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Limit cycles for a class of quintic [Z.sub.6]-equivariant systems without infinite critical points.

1 Introduction and main results

Hilbert XVIth problem represents one of the open question in mathematics and it has produced an impressive amount of publications throughout the last century. The study of this problem in the context of equivariant dynamical systems is a relatively new branch of analysis and is based on the development within the last twenty years of the theory of Golubitsky, Stewart and Schaeffer in [9, 10]. Other authors [6] have specifically considered this theory when studying the limit cycles and related phenomena in systems with symmetry. Roughly speaking the presence of symmetry may complicate the bifurcation analysis because it often forces eigenvalues of high multiplicity. This is not the case of planar systems; on the contrary, it simplifies the analysis because of the reduction to isotypic components. More precisely it allows us to reduce the bifurcation analysis to a region of the complex plane.

In this paper we analyze the [Z.sub.6]-equivariant system

[??] = dz/dt = ([p.sub.1] + i[p.sub.2])[z.sup.2] [bar.z] + ([s.sub.1] + i[s.sub.2])[z.sup.3] [[bar.z].sup.3] - [[bar.z].sup.5] = f (z), (1)

where [p.sub.1], [p.sub.2], [s.sub.1], [s.sub.2] [member of] R.

The general form of the [Z.sub.q]-equivariant equation is

[??] = zA([[absolute value of z].sup.2]) + B[[bar.z].sup.q-1] + O([[absolute value of z].sup.q+1]),

where A is a polynomial of degree [(q - 1)/2]. The study of this class of equations is developed in several books, see [3, 6], when the resonances are strong, i.e. q < 4 or weak q > 4. The special case [theta] = 4 is also treated in several other articles, see [1, 6, 13]. In these mentioned works it is said that the weak resonances are easier to study than the other cases, as the equivariant term [[bar.z].sup.q-1] is not dominant with respect to the function on [[bar.z].sup.3]. This is true if the interest lies in obtaining a bifurcation diagram near the origin, but it is no longer true if the analysis is global and involves the study of limit cycles. This is the goal of the present work: studying the global phase portrait of system (1) paying special attention to the existence, location and uniqueness of limit cycles surrounding 1, 7 or 13 critical points. As far as we know this is the first work in which the existence of limit cycles is studied for this kind of systems.

The main result of our paper is the following.

Theorem 1. Consider equation (1) with [p.sub.2] [not equal to] 0, [absolute value of [s.sub.2]] > 1 and define the quantities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, the following statements are true:

(a) If one of the conditions

(i) [p.sub.1] [not member of] ([[SIGMA].sup.-.sub.A], [[SIGMA].sup.+.sub.A]), (ii) [p.sub.1] [not member of] [[SIGMA].sup.-.sub.A]/2,[[SIGMA].sup.+.sub.A]/2)

is satisfied, then equation (1) has at most one limit cycle surrounding the origin. Furthermore, when the limit cycle exists it is hyperbolic.

(b) There are equations (1) under condition (ii) having exactly one limit cycle surrounding either 1, 7 or 13 critical points, and equations (1) under condition (i) having exactly one hyperbolic limit cycle surrounding either 7 critical points if [p.sub.1] [not equal to] [[SIGMA].sup.[+ or -].sub.A], or the only critical point if [p.sub.1] = [[SIGMA].sup.[+ or -].sub.A] .

The conditions (i) and (ii) of Theorem 1 hold except when [p.sub.1] lies in ([[SIGMA].sup.-.sub.A], [[SIGMA].sup.+.sub.A]) [intersection] ([[SIGMA].sup.-.sub.A]/2, [[SIGMA].sup.+.sub.A]/2) The intersection of the two intervals is often empty, but this is not always the case as the example of Figure 1 shows. Examples of the relative positions of the two intervals are also given in Figure 3 below.

Our strategy for proving Theorem 1 will be to transform the system (1) into a scalar Abel equation and to study it. Conditions (i) and (ii) define regions where one of the functions in the Abel equation does not change sign. Since these functions correspond to derivatives of the Poincare returnmap, this imposes an upper bound on the number of limit cycles. When either [p.sub.1] lies in the intersection of the two intervals, or [absolute value of [s.sub.2]] [less than or equal to] 1, this analysis is not conclusive, and the study of the equations would require other methods. The qualitative meaning of conditions of statement (a) of the previous theorem is also briefly explained and illustrated in Remark 3 and in Figure 3 below.

The paper is organized as follows. In Section 2 we state some preliminary results while in Section 3 the study of the critical points is performed. Section 4 is entirely devoted to the proof of the main theorem of the paper.

2 Preliminary results

We start by obtaining the symmetries of (1). Following [9, 10], a system of differential equations dx/dt = f (x) is said to be [GAMMA]-equivariant if it commutes with the group action of [GAMMA], ie. f ([gamma]x) = [gamma]f (x), [for all][gamma] [member of] [GAMMA]. Here [GAMMA] = [Z.sub.6] with the standard action on C generated by [[gamma].sub.1] = exp(2[pi]i/6) acting by complex multiplication. Applying this concept to equation (1) we have the following result.

