x.sup.2.sub.j-1] / 2, j = 1, 2, ... ,

it is clear that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is obvious that

[Y.sub.0] ([W.sup.s.sub.loc]([S.sub.+]([Y.sub.0]))) [subset] [W.sup.s.sub.loc]([S.sub.-]([Y.sub.0])), [Y.sup.2.sub.0] ([W.sup.s.sub.loc]([S.sub.+]([Y.sub.0]))) [subset] [W.sup.s.sub.loc]([S.sub.+]([Y.sub.0])).

Consider the set

{(x, z, y, t) : x = (1 - 2[epsilon])ww(w), y = a(a - 2) - [[omega].sup.2](w) - [w.sup.2]]/2, z = -w(w), t = w, 0 < w < a(a - 2)/2}.

Computing its [Y.sub.0]-image, one can easily verify that the given set is a semi-locally unstable manifold [W.sup.u.sub.sl]([S.sub.+]([Y.sub.0])) of [Y.sub.0] at [S.sub.+]([Y.sub.0]). In fact, the equalities

x = (1 - 2[epsilon])ww(w), y = a(a - 2) - [[omega].sup.2](w) - [w.sup.2]_/2,

where [omega](t) is the solution of the equation (2.2), are fulfilled for any (x, z, y, t) [member of] [W.sup.u.sub.sl]([S.sub.+]([Y.sub.0])). Therefore x = [omega](y). The latter means that

[Y.sub.0] ((x, z, y, t)) = (-(1 - 2[epsilon]) [bar.w][omega]([bar.w]), [omega]([bar.w]), [a(a - 2) - [[omega].sup.2]([bar.w]) - [[bar.w].sup.2]/2, [bar.w])

and

[Y.sup.2.sub.0] ((x, z, y, t)) = (-(1 - 2[epsilon]) [??][omega]([??]),[omega]([??]), a(a - 2) - [[omega].sup.2]([??]) - [[??].sup.2]]/2, [??]), where [??] = y and [??] = a(a - 2) - [[omega].sup.2]([bar.w]) - [[bar.w].sup.2]/2. Thus, [W.sup.u.sub.sl]([S.sub.+]([Y.sub.0])) is an invariant set with respect to [Y.sup.2.sub.0].

Let z O t be a factor space similar to x O y, namely z O t = {(x, z, x, t)} or simply {(z, t)}. Because projections of [W.sup.u.sub.sl]([S.sub.+]([Y.sub.0])) on x O y and z O t are stretched under the action of [Y.sub.0], the set [W.sup.u.sub.sl]([S.sub.+]([Y.sub.0])) is stretched under the action of [Y.sub.0] too. Therefore, [Y.sub.0] ([W.sup.u.sub.sl]([S.sub.+]([Y.sub.0]))) [contains] [W.sup.u.sub.sl]([S.sub.-]([Y.sub.0])) and [Y.sup.2.sub.0] (W.sup.u.sub.sl]([S.sub.+]([Y.sub.0]))) [contains] [W.sup.u.sub.sl]([S.sub.+]([Y.sub.0])).

Denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the globally unstable manifold of [Y.sub.0] at [S.sub.+]([Y.sub.0]). Let us show that there is a point where [W.sup.u]([S.sub.+]([Y.sub.0])) intersects [W.sup.s.sub.loc]([S.sub.+]([Y.sub.0])) transversely. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we observe that two eigenvalues [v.sub.j,] J = 1,2 of D[Y.sup.2.sub.0] at [S.sub.[+ or -]]([Y.sub.0]) are equal to 0 and the other two eigenvalues are nonzero non·ze·ro
Not equal to zero.

