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A universal constant for semistable limit cycles

Abstract

We consider one-parameter families of 2-dimensional vector fields Xµ having in a convenient region R a semistable limit cycle of multiplicity 2m when µ = 0, no limit cycles if µ < 0, and two limit cycles one stable and the other unstable if µ > 0. We show, analytically for some particular families and numerically for others, that associated to the semistable limit cycle and for positive integers n sufficiently large there is a power law in the parameter µ of the form µn ≈ Cnα< 0 with C, α ∈ R, such that the orbit of Xµn through a point of p ∈ R reaches the position of the semistable limit cycle of X0 after given n turns. The exponent α of this power law depends only on the multiplicity of the semistable limit cycle, and is independent of the initial point p ∈ R and of the family Xµ. In fact α = -2m/(2m - 1). Moreover the constant C is independent of the initial point p ∈ R, but it depends on the family Xµ and on the multiplicity 2m of the limit cycle Γ.

semistable limit cycle; semistable fixed point; universal constant; power law


A universal constant for semistable limit cycles

Joan C. ArtésI; Jaume LlibreI; Marco Antonio TeixeiraII

IDepartament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain E-mails: artes@mat.uab.cat / jllibre@mat.uab.cat

IIDepartamento de Matemática, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil E-mail: teixeira@ime.umcamp.br

ABSTRACT

We consider one-parameter families of 2-dimensional vector fields Xµ having in a convenient region R a semistable limit cycle of multiplicity 2m when µ = 0, no limit cycles if µ < 0, and two limit cycles one stable and the other unstable if µ > 0.

We show, analytically for some particular families and numerically for others, that associated to the semistable limit cycle and for positive integers n sufficiently large there is a power law in the parameter µ of the form µn ≈ Cnα< 0 with C, α ∈ R, such that the orbit of Xµn through a point of pR reaches the position of the semistable limit cycle of X0 after given n turns. The exponent α of this power law depends only on the multiplicity of the semistable limit cycle, and is independent of the initial point pR and of the family Xµ. In fact α = -2m/(2m - 1). Moreover the constant C is independent of the initial point pR, but it depends on the family Xµ and on the multiplicity 2m of the limit cycle Γ.

Mathematical subject classification: 58F14, 58F21, 58F30.

Key words: semistable limit cycle, semistable fixed point, universal constant, power law.

1 Introduction and statement of the main results

Let Xµ be an one-parameter family of 2-dimensional vector fields. An isolated periodic orbit of Xµ in the set of all periodic orbits of Xµ is called a limit cycle. For a definition of stable or unstable limit cycle, semistable limit cycle and multiplicity of a limit cycle, see for instance [3,6].

Suppose that X0 has a semistable limit cycle γ, that Xµ for µ 0 has no limit cycles in a given annular neighborhood A of γ, and that Xµ for µ 0 has two limit cycles near γ in A, one stable and the other unstable. So the family Xµ exhibits a bifurcation at µ = 0. We also assume that the flow of every Xµ with µ 0 enters into A through one of its boundaries B S1 = R/(2πR) and exits through the other. We denote by R the annular subregion of A limited by B and Γ.

We take the annular region R sufficiently narrow in such a way that for a point pR we can choose as coordinates (r, θ), where θ is the angular variable along the periodic orbit γ with θ ∈ S1, and r is the distance along the orthogonal segment to γ through p.

We now fix a point p = (r, θ) ∈ R \ γ. For µ 0 the orbit γµ of Xµ through p crosses γ in positive time. We define µn as the value of µ 0 for which γµ reaches the point q = (0, θ + 2πn) ∈γ, i.e. the orbit γµn starting at p reaches γ at the point q after doing exactly n turns. In this way we have a sequence of increasing values of the parameter µ:

tending to 0. We have the following conjecture.

Conjecture 1. Assume that we have an one-parameter family of 2-dimensional vector fields Xµ satisfying all the previous assumptions stated in this section. Then for positive integers n sufficiently large we have that

with CR. Moreover the constant C is independent of the initial point pR, but depends on the family Xµ and on the multiplicity 2m of the limit cycle γ.