Proposition 1. Equation (1) is [Z.sub.6]-equivariant.

Proof. Let [[gamma].sub.k] = exp(2[pi]ik/6), k = 0, ..., 5. Then a simple calculation shows that f ([[gamma].sub.k]z) = [[gamma].sub.k] f (z). This is true because the monomials in z, [bar.z] appearing in the expression of f are the following: [[bar.z].sup.5], which is [[gamma].sub.k] -equivariant, and monomials of the form [z.sup.l+1] [[bar.z].sup.l], that are [Z.sub.n]-equivariant for all n.

Equation (1) represents a perturbation of a Hamiltonian one and in the following we identify conditions that some parameters have to fulfill in order to obtain this Hamiltonian. We have the following result.

Theorem 2. The Hamiltonian part of [Z.sub.6]-equivariant equation (1) is

[??] = i([p.sub.2] + [s.sub.2]z [bar.z])[z.sup.2] [bar.z] - [[bar.z].sup.5].

Proof. An equation [??] = F(z, [bar.z]) is Hamiltonian if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] For equation (1) we have

[partial derivative]F/[partial derivative]z = 2([p.sub.1] + i[p.sub.2])z [bar.z] + 3([s.sub.1] + i[s.sub.2])[z.sup.2] [[bar.z].sup.3]

[partial derivative] [bar.F]/ [partial derivative] [bar.z] = 2([p.sub.1] - i[p.sub.2])z [bar.z] + 3([s.sub.1] - i[s.sub.2])[z.sup.2] [[bar.z].sup.3]

and consequently it is Hamiltonian if and only if [p.sub.1] = [s.sub.1] = 0.

As we have briefly said in the introduction, we reduce the study of system (1) to the analysis of a scalar equation of Abel type. The first step consists in converting equation (1) from cartesian into polar coordinates.

Lemma 1. The study of periodic orbits of equation (1) that surround the origin, for [p.sub.2] [not equal to] 0, reduces to the study of non contractible solutions that satisfy x(0) = x(2[pi]) of the Abel equation

dx/d[theta] = A([theta])[x.sup.3] + B([theta])[x.sup.2] + C([theta]q)x (2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Proof. Using the change of variables

z = [square root of r](cos([theta]) + i sin([theta]))

and the time rescaling, dt/ds = r, it follows that the solutions of equation (1) are equivalent to those of the polar system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

From equation (4) we obtain

Dr/d[theta] = 2r[p.sub.1] + 2[r.sup.2] ([s.sub.1] - cos(6[theta]))/[p.sub.2] + r ([s.sub.2] + sin(6[theta])).

Then we apply the Cherkas transformation x = r/[p.sub.2] + r ([s.sub.2] + sin(6[theta])), see [5], to get the scalar equation (2). Obviously the limit cycles that surround the origin of equation (1) are transformed into non contractible periodic orbits of equation (2), as they cannot intersect the set {[??] = 0}. For more details see [7].

As we have already mentioned in the introductory section, our goal in this work is to apply the methodology developed in [1] to study conditions for existence, location and unicity of the limit cycles surrounding 1, 7 and 13 critical points.

A natural way for proving the existence of a limit cycle is to show that, in the Poincare compactification, infinity has no critical points and both infinity and the origin have the same stability. Therefore, we would like to find the sets of parameters for which these conditions are satisfied. In the following lemma we determine the stability of infinity.

Lemma 2. Consider equation (1) in the Poincare compactification of the plane. Then:

(i) There are no critical points at infinity if and only if [absolute value of [s.sub.2]] > 1;

(ii) When [s.sub.2] > 1, infinity is an attractor (resp. a repellor) when [s.sub.1] > 0 (resp. [s.sub.1] < 0) and the opposite when [s.sub.2] < -1.

Proof. The proof follows the same steps as the Lemma 2.2 in [1]. After the change of variable R = 1/r in system (4) and reparametrization dt/ds = R, we get the system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

giving the invariant set {R = 0}, that corresponds to the infinity of system (4). Consequently, it has no critical points at infinity if and only if [absolute value of [s.sub.2]] > 1. To compute the stability of infinity in this case, we follow [11] and study the stability of {R = 0} in the system above. This stability is given by the sign of

[[integral].sup.2[pi].sub.0] -2 ([s.sub.1] - cos(6[theta]))/[s.sub.2] + sin(6[theta]) d[theta] = -sgn([s.sub.2])4[pi][s.sub.1]/[square root of [s.sup.2.sub.2] - 1],

and the result follows.

3 Analysis of the critical points

In this section we are going to analyze which conditions must be satisfied to ensure that equation (4) has one, seven or thirteen critical points. Obviously, the origin of the system is always a critical point. We are going to prove that it is monodromic: there is no trajectory of the differential equations that approaches the critical point with a definite limit direction.

