Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones

Jackson Itikawa,Regilene Oliveira,Jaume Llibre,Ana Cristina Mereu

doi:10.3934/dcdsb.2017136

Abstract

We apply the averaging theory of first order for discontinuous differential systems to study the bifurcation of limit cycles from the periodic orbits of the uniform isochronous center of the differential systems \begin{document} $\dot{x}=-y+x^2, \;\dot{y}=x+xy$ \end{document} , and \begin{document} $\dot{x}=-y+x^2y, \;\dot{y}=x+xy^2$ \end{document} , when they are perturbed inside the class of all discontinuous quadratic and cubic polynomials differential systems with four zones separately by the axes of coordinates, respectively. Using averaging theory of first order the maximum number of limit cycles that we can obtain is twice the maximum number of limit cycles obtained in a previous work for discontinuous quadratic differential systems perturbing the same uniform isochronous quadratic center at origin perturbed with two zones separately by a straight line, and 5 more limit cycles than those achieved in a prior result for discontinuous cubic differential systems with the same uniform isochronous cubic center at the origin perturbed with two zones separately by a straight line. Comparing our results with those obtained perturbing the mentioned centers by the continuous quadratic and cubic differential systems we obtain 8 and 9 more limit cycles respectively.

Highlights

Introduction and Statement of the MainResultsSuppose that q ∈ R2 is a center of a polynomial differential system in R2
Applying the averaging theory of first order we investigate the number of limit cycles which can bifurcate from the periodic orbits of the uniform isochronous center of the following quadratic and cubic differential systems x = −y + x2, y = x + xy, (2)
We recall that from Proposition 3 and from the fact that (−y + xf (x, y))2 + (x + yf (x, y))2 = (x2 + y2)(1 + f 2(x, y)) > 0 if (x, y) = (0, 0). It follows that the uniform isochronous center at the origin of coordinates is the only finite singular point of systems (2) and (3) ( this result is valid for any system (1)) and we only need to analyze the bifurcation of limit cycles from the periodic orbits of such center in those differential systems

Summary

Introduction and Statement of the Main Results

Suppose that q ∈ R2 is a center of a polynomial differential system in R2. Without loss of generality we can assume that q is at the origin of coordinates. For |ε| = 0 sufficiently small there exist discontinuous piecewise quadratic polynomial differential systems (4) with j = 2 having at least 10 limit cycles bifurcating from the periodic orbits of the uniform isochronous center of system (2), using the averaging theory of first order. In the same work the authors proved that for system (3), using the averaging method of first order, the maximum number of limit cycles that can bifurcate from the periodic solutions surrounding the center is 7, and this number can be reached In both cases studied in [13], the considered discontinuous systems were formed by two cubic polynomial differential systems separated by the straight line y = 0.

Preliminary results

Proof of Theorem 1

Proof of Theorem 2

Conclusion and future works