Lower bounds for the local cyclicity for families of centers

doi:10.1016/j.jde.2020.11.035

Journal of Differential Equations

Volume 275, 25 February 2021, Pages 309-331

https://doi.org/10.1016/j.jde.2020.11.035 Get rights and content

Abstract

In this paper, we are interested in how the local cyclicity of a family of centers depends on the parameters. This fact was pointed out in [21], to prove that there exists a family of cubic centers, labeled by $C D_{31}^{12}$ in [25], with more local cyclicity than expected. In this family, there is a special center such that at least twelve limit cycles of small amplitude bifurcate from the origin when we perturb it in the cubic polynomial general class. The original proof has some crucial missing points in the arguments that we correct here. We take advantage of a better understanding of the bifurcation phenomenon in nongeneric cases to show two new cubic systems exhibiting 11 limit cycles and another exhibiting 12. Finally, using the same techniques, we study the local cyclicity of holomorphic quartic centers, proving that 21 limit cycles of small amplitude bifurcate from the origin, when we perturb in the class of quartic polynomial vector fields.

Introduction

The study of limit cycles began at the end of the 19th century with Poincaré. Years later, in 1900, Hilbert presents a list of unsolved problems. From the original 23 problems of this list, the 16th is still open. The second part of this problem consists in determining a uniform bound of the maximal number of limit cycles (named $H (n)$ ), and their relative positions, of a planar polynomial system of degree n. However, there are also weak versions of 16th Hilbert's problem. Arnol'd in [1] proposed a version focused on studying the number of limit cycles bifurcating from the period annulus of Hamiltonian systems. In this paper, we are interested in providing the maximal number $M (n)$ of small amplitude limit cycles bifurcating from an elementary center or an elementary focus, in special for degrees 3 and 4. The main idea is to study the local cyclicity of families of centers depending on a finite number of parameters.

As it is well known, for $n = 2$ , Bautin proved in [2] that $M (2) = 3$ . The case $n = 3$ but without quadratic terms (homogeneous cubic perturbation) was studied in [3], [19] and solved in [23], then $M_{h} (3) = 5$ . In [24], [26] Żoła̧dek shown that $M (3) \geq 11$ . Christopher, in [5], gave a simple proof of Żoła̧dek's result perturbing another cubic center with a rational first integral, using only the linear parts of the Lyapunov constants. The interest of this result is that we can compute these linear parts [5], in a parallelized way [13], [17], near a center without having the complete expressions of the Lyapunov constants. Basically the used technique consists in to choose a point on the center variety and at this point consider the linear term of the Lyapunov constants, if the point is chosen on a component of the center variety of codimension k, then the first k linear terms of the Lyapunov constants are independent. This is a direct application of the Implicit Function Theorem to prove that $M (n) \geq k$ . Usually, we use this technique to provide lower bounds for the local cyclicity problem in the class of polynomial vector fields of degree n. In [10], [11], Giné presents a conjecture that the number $M (n)$ is bounded below by $n^{2} + 3 n - 7$ and studies the cyclicity of different families of centers presented in [9]. In [14], [13] new lower bounds for $M (n)$ and n small have been obtained. The new values are $M (4) \geq 20$ , $M (5) \geq 33$ , $M (6) \geq 44$ , $M (7) \geq 61$ , $M (8) \geq 76$ , and $M (9) \geq 88$ .

In [21], Yu and Tian point out that the 1-parameter family of centers labeled by $C D_{31}^{12}$ in [25] is quite special because it can exhibit one more limit cycle than expected in Giné's conjecture. This family has the next rational first integral $H (x, y) = \frac{{(x y^{2} + x + 1)}^{5}}{x^{3} {(x y^{5} + 5 x y^{3} / 2 + 5 y^{3} / 2 + 15 x y / 8 + 15 y / 4 + a)}^{2}}$ and it has, following Żoła̧dek computations, codimension 12. The original proof has some crucial missing points in the arguments that we correct here, proving effectively that there exist some special values of the parameter a in (1) such that 12 limit cycles of small amplitude bifurcate from the origin when we perturb in the class of complete cubic polynomial vector fields. This family was also studied by Christopher in [5] and it was the first clear proof about the existence of at least 11 limit cycles of small amplitude bifurcating from an equilibrium in polynomial vector fields of degree three.

The main result of this paper is the following.