For this purpose, let us define the generalized Lyapunov constants. Consider the solution of the following scalar equation

dr/d[theta] = [[infinity].summation over (i=1)] [R.sub.i] ([theta]) [r.sup.i], (5)

where [R.sub.i]([theta]), i [greater than or equal to] 1 are T-periodic functions. To define the generalized Lyapunov constants, consider the solution of (5) that for [theta] = 0 passes through [rho]. It may be written as

r([theta], [rho]) = [[infinity].summation over (i=1)] [u.sub.i] ([theta])[[rho].sup.i]

with [u.sub.1](0) = 1, [u.sub.k](0) = 0, [for all] k [greater than or equal to] 2. Hence, the return map of this solution is given by the series

[PI]([rho]) = [[infinity].summation over (i=1)] [u.sub.i](T)[[rho].sup.i] .

For a given system, in order to determine the stability of a solution, the only significant term in the return map is the first nonvanishing term that makes it differ from the identity map. Moreover, this term will determine the stability of this solution. On the other hand, if we consider a family of systems depending on parameters, each of the [u.sub.i](T) depends on these parameters. We will call kth generalized Lyapunov constant [V.sub.k] = [u.sub.k](T) the value of this expression assuming [u.sub.1](T) = 1, [u.sub.2](T) = ...,= [u.sub.k-1](T) = 0.

Lemma 3. The origin of system (4) is monodromic, if [p.sub.2] [not equal to] 0. Moreover, its stability if given by the sign of [p.sub.1], if it is not zero, and by the sign of [s.sub.1] if [p.sub.1] = 0.

Proof. To prove that the origin ismonodromic we calculate the arriving directions of the flow to the origin, see [2, Chapter IX] for more details. Concretely if we write system (4) in cartesian coordinates we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Then any solutions arriving at the origin will be tangent to the directions [theta] that are the zeros of r[??] = R(x, y) = xQ(x, y)-yP(x, y). Since the lowest order term of R(x, y) is [p.sub.2][([x.sup.2] + [y.sup.2]).sup.2], which is always different from zero away from the origin, it follows that the origin is either a focus or a center.

To know the stability of the origin, we compute the two first Lyapunov constants. As it is well-known, the two first Lyapunov constants of an Abel equation are given by

[V.sub.1] = exp([[integral].sup.2[pi].sub.0] C([theta])d[theta]) - 1, [V.sub.2] = [[integral].sup.2[pi].sub.0] B([theta])d[theta].

Applying this to equation (2) with the expressions given in (3) we get the following result:

[V.sub.1] = exp(4[pi] [p.sub.1]/[p.sub.2]) - 1,

and if [V.sub.1] = 0, then [V.sub.2] = 4[pi][s.sub.1], and we get the result.

Remark 1. It is clear that ([V.sub.1],[V.sub.2]) = (0, 0) if and only if ([p.sub.1], [s.sub.1]) = (0, 0) i.e. equation (1) is hamiltonian (see Theorem 2). Consequently, as the origin remains being monodromic, in this case it is a center.

In the next result we study the equilibria of equation (4) with r [not equal to] 0. They will be the non-zero critical points of the system.

Lemma 4. Let -[pi]/6 < [theta] < [pi]/6. Then the equilibria of system (4) with r [not equal to] 0 are given by:

r = -[p.sub.2]/[s.sub.2] + sin (2[[theta].sub.[+ or -]]), [[theta].sub.[+ or -]] = 1/3 arctan([[DELTA].sub.[+ or -]]), (7)

where [[DELTA].sub.[+ or -]] = [p.sub.1] [+ or -] u /[p.sub.2] - [p.sub.1][s.sub.2] + [p.sub.2][s.sub.1] and u = [square root of [p.sup.2.sub.1] + [p.sup.2.sub.2] - [([p.sub.1][s.sub.2] - [p.sub.2][s.sub.1]).sup.2]].

Proof. Let -[pi]/6 < q < [pi]/6. To compute the critical points of system (4), we have to solve the following nonlinear system:

0 = 2r[p.sub.1] + 2[r.sup.2] ([s.sub.1] - cos(6[theta]))

0 = [p.sub.2] + r ([s.sub.2] + sin(6[theta])) . (8)

Let x = 6[theta] and [tau] = tan x/2, so [tau] = tan 3[theta]. Then, doing some simple computations one gets

sin x = 2t/1 + [t.sup.2] cos x = 1 - [t.sup.2]/1 + [t.sup.2] . (9)

Eliminating r from equations (8) and using the previous formulas we get

(-[p.sub.2] + [p.sub.1][s.sub.2] - [p.sub.2][s.sub.1])[t.sup.2] + 2[p.sub.1]t + [p.sub.2] + [p.sub.1][s.sub.2] - [p.sub.2][s.sub.1] = 0.

Solving the previous equation for t, yields the result.

Finally, consider the interval -[pi]/6 [less than or equal] [theta] < [pi]/6, let x = 6[theta] and [tau] = cot x/2, so [tau] = cot 3[theta]. The same expressions (9) for sin x and cos x hold if t is replaced by [tau] = 1/t, hence the rest of the proof is applicable.

