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 in a topologically transitive invariant chaotic set can generate global map phase portrait 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. 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 that are homoclinic to the periodic points. These orbits appear suddenly and consist of points of transversal intersections of stable manifolds and unstable ones built up at the periodic points. Therefore, at a moment immediately just after the bifurcation, a countable set of periodic points and, moreover, a whole large invariant topologically transitive set 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 subject classification: 37G35, 37D45, 37E99.

Keywords: Bifurcation, dynamical systems, fixed points, coupled maps.

1. Introduction and Main Results

Although there are very many results concerning dynamical system properties, one can determine very seldom a priori 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 of appearance of new homoclinic orbits at maps used to study a synchronization phenomenon. These are so-called 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 a new transversal homoclinic orbit appearance and, as a corollary, generate global changes of the phase map portrait. 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) can differ very strongly from those for isolated fixed points.

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

Here [[PHI].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. This map family is known as one of coupled logistic maps and is used very often as a sample for researching of coupled map properties. It is easy to show that there are parameter values such that the map [??] has a nontrivial topologically transitive attractor on the "diagonal"[??] = [??]. Indeed, denote by [??] [subset] (0,4) a set of values of a such that [[??].sub.a] : t [right arrow] [[??].sub.a](t) has the aforesaid attractor provided a [member of] [??]. As is well known [6], [??] [not equal to] = [??]. Moreover, mes([??]) > 0, where mes(.) is the Lebesgue measure. Therefore, if a [member of] [??] and the "diagonal" is an asymptotically stable subspace, then [??] has the aforementioned 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.

Denote O = (0, 0), [LAMBDA] = [[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 1.1. There exists A [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that, for any a [member of] (A, 4) 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) 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] [??] 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 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 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 diffeomorphism.

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 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 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 expand along the whole chaotic set at once as soon as these manifolds appear.

From now on we assume that a > 3. Designate a closure by Cl(.), an interior by Int(.), a neighbourhood by U(.), an a-neighbourhood by [U.sub.[epsilon]](.) and an origin of coordinates 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 of DF at [P.sub.+], [S.sub.[+ or -]] = ([+ or -] [square rot 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 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 of F, these manifolds are analytic [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 with respect to the y-axis. Let x = [omega](y) be a function whose graph 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 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 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. 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 curve. Here and further we call a curve convex (concave) when it is a graph of a convex (concave) function.

Denote [kappa] = [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 its derivative 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 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 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 [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" 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].sub.n=0] is a monotone 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 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](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 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 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 > 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 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 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. 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 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, y = [OMEGA](x) is an analytic function. 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 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 (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 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 of [sigma]] < [10.sup.-3] be small enough. Denote by [S.sub.[psi]] 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 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 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 of the coordinate axes 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 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 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 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 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. 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 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 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 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, 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. 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 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 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 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 towards the line y = a - 2. Obviously this can be done. Then the shrinkage coefficient 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 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 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:

[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 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 the validity of the last statements. To do this, it is necessary to localize [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 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 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 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 that is presented above in figures than to describe those steps by words. That is why we depict 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 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 [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 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 "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, 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 [[product].sub.0] is located inside the cone [??]([[product].sub.0]). Notice that [Dom.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. 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 at x = 0 and has a finite continuous first derivative 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 rate of convergence 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 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 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 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]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 (3.1) reduces then to

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

Since the projection 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 

-(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. Since [v.sup.4](0) is the only eigenvalue 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 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 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 [absolute value of b].

Received September 29, 2006; Accepted April 14, 2007

References

[1] V. I.Arnold andYu. S. Ilyashenko. Ordinary differential equations [current problems in mathematics. fundamental directions, vol. 1, 7-149, Akad. Nauk SSSR,Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985; MR0823489 (87e:34049)]. In Dynamical systems, I, volume 1 of Encyclopaedia Math. Sci., pages 1-148. Springer, Berlin, 1988. Translated from the Russian by E. R. Dawson and D. O'Shea.

[2] V. A. Dobrynskii. Critical sets and unimodal mappings of a square, Dokl. Akad. Nauk, 341(4):442-445, 1995.

[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 mappings, J. Dynam. Control Systems, 5(2):227-254, 1999.

[5] Jack K. Hale and Xiao-Biao Lin. Symbolic dynamics and nonlinear semiflows, Ann. Mat. Pura Appl. (4), 144:229-259, 1986.

[6] M. Jacobson. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Commun. Math. Physics, 81(1):39-88, 1981.

[7] TienYien Li and James A.Yorke. Period three implies chaos, Amer. Math. Monthly, 82(10):985-992, 1975.

[8] Zbigniew Nitecki. Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971.

[9] Heinrich Steinlein and Han[S.sub.-]Otto Walther. Hyperbolic sets, transversal homoclinic trajectories, and symbolic dynamics for C1-maps in Banach spaces, J. Dynam. Differential Equations, 2(3):325-365, 1990.

(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 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 A. Dobrynskii Institute for Metal Physics of N.A.S.U., 36, Academician Vernadsky Blvd., 03680 Kiev-142, Ukraine E-mail: [e-mail]