Theorem 1.1

The number of limit cycles of small amplitude bifurcating from an equilibrium of monodromic type in the classes of polynomial vector fields of degrees 3 and 4 is $M (3) \geq 12$ and $M (4) \geq 21$ , respectively.

After the above result, clearly, the commented general lower bound for $M (n)$ should be updated to be $n^{2} + 3 n - 6$ . We remark that the total number of parameters for polynomial vector fields of degree n is $n^{2} + 3 n + 2$ . Then, the new conjecture removes 8 to this total number of parameters. Six corresponding to an affine change of variables that writes the linear part in its normal form, one corresponding to a rotation and another to a rescaling. The previous conjecture took into account that the number of limit cycles in a center component does not change. But this is only generically. In this work, we will provide examples where this property fails. Hence we establish the following conjecture.

Conjecture 1.2

The number of limit cycles of small amplitude bifurcating from an equilibrium of monodromic type in the class of polynomial vector fields of degree n is $M (n) = n^{2} + 3 n - 6$ .

The proof of the above theorem is based on an extension of Christopher results ([5]) for linear and higher-order studies when the considered center has parameters. This new result, Theorem 3.1, is proved in Section 3. For completeness we also include here the Christopher results, see Theorem 2.2, Theorem 2.4 in the next section, where we have also added a detailed description of how they should be used together. We remark that the parallelization algorithm introduced in [13], [17] is crucial to get the results because facilitates all the needed computations. In Section 4 we do the proof of the statement of Theorem 1.1 corresponding to degree 3 vector fields. Moreover, we study also the bifurcation diagrams of limit cycles of small amplitude bifurcating from three families of centers. The first is 1-parametric and it is the rational reversible center family labeled by $C R_{12}^{17}$ in [25]. The second is a 2-parameter holomorphic cubic center family. The third is labeled by $C D_{10}^{(11)}$ in [25], it has also 2 parameters and no maximal codimension 12 but, from it, also 12 limit cycles bifurcate from the center inside the cubic polynomial class. In Section 5 we study the bifurcation diagram for a 2-parameter center family of degree 4 that allows us to prove the statement of Theorem 1.1 corresponding to vector fields of degree 4. Finally, we also study partially the bifurcation diagram for a 4-parameter quartic holomorphic family of centers.

We have used a cluster of computers with 128 processors simultaneously with 725 GB of total ram memory. All the computations have been made with Maple [18].

Section snippets

Lyapunov constants and parallelization

Let us consider the system ${\begin{matrix} \dot{x} & = & - y + P_{n} (x, y), \\ \dot{y} & = & x + Q_{n} (x, y), \end{matrix}$ with $P_{n}$ and $Q_{n}$ polynomials of degree n in variables $x, y$ . Writing the system in complex coordinates, we have $R (z, \bar{z}) = i z + R_{n} (z, \bar{z}),$ where $R_{n} (z, \bar{z})$ are polynomials of degree n in variables $(z, \bar{z})$ . We seek for a first integral in the form $H (z, \bar{z})$ in a neighborhood of the origin such as $X (H) = \sum_{k = 0}^{\infty} v_{k} {(z \bar{z})}^{k + 1},$ where $X$ is the vector field associated to (2) and $v_{k}$ the k-Lyapunov constant. Clearly, the origin will be a center if, and only if $v_{n} (2 π) = 0$

Local cyclicity depending on parameters

This section is devoted to extend Theorem 2.2, Theorem 2.4 to families of centers that depend on some parameters. Let $(\dot{x}, \dot{y}) = (P_{c} (x, y, μ), Q_{c} (x, y, μ))$ be a family of polynomial centers of degree n depending on a parameter $μ \in R^{ℓ}$ , having a center equilibrium point at the origin. We consider the perturbed polynomial system ${\begin{matrix} \dot{x} & = & P_{c} (x, y, μ) + α y + P (x, y, λ), \\ \dot{y} & = & Q_{c} (x, y, μ) + α x + Q (x, y, λ), \end{matrix}$ with $P, Q$ polynomials of degree n having no constant nor linear terms. More concretely, $P (x, y, λ) = \sum_{k + l = 2}^{n} a_{k, l} x^{k} y^{l}, Q (x, y, λ) = \sum_{k + l = 2}^{n} b_{k, l}$