The expression Cnα for µn is called a power law with constant C and exponent α. Note that in particular the conjecture says that the exponent α depends only on the multiplicity of the semistable limit cycle γ, and is independent of the initial point p ∈ R and of the family Xµ. In this sense the exponent αis a universal constant for the semistable limit cycles.

Conjecture 1 is supported by some analytical and numerical results. The analytical result is the following one.

Theorem 2.Let (r, θ) be the polar coordinates on the cylinder R X S1. Consider the one-parameter family of 2-dimensional vector fields

Such a family for µ = 0 has the semistable limit cycle r = 0 of multiplicity 2m. Then we have that µn Cnα where

Theorem 2 is proved in Section 2.

Note that in Theorem 2 the constant C depends on the vector fields Xµ through the constant k, and also depends on the multiplicity of the semistable limit cycle, but it is independent on the initial point pR used for computing the values µn. All this will be detailed in the proof of Theorem 2.

The numerical results giving support to Conjecture 1 are described in Section 3.

It easy to check that when µn follows a power law Cnα the value of the limit

is always equal to 1. This quotient is the one studied by Feigembaum [4, 5] when he found the universal constant for a cascade of period doubling bifurcations. But for power laws this limit does not provide any information.

In this paper we have restricted our attention to one-parameter families of 2-dimensional vector fields Xµ, but if we consider the return map f0 : II associated to the semistable limit cycle γ of X0, and extend it to Xµ for µ in a neighborhood of µ = 0, we get an one-parameter family fµ : II of interval maps. Here I is a small segment transversal to the flow of X0 which intersects the semistable limit cycle γ at a point q in the interior of I. We denote by qi with i = 1, 2 the endpoints of the segment I.

By construction the family of interval maps fµ has a semistable fixed point of multiplicity 2m at µ = 0, no fixed points if µ 0, and two fixed points one stable and the other unstable if µ 0. We can translate the results obtained for the studied families of 2-dimensional vector fields Xµ, to their corresponding families of interval maps fµ.

We denote by R the subsegment of I with endpoints either q1 and q, or q and q2, such that the ω-limit of all the points of R under f0 are equal to q. As for Xµ we get for the semistable fixed point q and for positive integers n sufficiently large a power law in the parameter µ of the form µn

Cnα < 0 with C, α ∈ R, such that the orbit of fµn through a point of pR reaches the position q of the semistable fixed point of f0 after n iterates. Translating Conjecture 1 from the family of vector fields Xµ to the family of interval maps fµ, we have the conjecture.

Conjecture 3. Assume that we have an one-parameter family of interval maps fµ satisfying the previous assumptions. Then for positive integers n sufficiently large we have that

with C ∈ R. oreover C is independent of the initial point pR, but depends on the family fµ and on the multiplicity 2m of the semistable fixed point q.

Some open questions are:

(1) How to prove Conjectures 1 or 3?

(2) Does there exist a similar result or conjecture on the stable manifold of semistable limit cycles for one-parameter families of vector fields in diension larger than 2?

(3) Does there exist a similar result or conjecture on the stable manifold of semistable fixed points for one-parameter families of functions in dimension larger than 1?

2 Proof of Theorem 2

The solutions of the vector field Xµ = =(kr2m - µ, 1) with k > 0 on the cylinder (r, θ) ∈ R x S1 can be studied solving the differential equation

Clearly for µ < 0 we have that dr/dθ > 0, therefore the differential equation (1) has no periodic orbits. For µ = 0 the unique periodic orbit is r (θ) = 0, that is a semistable limit cycle of multiplicity 2m. It attracts the orbits of the half-cylinder r < 0, and repels the orbits of the other half-cylinder r > 0. Finally when µ > 0 equation (1) has exactly two limit cycles, namely r±(θ) = ±(µ/k)1/(2m). The limit cycle r- (θ) is stable, and the limit cycle r+(θ) is unstable.