nonzero

Not equal to zero.
and coincide with those of D[F.sub.2] at [S.sub.[+ or -]]. Therefore [square root of [v.sub.3](0)] < 1, [square root of [v.sub.4](0)] > 1. Thus, the dimension of the locally stable manifold of [Y.sub.0] adjoining to the point [S.sub.[+ or -]]([Y.sub.0]) is equal to 3 and that of the local unstable manifold of [Y.sub.0] at [S.sub.[+ or -]]([Y.sub.0]) is equal to 1. Further, considering the manifold projections on x O y, we find those that coincide with [W.sup.s]([S.sub.+]) and [W.sup.u]([S.sub.+]), respectively. According to Lemma 2.8, in any vicinity of U([S.sub.+]) {[S.sub.+]}, there are points of transverse intersection of [W.sup.s]([S.sub.+]) and [W.sup.u]([S.sub.+]). Let ([x.sub.[THETA]], [y.sub.[THETA]]) [member of] U([S.sub.+]) \ {[S.sub.+]} be one of such points. Because ([[x.sub.[THETA]]], [y.sub.[THETA]]) [member of] [W.sup.u]([S.sub.+]), there exist w, j such that [F.sup.j] (([[??].sub.[THETA]], [[??].sub.[THETA]])) = ([x.sub.[THETA]], [y.sub.[THETA]]), where ([??],[[??].sub.[THETA]]) [member of] [W.sup.u.sub.loc]([S.sub.+]), [[??].sub.[THETA]] = (1 - 2[epsilon])w[omega](w) and [[??].sub.[THETA]] = a(a - 2) - [[omega].sup.2](w) - [w.sup.2]]/2. Consider a point [Y.sup.j.sub.0] (([[??].sub.[THETA]], [[??].sub.[THETA]], [[??].sub.[THETA]], [[??].sub.[THETA]])), where [[??].sub.[THETA]] = - [omega](w), [[??].sub.[THETA]] = w. It is obvious that [Y.sup.j.sub.0] (([[??].sub.[THETA]], [[??].sub.[THETA]], [[??].sub.[THETA]], [[??].sub.[THETA]])) = ([x.sub.[THETA]], [z.sub.[THETA]], [y.sub.[THETA]], [t.sub.[THETA]]) [member of] [W.sup.u]([S.sub.+]([Y.sub.0])). On the other hand, ([x.sub.[THETA]], [y.sub.[THETA]]) [member of] [W.sup.s]([S.sub.+]) implies [y.sub.[THETA]] = [OMEGA]([x.sub.[THETA]]). Therefore ([x.sub.[THETA]], [z.sub.[THETA]], [y.sub.[THETA]], [t.sub.[THETA]]) [member of] [W.sup.s.sub.loc]([S.sub.+]([Y.sub.0])). It is easy to verify that this is a point of transverse intersection of [W.sup.s.sub.loc]([S.sub.+]([Y.sub.0])) and [W.sup.u]([S.sub.+]([Y.sub.0])).

Indeed, as is known, a curve determined by the equations x = x(w), z = z(w), y = y(w), t = t (w) is tangent to the hypersurface In mathematics, a hypersurface is some kind of submanifold.
• For differential geometry usage, see glossary of differential geometry and topology.
• In algebraic geometry, a hypersurface in projective space of dimension n
N(x, z, y, t) = 0 at the point (x([??]), z([??]), y([??]), t ([??])) if the following two conditions are fulfilled: N(x([??]), z([??]), y([??]), t ([??])) = 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

In doing so, at least one of the derivatives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] must be nonzero. Applying the aforesaid reasons to the case under examination, we find that N(x, z, y, t) = y - [OMEGA](x), [partial derivative partial derivative

In differential calculus, the derivative of a function of several variables with respect to change in just one of its variables. Partial derivatives are useful in analyzing surfaces for maximum and minimum points and give rise to partial differential
]N/[partial derivative]x = -[OMEGA](x), [partial derivative]N/[partial derivative]z = 0, [partial derivative]N/[partial derivative]y = 1, [partial derivative]/ [partial derivative]t = 0. Without loss of generality, one can assume that [??] is such that ([x.sub.[THETA]], [z.sub.[THETA]], [y.sub.[THETA]], [t.sub.[THETA]]) [equivalent to] (x([??]), z([??]), y([??]), t ([??])). The equality equality

Generally, an ideal of uniformity in treatment or status by those in a position to affect either. Acknowledgment of the right to equality often must be coerced from the advantaged by the disadvantaged. Equality of opportunity was the founding creed of U.S.
(3.1) reduces then to

dy([??]) / dw - d[OMEGA](x([??])) / dx dx([??]) / dw = 0.