We will prove now that simultaneous equilibria of the type (r, [theta]) = (r, 0), (r, [theta]) = ([??], [pi]/6) are not possible.

Lemma 5. If [absolute value of [s.sub.2]] > 1, then there are no parameter values for which there are simultaneous equilibria of system (4) for [theta] = 0 and [theta] = [pi]/6 different from the origin.

Proof. If we solve

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for [theta] = 0, we get [p.sub.1][s.sub.2] = [p.sub.2]([s.sub.1] - 1) with the restriction sign [p.sub.2] = -sign [s.sub.2], to have r well defined in the second equation. On the other hand, solving the same system for [theta] = [pi]/6, we get [p.sub.1][s.sub.2] = [p.sub.2]([s.sub.1] + 1) with the same restriction sign [p.sub.2] = -sign [s.sub.2]. This means [p.sub.2] = 0 that implies r = 0 and the result follows.

In the following we summarize the conditions that parameters have to fulfill in order that system (4) has exactly one, seven or thirteen critical points (see figure 2).

Lemma 6. Consider system (4) with [absolute value of [s.sub.2]] > 1. If [s.sub.2][p.sub.2] [greater than or equal to] 0, then the only equilibrium is the origin. If [s.sub.2][p.sub.2] < 0 then the number of equilibria of system (4) is determined by the quadratic form:

Q([p.sub.1], [p.sub.2]) = [p.sup.2.sub.1] + [p.sup.2.sub.2] - ([p.sub.1][s.sub.2] - [p.sub.2][s.sub.1])2 = (1 - [s.sup.2.sub.2])[p.sup.2.sub.1] + (1 - [s.sup.2.sub.1])[p.sup.2.sub.2] + 2[s.sub.1][s.sub.2][p.sub.1][p.sub.2] (10)

Concretely:

1. exactly one equilibrium (the origin) if Q([p.sub.1], [p.sub.2]) < 0;

2. exactly seven equilibria (the origin and one non-degenerate saddle-node per sextant) if Q([p.sub.1], [p.sub.2]) = 0;

3. exactly thirteen equilibria (the origin and two per sextant, a saddle and a node) if Q([p.sub.1], [p.sub.2]) > 0.

Proof. We will do the proof for the case [s.sub.2] > 1, the other case being analogous.

There will exist more critical points different from the origin if and only if the formulas (7) given in Lemma 4 are realizable with r > 0. This will not occur either when the expression for r is negative (that corresponds to the condition [p.sub.2] [greater than or equal to] 0) or when the discriminant in [[DELTA].sub.[+ or -]] is negative (that corresponds to Q([p.sub.1], [p.sub.2]) < 0).

In order to have exactly six more critical points, two conditions have to be satisfied: [p.sub.2] < 0 to ensure positive values of r, as [s.sub.2] > 1, and the quantities [[DELTA].sub.[+ or -]] have to coincide, i.e. the discriminant u in Lemma 4 has to be zero, that is the condition (2) given in the statement of the lemma. Hence, [r.sub.+] = [r.sub.-] and [[theta].sub.+] = [[theta].sub.-].

We prove now that these additional critical points are saddle-nodes. By symmetry, we only need to prove it for one of them. The Jacobian matrix of system (4) evaluated at ([r.sub.+], [[theta].sub.+]) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Evaluating the Jacobian matrix (11) at the concrete expression (7) of the critical point and taking into account the condition [p.sup.2.sub.1] + [p.sup.2.sub.2] = [([p.sub.1][s.sub.2] - [p.sub.2][s.sub.1]).sup.2], the eigenvalues of the matrix are

[[lambda].sub.1] = 0 [[lambda].sub.2] = 2[p.sub.1] - 6[p.sub.2]([p.sub.2] + [p.sub.2][s.sub.1] - [p.sub.1][s.sub.2])/([p.sub.2][s.sub.2] + [p.sub.1])(1 + [s.sub.1]) - [p.sub.1][s.sup.2.sub.2].

Therefore, ([r.sub.+], [[theta].sub.+]) has a zero eigenvalue. To show that these critical points are saddle-nodes we use the following reasoning. It is well-known, see for instance [2], that the sum of the indices of all critical points contained in the interior of a limit cycle of a planar system is +1. As under our hypothesis the infinity does not have critical points on it, it is a limit cycle of the system and it has seven singularities in its interior: the origin, that is a focus and hence has index +1, and 6 more critical points, all of the same type because of the symmetry. Consequently, the index of these critical points must be 0. As we have proved that they are semi-hyperbolic critical points then they must be saddle-nodes.

In order that equation (4) has exactly thirteen critical points it is enough that r > 0 (i.e. [p.sub.2] < 0) and the discriminant in [[DELTA].sub.[+ or -]] of Lemma 4 is positive, that is the condition (3) of the statement.