The differential equation (1) is of separable variables, so we can solve it and get that the solution r(θ) such that r(0) = r0 is

where F(a, b; c; x) is the hypergeometric function

Here we have used the notation

For more details on this hypergeometric function see [1].

In order to compute µn we must find the value of µ satisfying equation (2) when θ = 2πn and r(2πn) = 0 for a given value of r0 < 0, i.e. we must solve

with respect to µn. Developing equation (3) in Laurent series with respect to the small parameter µn, we obtain

Since µn→0 when n → ∞, for n sufficiently large we have that

From this expression the power law stated in Theorem 2 follows.

3 Numerical results for vector fields

In this section we shall compute the sequence (µn) numerically for several different kind of planar polynomial vector fields, and we shall see that all the sequences obtained provide support to Conjecture 1.

Once we have the numerical values of the sequence (µn) if for n sufficiently large this sequence follows a power law Cnα, we can compute αand C as follows

We divide the rest of this section in five subsections. In every one we study Conjecture 1 for a different family of polynomial vector fields Xµ.

3.1 A polynomial vector field of degree 5

In all this subsection X = (P, Q) will denote the following polynomial vector field of degree 5

Since in polar coordinates (r, θ), defined by (x, y) = (r cosθ, r sin θ), the vector field becomes

it follows that the phase portrait of X is formed by an unstable focus at the origin O and a semistable limit cycle γ at r = 1 of multiplicity 2, see Figure 1. If an orbit is contained in the annulus 0 < r < 1, then its α-limit is O and its β-limit is γ. If an orbit is contained in the annulus r > 1, then its α-limit is γ and its ω-limit is at infinity. Moreover, in the Poincaré compactification the circle of infinity is formed by a continuum of singular points. For more details on the Poincaré compactification see Chapter 5 of [3]. When we work with the Poincaré compactification of a polynomial vector field we shall use the notation introduced in that chapter. Thus the continuum of singular points at infinity follows from the fact that the polynomials F(z1) and G(z1) (defined in the mentioned Chapter 5) and whose zeros provide the singular points at infinity are identically zero.


See Chapter 7 of [3] for a summary on the dependence of the limit cycles with respect to the parameter of a rotated family of vector fields. The family of vector fields

forms a rotated parameter family with respect to the parameter µ ∈ R, because the singular points of Xµ remain the same for all µ, the determinant

if µ1 < µ2; and the equality cannot hold on an entire periodic orbit of Xµ with µ = µi, for i = 1, 2. To verify this, it is sufficient to see that the curve P(x,y) = 0 cannot contain an oval. Since in polar coordinates this curve becomes the origin union the curve tan θ = (r2 - 1 )2, it follows that it cannot contain an oval surrounding the origin, the unique singular point of the system, because tan θ cannot take negative values. We note that in the definition of our rotated family of vector fields the interval coincides with the real line.

Of course X0 = X. So, by the semistable property (see Chapter 7 of [3]), µ = 0 is a bifurcation value of the parameter µ because for µ 0 there are two limit cycles (one stable and the other unstable) since the multiplicity of the semistable limit cycle x2 + y2 = 1 of the vector field X is two, and for µ 0 there are no limit cycles.

Using the program P4 see the last chapter of [3] (which allows to draw the compactified phase portrait of a polynomial vector field in the Poincaré disc), we know that for µ < 0 small (from now on small means sufficiently small) the vector field Xu has no limit cycles in the whole plane R2.

We are only interested in studying the phase portrait of Xµ for µ < 0 and small. For µ < 0 and small every orbit starting at a point (x, y) with x2 + y2 > 0 goes to infinity in forward time giving finitely many turns around the origin, and goes to the origin in backwards time giving infinitely many turns around it. Now we will prove this claim.

First we study the infinity of Xµ with µ < 0 and small. Using the notation of Chapter 5 of [3] we have that

Therefore the unique infinite singular points of Xµ are the origins of the local charts U2 and V2; i.e. the endpoints of the y-axis. The eigenvalues at these singular points are -1 and 0. Therefore, applying Theorem 2.19 of [3], we know that these singular points only can be a node, a saddle or a saddle-node with topological indices 1, -1 and 0, respectively. Of course, since the local phase portrait at these two diametrally opposite singular points at infinity are symmetric with respect the origin of the Poincaré sphere S2 (due to the construction of the Poincaré compactification), they have the same topological index.