Since the projection projection, in psychology: see defense mechanism.

See rear-projection TV, front-projection TV and LCD panel.

(theory) projection - In domain theory, a function, f, which is (a) idempotent, i.e.
of [W.sup.u]([S.sub.+]([Y.sub.0])) on x O y coincides with [W.sup.u]([S.sub.+]), at least one of the derivatives dx([??])/dw, dy([??])/dw should be nonzero. Let it be dx([??])/dw. Taking into account that [absolute value of dy([??])/dw / dx([??])/dw] = [absolute value ov 1 / [omega]'([y.sub.[THETA]])| > 1/[tau] [much greater than] 1, we find that

dy([??]) / dw - d[OMEGA](x([??])) / dx dx([??]) / dw = dx([??]) / dw [1 / [omega]'([y.sub.[THETA]]) - [OMEGA]'([x.sub.[THETA]])] [not equal to] 0

because [absolute value of [OMEGA]([x.sub.[THETA]])] [much less than] 1 at ([x.sub.[THETA]], [y.sub.[THETA]]). Therefore, ([x.sub.[THETA]], [z.sub.[THETA]], [y.sub.[THETA]], [t.sub.[THETA]]) is a point where [W.sup.u]([S.sub.+]([Y.sub.0])) intersects [W.sup.s.sub.loc]([S.sub.+]([Y.sub.0])) transversely.

Consider now [Y.sub.b]. Since D[Y.sub.b] = (1-2[eta])[b.sup.2] [not equal to] 0, [Y.sub.b] is a diffeomorphism of [R.sup.4] into itself.

[P.sub.+]([Y.sub.b]) [not equal to] (0, 0, [square root of (a(a - 2) + (1 - [b.sup.2]))] - (1 - b), [square root of (a(a - 2) + (1 - [b.sup.2]))] - (1 - b)) is a fixed point of [Y.sub.b] and [DELTA]([Y.sub.b]) [equivalent to] Cl ([W.sup.u]([P.sub.+]([Y.sub.b])) [intersection] {(x, z, y, t) : z = t = 0}). At [epsilon] = [[epsilon].sub.ab[eta]], where [[epsilon].sub.ab[eta]] is a solution of the equation [1 - (1 - 2[eta])b]/(1 - 2[[eta].sub.ab[eta]]) = [square root of (a(a - 2) + (1 - [b.sup.2]))] - (1-b), the point [P.sub.+]([Y.sub.b]) bifurcates in a direction transverse to the plane x = z = 0 and two periodical of period 2 points [S.sub.[+ or -]]([Y.sub.b]) [equivalent to] ([+ or -][x.sub.S], [+ or -][z.sub.S], [y.sub.S], [t.sub.S]) appear. Here

[x.sub.S] = -[z.sub.S] = [square root of (a[(a - 2) - [1 - (1 - 2[eta])b].sup.2][(1 - 2[epsilon]).sup.-2] - 2(1 - 2[eta])[1 - (1 - 2[eta])b]/(1 - 2[epsilon])

and [y.sub.S] = [t.sub.S] = [1 - (1 - 2[eta])b]/(1 - 2[epsilon]). The points [S.sub.[+ or -]]([Y.sub.b]) exist as long as the values of a, [epsilon], b and [eta] satisfy the inequalities This page lists Wikipedia articles about named mathematical inequalities. Pure mathematics
• Abel's inequality
• Barrow's inequality
• Berger's inequality for Einstein manifolds
• Bernoulli's inequality
• Bernstein's inequality (mathematical analysis)

-(1 - b) - [square root of a(a - 2) + (1 - [b.sup.2])] [less than or equal to] [1 - (1 - 2[eta])b]/(1 - 2[epsilon]) [less than or equal to] [square root of a(a - 2) + (1 - [b.sup.2]) - (1 - b).

Of course, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is clear that [Y.sub.b] converges to [Y.sub.0] uniformly as [absolute value of b] [right arrow] 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The latter implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [v.sub.j] (b) are the eigenvalues of D[Y.sup.2.sub.b] at [S.sub.[+ or -]]([Y.sub.b]). Therefore, for [absolute value of b] small enough, [Y.sub.b] has the 3-dimensional stable [W.sup.s]([S.sub.+](Y)) manifold and the 1-dimensional unstable [W.sup.u([S.sub.+]([Y.sub.b])) manifold. We show there exist points where these manifolds intersect each other transversely.

