On the limit cycles for a class of discontinuous piecewise cubic polynomial differential systems

In this paper, we bound the number of limit cycles for a class of cubic reversible isochronous system inside the class of all cubic polynomial differential systems. By applying the averaging method of second order to this system, it is proved that at most eight limit cycles can bifurcate from the period annulus. Moreover, this bound is sharp.

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A number of limit cycle of sextic polynomial differential systems via the averaging theory

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.41922 ◽

2021 ◽

Vol 39 (4) ◽

pp. 181-197

Author(s):

Amour Menaceur ◽

Salah Boulaaras

Keyword(s):

Limit Cycles ◽

Vector Fields ◽

Averaging Theory ◽

Polynomial Vector ◽

Polynomial Vector Fields ◽

Differential Systems ◽

Polynomial Differential Systems ◽

Period Annulus ◽

Cubic System ◽

Cubic Polynomial

The main purpose of this paper is to study the number of limit cycles of sextic polynomial differential systems (SPDS) via the averaging theory which is an extension to the study of cubic polynomial vector fields in (Nonlinear Analysis 66 (2007), 1707--1721), where we provide an accurate upper bound of the maximum number of limit cycles that SPDS can have bifurcating from the period annulus surrounding the origin of a class of cubic system.

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On the Limit Cycles for Continuous and Discontinuous Cubic Differential Systems

Discrete Dynamics in Nature and Society ◽

10.1155/2016/4939780 ◽

2016 ◽

Vol 2016 ◽

pp. 1-11

Author(s):

Ziguo Jiang

Keyword(s):

Periodic Orbits ◽

Limit Cycles ◽

Quadratic Polynomial ◽

Averaging Theory ◽

Isochronous Center ◽

Discontinuous Systems ◽

Differential Systems ◽

First Order ◽

Polynomial Differential Systems ◽

Cubic Polynomial

We study the number of limit cycles for the quadratic polynomial differential systemsx˙=-y+x2,y˙=x+xyhaving an isochronous center with continuous and discontinuous cubic polynomial perturbations. Using the averaging theory of first order, we obtain that 3 limit cycles bifurcate from the periodic orbits of the isochronous center with continuous perturbations and at least 7 limit cycles bifurcate from the periodic orbits of the isochronous center with discontinuous perturbations. Moreover, this work shows that the discontinuous systems have at least 4 more limit cycles surrounding the origin than the continuous ones.

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Fifteen Limit Cycles Bifurcating from a Perturbed Cubic Center

Discrete Dynamics in Nature and Society ◽

10.1155/2021/8178729 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Amor Menaceur ◽

Mufda Alrawashdeh ◽

Sahar Ahmed Idris ◽

Hala Abd-Elmageed

Keyword(s):

Limit Cycles ◽

Averaging Theory ◽

Bifurcation Of Limit Cycles ◽

Differential Systems ◽

First Order ◽

Polynomial Differential Systems ◽

Period Annulus ◽

Cubic Polynomial

In this work, we study the bifurcation of limit cycles from the period annulus surrounding the origin of a class of cubic polynomial differential systems; when they are perturbed inside the class of all polynomial differential systems of degree six, we obtain at most fifteenth limit cycles by using the averaging theory of first order.

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