To get the stability of the twelve critical points we evaluate the Jacobian matrix (11) at these critical points, and taking into account the condition [p.sup.2.sub.1] + [p.sup.2.sub.2] > ([p.sub.1][s.sub.2] - [p.sub.2][s.sub.1])2, the eigenvalues of the critical points ([r.sub.+], [[theta].sub.+]) are [[lambda].sub.1,2] = [R.sub.+] [+ or -] [S.sub.+], while the eigenvalues of ([r.sub.-], [[theta].sub.-]) are [[alpha].sub.1,2] = [R.sub.-] [+ or -] [S.sub.-], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the following we will show that one of the critical points has index +1, while the other is a saddle.

Doing some computations one gets that the product of the eigenvalues of ([r.sub.+], [[theta].sub.+]) and ([r.sub.-], [[theta].sub.-]) are, respectively

[R.sup.2.sub.+] - [S.sup.2.sub.+] = -12([p.sup.2.sub.1] + [p.sup.2.sub.2])u/u - [p.sub.1][s.sub.1] - [p.sub.2][s.sub.2]

[R.sup.2.sub.-] - [S.sup.2.sub.-] = -12([p.sup.2.sub.1]+ [p.sup.2.sub.2])u/u + [p.sub.1][s.sub.1] + [p.sub.2][s.sub.2].

The numerator of both expressions is negative.

If [p.sub.1][s.sub.1] + [p.sub.2][s.sub.2] > 0 then [S.sup.2.sub.-] > 0 and [R.sup.2.sub.-] - [S.sup.2.sub.-] < 0. Consequently the critical point ([r.sub.-], [[theta].sub.-]) is a saddle. On the other hand, the denominator of [R.sup.2.sub.+] - [S.sup.2.sub.+] is negative. This is true as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that is always true. Hence, [R.sup.2.sub.+] - [S.sup.2.sub.+] > 0 and ([r.sub.+], [[theta].sub.+]) has index +1.

If [p.sub.1][s.sub.1] + [p.sub.2][s.sub.2] < 0, doing similar reasonings one gets that the critical point ([r.sub.+], [[theta].sub.+]) is a saddle while ([r.sub.-], [[theta].sub.-]) has index +1.

Note that Q is a quadratic form on [p.sub.1], [p.sub.2], and its determinant 1 - [s.sup.2.sub.1]- [s.sup.2.sub.2], is negative if [s.sub.2] > 1. Hence, for each choice of [s.sub.1], [s.sub.2] with [s.sub.2] > 1, the points where Q([p.sub.1], [p.sub.2]) is negative lie on two sectors, delimited by the two perpendicular lines where Q([p.sub.1], [p.sub.2]) = 0. Since Q(0, [p.sub.2]) = (1 - [s.sup.2.sub.2])[p.sup.2.sub.2] < 0 for [s.sub.2] > 1, then the sectors where there are two equilibria in each sextant do not include the [p.sub.2] axis, as in Figure 2.

Lemma 7. Consider [absolute value of [s.sub.2]] > 1 and define the following two numbers:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let A([theta]) be the function given in Lemma 1. Then the function A([theta]) changes sign if and only if [p.sub.1] [member of] ([[SIGMA].sup.-.sub.A], [[SIGMA].sup.+.sub.A]).

Proof. Writing x = sin(6[theta]), y = cos(6[theta]), the function A([theta]) in (3) becomes

A(x, y) = 2/[p.sub.2] ([p.sub.1] - [p.sub.2][s.sub.1][s.sub.2] + [p.sub.1][s.sup.2.sub.2] + (2[p.sub.1][s.sub.2] - [p.sub.2][s.sub.1])x + ([p.sub.2]x - [p.sub.1]y + [p.sub.2][s.sub.2])y).

Next we solve the set of equations

A(x, y) = 0, [x.sup.2] + [y.sup.2] = 1.

to get the solutions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Observe now the the two first pairs of solutions ([x.sub.1], [y.sub.1]), ([x.sup.2], [y.sup.2]) cannot be solutions of our equation A([theta]) = 0 since x = sin(6[theta]) = -[s.sub.2] < -1.

On the other hand, if we look for the intervals where the function A([theta]) does not change sign we have two possibilities: either [absolute value of [x.sub.[+ or -]]] > 1 (and consequently [absolute value of [y.sub.[+ or -]]] > 1) or the discriminant of [x.sub.[+ or -]], [DELTA] = [p.sup.2.sub.1] + [p.sup.2.sub.2] - ([p.sub.2][s.sub.1] - [p.sub.1][s.sub.2])2 is negative or zero. In the case [DELTA] < 0, the solutions will be complex non-real, and in the second case [DELTA] = 0, the function will have a zero but a double one, and it will not change its sign.

The first possibility turns out to be impossible in our region of parameters. The second possibility leads to the region [p.sub.1] [member of] R \ ([[SIGMA].sup.+.sub.A], [[SIGMA].sup.-.sub.A]).