Let a, [epsilon] be such that Lemma 2.10 is valid for F. As is known [8], a stable/unstable manifold of diffeomorphism at its hyperbolic fixed point depends continuously (even [C.sup.k] smoothly, where k is the degree of diffeomorphism smoothness) on the diffeomorphism. Analyzing the proof of the given fact which is presented in [8], it is not difficult to Observe that this proof remains true for a wide map class. This is because what is really used there is a non-degeneracy of differential on its contracting/expanding tangent subspace rather than in the whole tangent space In mathematics, the tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. . Since [v.sup.4](0) is the only eigenvalue eigenvalue

In mathematical analysis, one of a set of discrete values of a parameter, k, in an equation of the form Lx = kx. Such characteristic equations are particularly useful in solving differential equations, integral equations, and systems of
of D[Y.sup.2.sub.0] whose absolute value exceeds 1, we see that D[Y.sup.2.sub.0] ([S.sub.+]([Y.sub.0])) is an operator which is non-degenerate on its expanding tangent subspace and, therefore, the aforementioned theorem is true for [W.sup.u] ([S.sub.+]([Y.sub.0])) too. The latter means that an arbitrary compact piece of [W.sup.u] ([S.sub.+]([Y.sub.0])) can be with any desirable accuracy approximated (as b [right arrow] 0) by compact pieces of [W.sup.u] ([S.sub.+]([Y.sub.b])). Unfortunately, a similar general result with regard to properties of stable manifolds at the hyperbolic fixed point of endomorphisms whose differential is degenerate degenerate /de·gen·er·ate/ (de-jen´er-at) to change from a higher to a lower form.
degenerate /de·gen·er·ate/ (de-jen´er-at) characterized by degeneration.
on its tangent contracting subspace is unknown to the author. Therefore, in what follows, the continuous dependence of the stable manifold of [W.sup.s] ([S.sub.+]([Y.sub.0])) on the map will be shown. In other words, we show that [W.sup.s.sub.loc] ([S.sub.+]([Y.sub.b])) tends uniformly to [W.sup.s.sub.loc] ([S.sub.+]([Y.sub.0])) as b [right arrow] 0.

Let us look for the locally stable manifold of [Y.sub.b] at [S.sub.+]([Y.sub.b]) as a hypersurface y = Pb(x, z, t). Due to symmetry symmetry, generally speaking, a balance or correspondence between various parts of an object; the term symmetry is used both in the arts and in the sciences.  of [W.sup.s] ([S.sub.+]([Y.sub.b])), [W.sup.s] ([S.sub.-]([Y.sub.b])) with respect to opposite values of x and z, it is clear that Pb(x, z, t) = Pb(-x,-z, t). Let (x, z, y, t) [member of][W.sup.s] ([S.sub.+]([Y.sub.b])). As [Y.sub.b] ([W.sup.s] ([S.sub.+]([Y.sub.b]))) = [W.sup.s] ([S.sub.-]([Y.sub.b])), we have ([??], [??], [??], [??]) [member of] [W.sup.s] ([S.sub.-]([Y.sub.b])), where [??] = (1 - 2[epsilon])x Pb (x,z, t) + (1 - 2c)bz, [??] = x, [??] = [a(a - 2)- [P.sup.2.sub.b](x, z, t) - [x.sup.2]]/2 + bt, [??] = Pb(x, z, t) and, in doing so, [??] = Pb([??], [??], [??]). Since [??] = Pb([??], [??], [??]) = Pb(-[??], -[??], [??]), what is said above can be expressed in view of an equation with respect to Pb:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Differentiation of the given equation with respect to x, z, t gives us three equations Involving [partial derivative]Pb/[partial derivative]x,[partial derivative]Pb/[partial derivative]z, [partial derivative]Pb/[partial derivative]t. Substituting the coordinates of [S.sub.+]([Y.sub.b]) to these equations, we obtain equations for

[partial derivative]Pb([x.sub.S], [z.sub.S], [t.sub.S])/[partial derivative]x, [partial drivative]Pb([x.sub.S], [z.sub.S], [t.sub.S])/[partial derivative]z, [partial derivative]Pb([x.sub.S], [z.sub.S], [t.sub.S])/[partial derivative]t.