Bifurcation diagrams for local cyclicity in families of cubic centers

In this section we use Theorem 3.1 to study the bifurcation diagram for some families of cubic centers, lying in components of the center variety of codimension 11, 10, and 9. The first, in Proposition 4.1, is the family labeled $C D_{31}^{(12)}$ that has generically cyclicity 11 and was studied previously by Christopher in [5], for only one parameter value $a = 2$ in (1), and by Yu and Tian in [21]. This proposition proves partially the main Theorem 1.1. The family labeled as $C R_{17}^{(12)}$ in [25], which

Bifurcation diagrams for local cyclicity in families of quartic centers

This section is devoted to proving the second part of the statement of our main result, Theorem 1.1. It follows from the next proposition. We provide the bifurcation diagram of the local cyclicity of the cubic center given by Bondar and Sadovskiĭ in [4] adding a straight line of equilibria. This problem can be studied to get 19 limit cycles. Here show a curious fact, the cyclicity depends on the selected straight line. We work with two parameters $(a, b)$ , showing the existence of a curve with

Final comments

The computations in this work are quite high although basically we have worked only with developments of order 1 in the Lyapunov constants but center depending on parameters. This is because the existence of parameters in the unperturbed centers makes the things more complicated. Before the simplifications, the polynomials appearing as coefficients of the perturbation parameters are of very high degree and with rational coefficients with a high number of digits. In fact, this is why we have

Accurate interval analysis

Next two technical results will help us to find upper and lower bounds for a polynomial of n variables in a n-dimensional cube. The proofs of them can be found in [6].

Lemma 7.1

[6]

Consider $h > 0$ , $p > 0$ , q real numbers such that $p \in [\underline{p}, \overline{p}]$ , with $\underline{p} \overline{p} > 0$ , and $q \in [\underline{q}, \overline{q}]$ , with $\underline{q} \overline{q} > 0$ .

(i)
Then, $σ^{ℓ} (q, p) \leq q p \leq σ^{r} (q, p)$ , where $σ^{ℓ} (q, p) = {\begin{matrix} q \underline{p}, & if q > 0, \\ q \overline{p}, & if q < 0, \end{matrix} σ^{r} (q, p) = {\begin{matrix} q \overline{p}, & if q > 0, \\ q \underline{p}, & if q < 0 . \end{matrix}$
(ii)
If $u_{j} \in [- h, h]$ , for $j = 1, \dots, n$ and denoting $u^{i} = u_{1}^{i_{1}} \dots u_{n}^{i_{n}}$ , for the multiindex $i = (i_{1}, \dots, i_{n}) \neq 0$ , we have $X^{ℓ} (q, u^{i}) \leq q u^{i} \leq X^{r} (q, u^{i})$ , where $X^{ℓ} (q, u^{i}) = {\begin{matrix} 0, & if q > 0 and i_{k} \end{matrix}$

Acknowledgements

We would like to thank the referee for his/her careful reading to help us to improve the clarity in the exposition of the results.

This work has been realized thanks to the Catalan AGAUR 2017SGR1617 and 2017SGR1276 grants, the Spanish Ministerio de Ciencia, Innovación y Universidades - Agencia estatal de investigación MTM2017-84383-P (FEDER) and PID2019-104658GB-I00 (FEDER) grants, the European Community Marie Skłodowska-Curie H2020-MSCA-RISE-2017-777911 grant and the Brazilian CNPq 200484/2015-0

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Lower bounds for the local cyclicity for families of centers

Abstract

Introduction

Section snippets

Lyapunov constants and parallelization

Local cyclicity depending on parameters

Bifurcation diagrams for local cyclicity in families of cubic centers

Bifurcation diagrams for local cyclicity in families of quartic centers

Final comments

Accurate interval analysis

[6]

Acknowledgements

J. Differ. Equ.

Appl. Math. Comput.

Appl. Math. Comput.

J. Differ. Equ.

J. Differ. Equ.

Commun. Nonlinear Sci. Numer. Simul.

Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields

Funct. Anal. Appl.

On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type

Am. Math. Soc. Transl.

The number of limit cycles of certain polynomial differential equations

Proc. R. Soc. Edinb. A

On a theorem of Zoladek

Differ. Uravn.

Estimating limit cycle bifurcations from centers

Local and global phase portrait of equation z˙=f(z)

Discrete Contin. Dyn. Syst.

Über die abgrenzung der eigenwerte einer matrix

Izv. Akad. Nauk SSSR Otd. Fiz.-Mat.

Local and global phase portrait of equation $\dot{z} = f (z)$