Lemma 8. Consider [absolute value of [s.sub.2]] > 1 and define the following two numbers:

[[SIGMA].sup.[+ or -].sub.B] = [[SIGMA].sup.[+ or -].sub.A]/2.

Let B([theta]) be the function given in Lemma 1. Then the function B([theta]) changes sign if and only if [p.sub.1] [member of] ([[SIGMA].sup.-.sub.B], [[SIGMA.sup.+.sub.B]).

Proof. By direct computations with the same substitution as the one in the proof of the previous lemma, x = sin(6[theta]), y = cos(6[theta]), we get that the zeroes of the system

B(x, y) = 0, [x.sup.2] + [y.sup.2] = 1,

are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Applying arguments similar to those in the previous proof, we get that the function B([theta]) will not change sign if and only if [p.sub.1] 6 [member of] ([[SIGMA].sup.+.sub.B], [[SIGMA].sup.-.sub.B]).

Remark 2. * In general systems there are examples for which the function A([theta]) changes sign but B([theta]) does not and vice-versa.

* The condition given in Lemma 7 under which the function A([theta]) does not change sign is closely related to the number of critical points of system (4). More precisely, the condition [DELTA] < 0 in the proof of Lemma 7 is the same as the condition for existence of a unique critical point in system (4), while the condition [DELTA] = 0, together with [p.sub.2] < 0, is equivalent to the existence of exactly 7 critical points.

In the following we will need some results on Abel equations proved in [12] and [8]. We summarize them in a theorem.

Theorem 3. Consider the Abel equation (2) and assume that either A([theta]) [not equivalent to] 0 or B([theta]) [not equivalent to] 0 does not change sign. Then it has at most three solutions satisfying x(0) = x(2[pi]), taking into account their multiplicities.

Remark 3. Condition (a) of Theorem 1 implies that one of the functions A([theta]), B([theta]) of the Abel equation does not change sign. If condition (i) is satisfied then A([theta]) does not change sign and, as the third derivative of the Poincare return map of the Abel equation is this function A([theta]) then, the Abel equation can have at most 3 limit cycles. In our case, one of them is the origin and another one is infinity. Consequently only one non-trivial limit cycle can exist, both for the scalar equation and for the planar system.

If condition (ii) is verified then it is the function B([theta]) that does not change sign and, by a change of coordinates, one can get another scalar equation for which the third derivative of the Poincare map is the function B([theta]), getting the same conclusions as before.

4 Limit cycles

Proof of Theorem 1. We first define the function c([theta]) = [s.sub.2] + sin(6[theta]) and the set [THETA] := {(r, [theta]) :[??] = [p.sub.2] + ([s.sub.2] + sin(6[theta]))r = 0}. Since [absolute value of [s.sub.2]] > 1, we have c([theta]) [not equal to] 0, [for all]q [member of] [0, 2p].

(a) Let's assume first that condition (i) is satisfied. By Lemma 1, we reduce the study of the periodic orbits of equation (1) to the analysis of the non contractible periodic orbits of the Abel equation (2). Since [p.sub.1] [not member of] ([[SIGMA].sup.-.sub.A], [[SIGMA].sup.+.sub.A]), by condition (i) of Lemma 7, we know that function A([theta]) in the Abel equation does not change sign. Hence, from Theorem 3, the maximum number of solutions satisfying x(0) = x(2[pi]) in system (2) taking into account their multiplicities, is three. One of them is trivially x = 0. Since c([theta]) [not equal to] 0, by simple calculations we can prove that the curve x = 1/c([theta]) is a second solution satisfying this condition. As shown in [1], undoing the Cherkas transformation we get that x = 1/c([theta]) is mapped into infinity of the differential equation. Then, by Lemma 1, the maximum number of limit cycles of equation (1) is one. Moreover, from the same lemma it follows that the limit cycle is hyperbolic. From the symmetry, it follows that a unique limit cycle must surround the origin.

(b) We follow the same analysis method as in [1]. When [p.sub.1] > [[SIGMA].sup.+.sub.A], by Lemma 3, both the origin and infinity in the Poincare compactification are repelors. In particular the origin is an unstable focus. On the other hand, from Lemma 7, A([theta]) does not vanish when [p.sub.1] > [[SIGMA].sup.+.sub.A], and the origin is the unique critical point. It is easy to see that, since [p.sub.2]A([theta]) > 0, then the exterior of the closed curve [THETA] is positively invariant and therefore, by applying the Poincare-Bendixson Theorem and part (a) of this theorem, there is exactly one hyperbolic limit cycle surrounding the curve [THETA]. Moreover, this limit cycle is stable.

When [p.sub.1] = [[SIGMA].sup.+.sub.A], six more semi-elementary critical points appear and they are located on [THETA]. They are saddle-nodes as shown in Lemma 6. We will show that at this value of [p.sub.1] the periodic orbit still exists and it surrounds the seven critical points. We will prove this by constructing a polygonal line with no contact with the flow of the differential equation. On the polygonal, the vector field points outside, and consequently, as the infinity is a repelor, the w-limit set of the unstable separatrices of the saddle-nodes must be a limit cycle surrounding [THETA], see Figure 4 and Example 1 below.