By the Poincaré-Hopf Theorem (see, for instance, Chapter 6 of [3]), if p(Xµ) S2 has finitely many singular points (as it is the case for µ < 0), then the sum of all their topological indices is 2. Now, the sum of the topo-logical indices of the finite singular point of Xµ is 1, because it is a focus for µ < 0 and small. Therefore the sum of the indices of the Poincaré compact-ification p(Xµ) in S2 \ S1 is 2. Consequently the sum of the indices of the two infinite singular points must be zero. Hence, the indices of those points are 0, and consequently they are saddle-nodes. Again, using Theorem 2.19 of [3], we get that their local phase portraits are those of Figure 2.


In short we know for all the singular points of Xµ, located at infinity or not, their local phase portraits, we also know that the vector fields Xµ form a rotated family with respect to µ, consequently the phase portrait of Xµ for µ < 0 and small has no limit cycles near x2 + y2 = 1 (the unique possible place for the limit cycles). Hence the phase portrait of Xµ for µ < 0 and small is the one given in Figure 2. Therefore the claim is proved.

From Figure 2 it is clear that Xµ for µ < 0 and small has only two separatrices and contained in R2, the stable ones of the saddle-nodes located at the origins of the local charts U2 and V2; i.e. at the endpoints of the y-axis. Here a separatrix is an orbit in the boundary of a hyperbolic sector, see for more details [3]. Of course, the ω-limit of is the origin of U2, while the ω-limit of is the origin of V2, and their α-limit is the origin O of R2.

Since Xµ for µ = 0 has a semistable limit cycle γ at x2 + y2 = 1, for µ < 0 and µ small it follows that the number of turns of or around O from the infinity to a point of the annulus R = {(x, y) : 1/2 < x2 + y2 < 1} increases when µ increases, and tends to ∞ when µ 0.

Now we fix a point p0 = (x0, y0) ∈ R and we shall use the separatrix Γµ = Γµ1 run in backward time for computing the sequences (µn). More precisely, we compute the intersection point rµ of separatrix Γµ with the circle x2 + y2 = 1, i.e. with Γ. After we look for the value µn such Γµ runs in backward time from rµ to the point p0 given n turns and less than n + 1. Due to the existence of the semistable limit cycle for µ = 0, there is a sequence of increasing values

tending to 0 for which Γµn passes through p0 exactly after doing n turns around O and less than n + 1. Of course, these µn does not coincide exactly with the ones defined in Section 1, but their differences tend to zero when n tends to infinity.

We estimate the values µn and (4) for different points p0, and there is numerical evidence that the limit αof the sequence αn is -2 = -2/1, independently of the point p0. More precisely, for the point p0 = (0,3/4) the values of µnn and Cn appear in Table 1, and for p0 = (0,1/2) in Table 2. We have also done the study starting from the point (0,7/8) but we do not add the table here. We will just summarize the results of all them in Table 5 at the end of the paper. Just to note that for this last case α404 = -2,000650974 and C404 = -0,031405.

Similar results to those of Tables 1 and 2 are obtained if we do the computations with the other separatrix or with any other initial point p0.

Looking to these results one could think that αmay be -2 for all the considered cases independently of the initial point, and also we can think that the constant C is independent of the initial point p0. When we compare this constant C with the ones obtained for different families of vector fields, we will see that it must depend on the family and/or the multiplicity. This dependence becomes clear in the analytic example of Section 2. But again the constant C looks independent of the initial point p0, this can be see clearly in Table 5 where we summarize the results of the first three subsections.

3.2 A polynomial vector field of degree 9

In this subsection X = (P, Q) will denote the polynomial vector field of degree 9 given by

In polar coordinates (r, θ) the vector field becomes

Therefore the phase portrait of Χ is formed by an unstable focus O at the origin and a semistable limit cycle Γ at r = 1 of multiplicity 4. This phase portrait is topologically equivalent to the one of Figure 1.