Solving them and taking then limits, we find that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The latter means that, in a small neighbourhood of [S.sub.+]([Y.sub.0]), a piece of [W.sup.s] ([S.sub.+]([Y.sub.0])) adjoining to [S.sub.+]([Y.sub.0]) can be with any desirable accuracy approximated as b [right arrow] 0 by [W.sup.s.sub.loc] ([S.sub.+]([Y.sub.b])). In view of structural stability of transverse intersections, this implies that if there is a point of transverse intersection of [W.sup.s.sub.loc] ([S.sub.+]([Y.sub.0])) with Wu ([S.sub.+]([Y.sub.0])) {[S.sub.+]([Y.sub.0])}, then one can find [??] > 0 such that the manifolds [W.sup.s.sub.loc] ([S.sub.+]([Y.sub.b])) and [W.sup.u] ([S.sub.+]([Y.sub.b]))\{[S.sub.+]([Y.sub.b])} intersect each other transversely in a neighbourhood of the given point when b [member of] (-[??], [??]). The existence of an orbit of F which is transverse homoclinic to [S.sub.+] results in the existence of a similar orbit of [Y.sub.b] which is transverse homoclinic to [S.sub.+]([Y.sub.b]). The latter means that Theorem 1.3 is valid for sufficiently small sufficiently small - suitably small  [absolute value of b].

Received September September: see month.  29, 2006; Accepted April 14, 2007

References

[1] V. I.Arnold andYu. S. Ilyashenko. Ordinary differential equations ordinary differential equation

Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function).
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Reference work that contains information on all branches of knowledge or that treats a particular branch of knowledge comprehensively. It is self-contained and explains subjects in greater detail than a dictionary.
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a North American term commonly used to describe heifers close to term with their first calf.
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associated in some way with Russia.

Russian blue
a breed of cats with short, dense, silver-tipped blue-colored coat and vivid green eyes.
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[3] V. A. Dobrynskii. Critical sets and unimodal mappings of a square, Mat. Zametki, 58(5):669-680, 1995.

[4] V. A. Dobrynskii. Critical sets and properties of endomorphisms built by coupling of two identical quadratic quadratic, mathematical expression of the second degree in one or more unknowns (see polynomial). The general quadratic in one unknown has the form ax2+bx+c, where a, b, and c are constants and x is the variable.  mappings, J. Dynam Dy´nam

n. 1. A unit of measure for dynamical effect or work; a foot pound. See Foot pound.
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[5] Jack K. Hale and Xiao-Biao Lin Lin   , Maya Ying Born 1959.

American sculptor and architect whose public works include the Vietnam Veterans Memorial in Washington, D.C. (1982).

Noun 1.
. Symbolic dynamics Introduction
In mathematics, symbolic dynamics is the practice of modelling a dynamical system by a space consisting of infinite sequences of abstract symbols, each symbol corresponding to a state of the system, and a shift operator corresponding to the dynamics.
and nonlinear A system in which the output is not a uniform relationship to the input.

nonlinear - (Scientific computation) A property of a system whose output is not proportional to its input.
semiflows, Ann ANN, Scotch law. Half a year's stipend over and above what is owing for the incumbency due to a minister's relict, or child, or next of kin, after his decease. Wishaw. Also, an abbreviation of annus, year; also of annates. In the old law French writers, ann or rather an, signifies a year. . Mat. Pura PURA PACOM Utilization & Redistribution Agency
PURA Public Utility Regulatory Act
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[6] M. Jacobson Jacobson is a surname with several variants. Some people with this name include:
• Amy Jacobson Television reporter for WMAQ News in Chicago
• Bill Jacobson (born 1955), an American photographer
• Carl Robert Jakobson (1841-1882), Estonian writer and teacher
. Absolutely continuous invariant measures In mathematics, an invariant measure is a measure that is preserved by some function. Invariant measures are of great interest in the study of dynamical systems. The Krylov-Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and  for one-parameter families of one-dimensional one-di·men·sion·al
1. Having or existing in one dimension only.

