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Limit cycles of cubic polynomial vector fields via the averaging theory. (English) Zbl 1127.34025

This paper deals with the number of limit cycles that can bifurcate from the period annulus surrounding the origin of real, planar, cubic differential systems. The method used to do this study is the averaging method at first order in the perturbation parameter $\epsilon$. In particular the considered systems are of the form:

$\stackrel{˙}{x}=-yf\left(x,y\right)+\epsilon P\left(x,y\right),\phantom{\rule{1.em}{0ex}}\stackrel{˙}{y}=xf\left(x,y\right)+\epsilon Q\left(x,y\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $f\left(x,y\right)=0$ is a conic such that either $f\left(0,0\right)\phantom{\rule{0.166667em}{0ex}}\ne \phantom{\rule{0.166667em}{0ex}}0$ or the origin is an isolated singular point of $f\left(x,y\right)=0$ in the real plane, $\epsilon$ is the real perturbation parameter and $P\left(x,y\right)$, $Q\left(x,y\right)$ are real polynomials of degree at most three and such that $P\left(0,0\right)=Q\left(0,0\right)=0$. We remark that, under these assumptions, the origin of the unperturbed system, that is system (1) with $\epsilon =0$, is a center. The problem tackled in this paper is the number of limit cycles which bifurcate from the period annulus of the origin of the unperturbed system inside the class of cubic systems of the form (1).

The introduction of the paper contains a complete description of the state of the art of the posed and related problems with the corresponding references. As it is explained in this introduction, there are several ways to study the bifurcation of limit cycles from a period annulus which can be equivalent in certain situations. One of the first results of the paper establishes that, when $f\left(0,0\right)\ne 0$, the first order averaging method coincides with the Abelian integral method for studying the set forth problem. In fact, the proof of this result comes from the fact that the application of the first order averaging method to system (1) reduces to computing the number of simple zeros of the function ${f}^{0}\left(r\right)$ defined by:

${f}^{0}\left(r\right)=\frac{1}{2\pi }{\int }_{0}^{2\pi }\frac{cos\theta \phantom{\rule{0.166667em}{0ex}}P\left(rcos\theta ,rsin\theta \right)+sin\theta \phantom{\rule{0.166667em}{0ex}}Q\left(rcos\theta ,rsin\theta \right)}{f\left(rcos\theta ,rsin\theta \right)}\phantom{\rule{0.166667em}{0ex}}d\theta ·\phantom{\rule{2.em}{0ex}}\left(2\right)$

The same integral needs to be computed when applying the Abelian integral method, so that both methods are equivalent.

In order to study the considered problem, the conics $f\left(x,y\right)=0$ are classified according to its geometrical properties. When $f\left(x,y\right)$ is a constant, which can be scaled to 1, a corollary of a result given in [Nonlinearity 9, 501–516 (1996; Zbl 0886.58087)] gives that at most 1 limit cycle bifurcates from the considered period annulus in system (1) at first order in $\epsilon$. When $f=0$ is a straight line, which can be translated to $f\left(x,y\right)=1+x$ by an affine change of coordinates, a corollary of a result given in [Nonlinear Anal., Theory Methods Appl. 46, 45–51 (2001; Zbl 0992.34024)] gives that at most 3 limit cycles bifurcate from the considered period annulus in system (1) at first order in $\epsilon$. The following nine cases are the ones considered in this paper:

(E) Ellipse: $f={\left(x+a\right)}^{2}+{\left(y+b\right)}^{2}-1=0$ with ${a}^{2}+{b}^{2}\ne 1$.

(CE) Complex ellipse: $f={\left(x+a\right)}^{2}+{\left(y+b\right)}^{2}+1=0$.

(H) Hyperbola: $f={\left(x+a\right)}^{2}-{y}^{2}-1=0$ with ${a}^{2}\ne 1$.

(CL) Two complex straight lines intersecting in a real point $f={\left(x+a\right)}^{2}+{\left(y+b\right)}^{2}=0$.

(RL) Two real straight lines intersecting in a point $f=\left(x+a\right)\left(y+b\right)=0$ with $ab\ne 0$.

(P) Parabola $f=x-a-{y}^{2}=0$ with $a\ne 0$.

(RPL) Two real parallel straight lines $f={\left(x+a\right)}^{2}-1=0$ with ${a}^{2}\ne 1$.

(CPL) Two complex parallel straight lines $f={\left(x+a\right)}^{2}+1=0$.

(DL) One double invariant real straight line $f={\left(x+a\right)}^{2}=0$ with $a\ne 0$.

A summary of the main results established in the paper is the following. System (1) with the conic $f=0$

– in the cases (E), (CE) and (CL) has at most five limit cycles,

– in the case (RL) has at most seven limit cycles,

– in the cases (RPL) and (CPL) has at most eight limit cycles,

– in the case (DL) has at most four limit cycles, up to first order in $\epsilon$ bifurcating from the period annulus surrounding the origin of the unperturbed system.

Moreover, there exist values of the cubic polynomials $P$ and $Q$ in system (1) with the conic $f=0$

– in the cases (E), (CE) and (CL) with five hyperbolic limit cycles,

– in the cases (H) and (P) with at least five limit cycles,

– in the case (RL) with at least six hyperbolic limit cycles,

– in the cases (RPL) and (CPL) with at least five hyperbolic limit cycles,

– in the case (DL) with at least three hyperbolic limit cycles,

bifurcating from the period annulus surrounding the origin of the unperturbed system.

The proof of this result consists in computing the function ${f}^{0}\left(r\right)$, either by direct calculation or by the Residue Theorem, and then studying the maximum number of its isolated zeros and when these zeros are hyperbolic.

##### MSC:
 34C29 Averaging method 34C23 Bifurcation (ODE) 34C05 Location of integral curves, singular points, limit cycles (ODE)