Repeating the arguments of Subsection 3.1, we obtain that the vector fields Xµ,(x, y) = (P(x, y), Q(x, y) + µP(x, y)) form a rotated family with respect to the parameter µ ∈ R. Again, X0 = Χ, and µ = 0 is a bifurcation value for the parameter µ in such a way that for µ < 0 and small the vector field Xµ has no limit cycles. For µ < 0 and small every orbit starting at a point (x, y) with x2 + y2 > 0 goes to infinity in forward time giving finitely many turns around the origin and goes to the origin in backward time giving infinitely many turns around it, see Figure 2. From this figure, it is clear that Xµ for µ < 0 and small has only two separatrices Γµ1 and Γµ2 contained in R2.

We fix a point p0 = (x0, y0) in R = {(x, y) : 1/2 < x2 + y2 < 1} and we compute the sequence (µΗ) as in the previous subsection.

Again for different points p0, there is numerical evidence that the limit of αas n ->∞ can be -4/3 = -1.33333... depending only on the multiplicity 4 of Γ. If we start from p = (0,3/4) (respectively p = (0,7/8)) we get that α404 = -1,353415... (respectively α404 = -1,345133...). Increasing the multiplicity of the semistable limit cycle increases the time of the computations for obtaining the same precision in the values of µn.

3.3 A polynomial vector field of degree 13

In this subsection X = (P, Q) will denote the polynomial vector field of degree 13 given by

In polar coordinates (r, θ) the vector field becomes

Therefore the phase portrait of X is formed by an unstable focus O at the origin and a semistable limit cycle Γ at r = 1 of multiplicity 6. This phase portrait is topologically equivalent to the one of Figure 1.

We construct the family of vector fields Xµ with X0 = X as in Subsection 3.2, and compute the sequence (µn) as in that subsection.

Again for different points p0, there is numerical evidence that the limit of µn is -6/5 = -1.2 as n -> ∞ depending only on the multiplicity 6 of the semistable limit cycle, because α404 = - 1, 234375.... If we start from p = (0, 3/4) (respectively p = (0, 7/8)) we get that α404 = -1 , 228354... (respectively α404 = -1 , 159198...). Now the differences between them become more evident, but we think that if we go to bigger values in n for αn, when the multiplicity increases these differences in αn changing the initial point p0 will disappear. In any case increasing n and the multiplicity the time for the computations increases strongly.

Gathering all the results from Subsections 3.1 to 3.3 we get Table 5.

3.4 A polynomial vector field of degree 7

In order to not limit our study to a single family of phase portraits with a multiple semistable limit cycles, we will study now a completely different system which has more singular points than the origin.

In this subsection X = (P, Q) will denote the polynomial vector field of degree 7 given by

It is easy to check that

is a first integral of X; i.e.

and consequently H is constant on the orbits of the vector field X. This first integral or the program P4 allows to show that the phase portrait of Χ in the Poincaré disc is the one of Figure 3. We note that X has exactly three singular points, an unstable focus at the origin (0, 0), a stable focus at (0, 2), and a saddle at


Since f = x2 + y2 - 1 satisfies the equality

it follows that f = 0 is an algebraic solution of Χ. Due to the fact that on f = 0 there are no singular points, f = 0 is a periodic orbit. Using the first integral we can see that it is isolated in the set of all periodic orbits, so it is a limit cycle, that we denote by Γ. Again, using the first integral, we can check that Γ is a semistable limit cycle. Using results of [2] we can show that Γ has multiplicity 2.

Repeating the arguments of Subsection 3.1, we obtain that the vector fields Xµ,(x, y) = (P(x, y), Q(x, y) + µP(x, y)) form a rotated family with respect to the parameter µ ∈ R. Again, Χ0 = Χ, and µ = 0 is a bifurcation value for the parameter µ in such a way that for bµ < 0 and small the vector field Xµ has no limit cycles. By the non-intersection property (see Chapter 7 of [3]), there are no periodic orbits of Xµ in the region occupied by the period annulus around the infinity of system X0.