The [Z.sub.6]-equivariance of system (1) allows us to study the flow in only one sextant of the phase space in cartesian coordinates, the behaviour in the rest of the phase space being identical. The polygonal line will join the origin to one of the saddle-nodes as in Figure 4.

We explain the construction of the polygonal line when 0 < [s.sub.1] [less than or equal to] 1, [s.sub.2] > 1 and [p.sub.2] < 0. In this case, [p.sub.1] = [S.sub.+] A implies that [p.sub.1] > 0 and, in the notation of Lemma 4, we have u = 0. Hence

1/[[DELTA].sub.[+ or -]] = ([s.sup.2.sub.2]- 1)(1 + [s.sub.1])/[s.sub.1][s.sub.2] -[square root of [s.sup.2.sub.1] + [s.sup.2.sub.2]- 1] - [s.sub.2] < -1

and using the expression in Lemma 4, we find that the angular coordinate [[theta].sub.0] of the saddle-node satisfies -1 < tan 3[[theta].sub.0] < 0, and therefore [pi]/4 < [[theta].sub.0] < [pi]/3. The first segment in the polygonal is the line [theta] = [pi]/4. Using the expression (4) we obtain that on this line[??] = [p.sub.2] + r([s.sub.2] - 1) which is negative for r between 0 and [r.sub.1] = -[p.sub.2]/([s.sub.2] - 1) > 0. Thus, the vector field is transverse to this segment and points away from the saddle-node on it.

Another segment in the polygonal is obtained using the eigenvector [upsilon] corresponding to the non-zero eigenvalue of the saddle-node [z.sub.0]. The line [z.sub.0] + t[upsilon] is the tangent to the separatrix of the hyperbolic region of the saddle-node. Let [t.sub.0] be the first positive value of t for which the vector field fails to be transverse to this line.

If the lines [z.sub.0] + t[upsilon] and [theta] = [pi]/4 cross at a point with 0 < t < [t.sub.0] and with 0 < r < [r.sub.1], then the polygonal consists of the two segments. If this is not the case, then the segment joining [z.sub.0] + [t.sub.0][upsilon] to [r.sub.1] ([square root of 2]/2,[square root of 2]/2) will also be transverse to the vector field, and the three segments will form the desired polygonal.

By using the same arguments presented above and the fact that infinity is a repelor, it follows by the Poincare-Bendixson Theorem that the only possible [omega]-limit for the unstable separatrix of the saddle-node is a periodic orbit which has to surround the six saddle-nodes, see again Figure 4.

If [p.sub.1] = [[SIGMA].sup.+.sub.A] then the function A([theta]) does not change sign; therefore the arguments presented in part (a) of the proof of this theorem assure the hyperbolicity of the limit cycle.

When we move [p.sub.1] towards zero but still very close to [[SIGMA].sup.+.sub.A], then B([theta]) is strictly positive because [[SIGMA].sup.+.sub.A] > [[SIGMA.sup.+.sub.B] and there are 13 critical points as shown in Lemma 6: the origin (which is a focus), six saddles and six critical points of index +1 on [THETA]. Applying one more time part (a) of the proof of this theorem, we know that the maximum number of limit cycles surrounding the origin is one. If [p.sub.1] = [[SIGMA].sup.+.sub.A] the limit cycle is hyperbolic and it still exists for the mentioned value of [p.sub.1]. Then we have the vector field with B([theta]) not changing sign, 12 non-zero critical points and a limit cycle which surrounds them together with the origin.

Example 1. As an example of the construction of the polygonal line in the proof of Theorem 1 we present a particular case, done with Maple. Let's fix parameters [p.sub.2] = -1, [s.sub.1] = -0.5, [s.sub.2] = 1.2.With these values of the parameters, we have [[SIGMA].sup.-.sub.A] = -0.52423, [[SIGMA].sup.+.sub.A] = 3.25151.

To construct the polygonal line we work in cartesian coordinates. A key point in the process of identifying the three segments of the polygonal line is knowing explicitly the eigenvector corresponding to the non-zero eigenvalue of the saddle-node (x0, y0) = (1.358, 1.5). This eigenvector is v = (-0.8594,-0.5114) so the slope of the tangent to the hyperbolic direction of the saddle-node is 0.5114/0.8594 and the straight line of this slope passing through the saddle-node is R [equivalent to] {y = 1.5 + 0.5114/0.8594 (x - 1.358)}. The tangent to the separatrix of the hyperbolic region of the saddle-node is locally transverse to this line. The scalar product of the vector field associated to equation (8) with the normal vector to R, (0.5114,-0.8594), is given by the equation

-2.39191647949065[x.sup.5] + 2.34410741916533[x.sup.4] + 4.86235167862649[x.sup.3] -2.71272659052423[x.sup.2] - 2.33924612305747x- 0.92289951077311

when evaluated on the straight line R; its unique real root is x [equivalent] -1.1737 and the flow is transversal from inside out through R for any x > -1.1737. Let us now define the polygonal line as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let's now consider the point ([x.sub.1], [y.sub.1]) = (1.4250, 1.5399), which stands at the intersection between the second and third segments of the polygonal line. The scalar product between the normal to each segment and the flow of the differential equation is negative, when evaluated on the corresponding segments. To show the calculation, we will exemplify it for the first segment, the remaining cases being treated similarly.