2. Lacking depth; superficial.

one-dimensional

1. having one dimension

2.
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[7] TienYien Li and James James, person in the Bible
James, in the Gospel of St. Luke, kinsman of St. Jude. The original does not specify the relationship.
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[8] Zbigniew Nitecki. Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, The M.I.T. Press, Cambridge Cambridge, city, Canada
Cambridge (kām`brĭj), city (1991 pop. 92,772), S Ont., Canada, on the Grand River, NW of Hamilton. It was formed in 1973 with the amalgamation of Galt, Hespeler, and Preston, all founded in the early 19th cent.
, Mass.-London, 1971.

[9] Heinrich Heinrich is a male given name or surname of Germanic origin. Equivalents in other languages are Henry (English), Enrico (Italian), Henri (French), Enrique (Spanish), and Henrique (Portuguese).  Steinlein and Han Han, Chinese dynasty
Han (hän), dynasty of China that ruled from 202 B.C. to A.D. 220. Liu Pang, the first Han emperor, had been a farmer, minor village official, and guerrilla fighter under the Ch'in dynasty.
[S.sub.-]Otto Otto, Austrian archduke
Otto: see Hapsburg, Otto von.
Walther Walther is a German male name meaning "powerful warrior" or "ruler of the world", from Old High German „waltan“ (to rule) and „heri“ (Army), similar to Harald. . Hyperbolic sets In mathematics, a subset of a manifold is said to have hyperbolic structure when its tangent bundle may be split into two invariant subbundles, one of which is contracting, and the other expanding. , transversal homoclinic trajectories, and symbolic dynamics for C1-maps in Banach spaces (mathematics) Banach space - A complete normed vector space. Metric is induced by the norm: d(x,y) = ||x-y||. Completeness means that every Cauchy sequence converges to an element of the space. , J. Dynam. Differential Equations differential equation

Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions.
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(1) Recall that the orbit Orb([??]) is termed homoclinic to the fixed point [S.sub.+] if it is biasymptotic to the given point, i.e., when Orb([??]) [subet] [W.sup.s]([S.sub.+]) [intersection] [W.sup.u]([S.sub.+]). The homoclinic orbit Orb([??]) is called the transverse homoclinic one when, first, [S.sub.+] is the hyperbolic fixed point and, second, for any sufficiently large In mathematics, the phrase sufficiently large is used in contexts such as:
is true for sufficiently large
natural m and n such that [[bar.F].sup.-m]([??]) [member of] [W.sup.u.sub.loc]([S.sub.+]), [[bar.F.sup.n]([??]) [member of] [W.sup.s.sub.loc]([S.sub.+]), there is a disc of [W.sup.u.sub.loc]([S.sub.+]) possessing the following properties: [iota]) the disc contains the point [[bar.F.sup.-m]([??]); [iota][iota]) [[bar.F].sup.m+n] is a diffeomorphism of the disc into its [[bar.F.sup.m+n]-image; [??]) the aforementioned [[bar.F].sup.m+n]-image of the disc intersects [W.sup.s.sub.loc]([S.sub.+]) at [[bar.F].sup.n]([??]) transversely.

Vladimir Vladimir (vlədyē`mĭr), city (1989 pop. 350,000), capital of Vladimir region, W central European Russia, on the Klyazma River. A rail junction, it has industries producing machinery, chemicals, cotton textiles, and plastics.  A. Dobrynskii Institute for Metal Physics of N.A.S.U., 36, Academician Vernadsky
Blvd Blvd abbr (= boulevard) → Bd ., 03680 Kiev-142, Ukraine Ukraine (y`krān, ykrān`), Ukr. Ukraina, republic (2005 est. pop.  E-mail: dobry@imp.kiev Kiev (kē`ĕf), Ukrainian Kyyiv, Rus. Kiyev, city (1990 est. pop. 2,600,000) and municipality with the status of a region (oblast), capital of Ukraine and of Kiev region, a port on the Dnieper River. .ua
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