In fact using the program P4 we see that the phase portrait of Xµ for µ < 0 and small is the one given in Figure 4. For these vector fields the infinity is a periodic orbit, this follows using Chapter 5 of [3] and checking that there are no infinite singular points.


Looking at Figure 4, we denote by Γ0 the stable separatrix of the saddle S whose α-limit for X0 is the semistable limit cycle f = 0. Then, for µ < 0 and small, we denote by Γµ the stable separatrix of the saddle S whoseα-limit for Xµ is the origin and such that Γµ tends to Γ0 when µ 0, see Figure 4.

We fix a point p0 = (x0, y0) in R = {(x, y) : 1/2 < x2 + y2 < 1}, and use the separatrix Γµ as we have used the separatrix Γµ1 in Subsection 3.1 for computing the sequence (µn).

For different points p0 there is numerical evidence that the limit of µn is -2 as n → ∞ independently of the point p0, because α 404 = -2,000427.... If we start from p = (0,3/4) (respectively p = (0,7/8)) we get that α404 = - 1,999850... (respectively α404 = -1,999054...). Again these numbers indicate that α seems to be independent of the differential system we take, and only depends on the multiplicity of the semistable limit cycle.

3.5 A polynomial vector field of degree 11

Finally we take a new example similar to the previous one, but this time having a semistable limit cycle of multiplicity 4.

In this section X = (P, Q) will denote the polynomial vector field of degree 11 given by

It is easy to check that

is a first integral of X; i.e.

and consequently H is constant on the orbits of the vector field X.

This system is already too complicated for P4 to study it but we know for sure (from the first integral) that it has a limit cycle of multiplicity 4 at the unity circle and thus we can do the same study as we have done up to now.

Since f = x2 + y2 - 1 satisfies the equality

it follows that f = 0 is an algebraic solution of Χ. Due to the fact that on f = 0 there are no singular points, f = 0 is a periodic orbit. Using the first integral we can see that it is isolated in the set of all periodic orbits, so it is a limit cycle, that we denote by Γ. Again, using the first integral, we can check that Γ is a semistable limit cycle. Using results of [2] we can show that Γ has multiplicity 4.

Repeating the arguments of Subsection 3.1, we obtain that the vector fields Xµ,(x, y) = (P(x, y), Q(x, y) + µP(x, y)) form a rotated family with respect to the parameter µ ∈ R. Again, X0 = X, and µ = 0 is a bifurcation value for the parameter µ in such a way that for µ < 0 and small the vector field Xµ has no limit cycles. By the non-intersection property, there are no periodic orbits of Xµ in the region occupied by the period annulus around the infinity of system X0.

We fix a point p0 = (x0, y0) in R = {(x, y) : 1/2 < x2 + y2 < 1}, and uses as in the previous subsection the separatrix Γµ in backward time for computing the sequence (µn).

Again for different points p0, there is numerical evidence that the limit of αn is -4/3 as n → ∞ independently of the point p0, because α404 = - 1,347251.... If we start from p = (0,3/4) (respectively p = (0,7/8)) we get that α404 = -1,344790... (respectively α404 = -1,327778...).

Received: 06/VII/10.

Accepted: 18/XI/10.

#CAM-233/10.

* The first two authors are partially supported by a MEC/FEDER grant BFM2008-03437, and a CIRIT grant number 2009SGR 410. The third author is partially supported by a grant FAPESP-2007/06896-5. All authors are also supported by the joint project CAPES-MECD grant HBP-2009-0025-PC.

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  • [3] F. Dumortier, J. Llibre and J.C. Artés, Qualitative theory of planar differential systems. UniversiText, Springer-Verlag, New York (2006).
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Publication Dates

  • Publication in this collection
    27 July 2011
  • Date of issue
    2011

History

  • Received
    06 July 2010
  • Accepted
    18 Nov 2010
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