We elected the segment of the line L [equivalent to] {y = x} and when substituting it into the system in cartesian coordinates (6) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A normal vector to the line L is (-1, 1) and the scalar product of ([??], [??]) with (-1, 1) yields f (x) = [x.sup.3](4[p.sub.2] + [x.sup.2](9[s.sub.2] - 8)). Solving this last equation leads to f (x) < 0 for -2[square root of -[p.sub.2]/9[s.sub.2]-8] [less than or equal to] x [less than or equal to] 2[square root of [p.sub.2]/9[s.sub.2]-8] . So we choose 0 [less than or equal to] x [less than or equal to] 2[square root of -[p.sub.2]/9[s.sub.2]-8], and we get that this scalar product is negative in the region where the polygonal line is defined as (t, t).

By using the same arguments presented above it follows that the only possible w-limit for the unstable separatrix of the saddle-node is a periodic orbit which has to surround the six saddle-nodes, see again Figure 4.

Acknowledgements An important part of the work was carried out during a FCT-supported visit of A.C. Murza to the Departament Ciencies Matematiques i Informatica de la Universitat de les Illes Balears, under the supervision of Professors Maria J. Alvarez and Rafel Prohens, to whom A.C.Murza is deeply thankful for their kind guidance.

References

[1] M.J. A LVAREZ, A. GASULL, R. PROHENS, Limit cycles for cubic systems with a symmetry of order 4 and without infinite critical points, Proc. Am. Math. Soc. 136, (2008), 1035-1043.

[2] A.A. ANDRONOV, E.A. LEONTOVICH, I.I. GORDON, A.G. MAIER, Qualitative theory of second-order dynamic systems, John Wiley and Sons, New-York (1973).

[3] V. ARNOLD, Chapitres supplementaires de la theorie des equations differentielles ordinaires, Editions MIR, Moscou, (1978).

[4] M. CARBONELL, J. LLIBRE Limit cycles of polynomial systems with homogeneous nonlinearities, J. Math. Anal. Appl. 142, (1989), 573-590.

[5] L.A. CHERKAS On the number of limit cycles of an autonomous second-order system, Diff. Eq. 5, (1976), 666-668.

[6] S.N CHOW, C. LI, D. WANG, Normal forms and bifurcation of planar vector fields, Cambridge Univ. Press, (1994).

[7] COLL, B.; GASULL, A.; PROHENS, R. Differential equations defined by the sum of two quasi-homogeneous vector fields. Can. J. Math., 49 (1997), pp. 212-231. [8] A. GASULL, J. LLIBRE Limit cycles for a class of Abel Equation, SIAM J. Math. Anal. 21, (1990), 1235-1244.

[9] M. GOLUBITSKY, D.G. SCHAEFFER, Singularities and groups in bifurcation theory I, Applied mathematical sciences 51, Springer-Verlag, (1985).

[10] M. GOLUBITSKY, I. STEWART, D.G. SCHAEFFER, Singularities and groups in bifurcation theory II, Applied mathematical sciences 69, Springer-Verlag, (1988).

[11] N.G. LLOYD A note on the number of limit cycles in certain two-dimensional systems, J. London Math. Soc. 20, (1979), 277-286.

[12] V.A. PLISS, Non local problems of the theory of oscillations, Academic Press, New York, (1966).

[13] ZEGELING, A. Equivariant unfoldings in the case of symmetry of order 4. Bulgaricae Mathematicae publicationes, 19 (1993), pp. 71-79.

Departament de Matematiques i Informatica, Universitat de les Illes Balears, 07122, Palma de Mallorca, Spain

email:chus.alvarez@uib.es

Centro de Matematica da Universidade do Porto. Rua do Campo Alegre 687, 4169-007 Porto, Portugal

emails:islabour@fc.up.pt, adrian.murza@fc.up.pt

M.J. Alvarez I.S. Labouriau A.C. Murza *

* M.J.A. was partially supported by grant MTM2008-03437. I.S.L. and A.C.M. were partially supported by the European Regional Development Fund through the programme COMPETE and through the Fundacao para a Ciencia e a Tecnologia (FCT) under the project PEstC/MAT/UI0144/2013. A.C.M. was also supported by the grant SFRH/ BD/ 64374/ 2009 of FCT

Received by the editors in October 2013.

Communicated by P. De Maesschalck.

2010 Mathematics Subject Classification : Primary: 34C07, 34C14; Secondary: 34C23, 37C27.
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