Limit Cycles on Piecewise Smooth Vector Fields with Coupled Rigid Centers

We provide an upper bound for the maximum number of limit cycles of the class of discontinuous piecewise differential systems formed by two differential systems separated by a straight line presenting rigid centers. These two rigid centers are polynomial differential systems with a linear part and a nonlinear homogeneous part. We study the maximum number of limit cycles that such a class of piecewise differential systems can exhibit.

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On the number of limit cycles of a class of polynomial differential systems

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2011.0741 ◽

2012 ◽

Vol 468 (2144) ◽

pp. 2347-2360 ◽

Cited By ~ 7

Author(s):

Jaume Llibre ◽

Clàudia Valls

Keyword(s):

Periodic Orbits ◽

Limit Cycles ◽

Upper Bound ◽

Second Order ◽

Averaging Theory ◽

Differential Systems ◽

Polynomial Differential Systems ◽

Generalized Polynomial

We study the number of limit cycles of polynomial differential systems of the form where g 1 , f 1 , g 2 and f 2 are polynomials of a given degree. Note that when g 1 ( x )= f 1 ( x )=0, we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that the above system can have bifurcating from the periodic orbits of the linear centre , using the averaging theory of first and second order.

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Birth of Isolated Nested Cylinders and Limit Cycles in 3D Piecewise Smooth Vector Fields with Symmetry

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420500984 ◽

2020 ◽

Vol 30 (07) ◽

pp. 2050098

Author(s):

Tiago Carvalho ◽

Bruno Rodrigues de Freitas

Keyword(s):

Vector Field ◽

Normal Form ◽

Periodic Orbits ◽

Limit Cycles ◽

Vector Fields ◽

Piecewise Smooth ◽

Initial Model ◽

Smooth Vector ◽

Piecewise Smooth Vector Fields ◽

Piecewise Smooth Vector Field

Our start point is a 3D piecewise smooth vector field defined in two zones and presenting a shared fold curve for the two smooth vector fields considered. Moreover, these smooth vector fields are symmetric relative to the fold curve, giving rise to a continuum of nested topological cylinders such that each orthogonal section of these cylinders is filled by centers. First, we prove that the normal form considered represents a whole class of piecewise smooth vector fields. After we perturb the initial model in order to obtain exactly [Formula: see text] invariant planes containing centers, a second perturbation of the initial model is also considered in order to obtain exactly [Formula: see text] isolated cylinders filled by periodic orbits. Finally, joining the two previous bifurcations we are able to exhibit a model, preserving the symmetry relative to the fold curve, and having exactly [Formula: see text] limit cycles.

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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 NUMBER OF LIMIT CYCLES FOR A GENERALIZATION OF LIÉNARD POLYNOMIAL DIFFERENTIAL SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741350048x ◽

2013 ◽

Vol 23 (03) ◽

pp. 1350048 ◽

Cited By ~ 3

Author(s):

JAUME LLIBRE ◽

CLAUDIA VALLS

Keyword(s):

Small Parameter ◽

Periodic Orbits ◽

Differential System ◽

Limit Cycles ◽

Upper Bound ◽

Averaging Theory ◽

Third Order ◽

Differential Systems ◽

Polynomial Differential Systems

We study the number of limit cycles of the polynomial differential systems of the form [Formula: see text] where g1(x) = εg11(x) + ε2g12(x) + ε3g13(x), g2(x) = εg21(x) + ε2g22(x) + ε3g23(x) and f(x) = εf1(x) + ε2 f2(x) + ε3 f3(x) where g1i, g2i, f2i have degree k, m and n respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when g1(x) = 0 we obtain the generalized Liénard polynomial differential systems. We provide an upper bound of the maximum number of limit cycles that the previous differential system can have bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = -x using the averaging theory of third order.

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On the Number of Limit Cycles for Discontinuous Generalized Liénard Polynomial Differential Systems

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741550131x ◽

2015 ◽

Vol 25 (10) ◽

pp. 1550131 ◽

Cited By ~ 2

Author(s):

Fangfang Jiang ◽

Junping Shi ◽

Jitao Sun

Keyword(s):

Differential System ◽

Limit Cycles ◽

Upper Bound ◽

Averaging Theory ◽

Differential Systems ◽

First Order ◽

Polynomial Differential Systems ◽

Polynomial Differential System

In this paper, we investigate the number of limit cycles for a class of discontinuous planar differential systems with multiple sectors separated by many rays originating from the origin. In each sector, it is a smooth generalized Liénard polynomial differential system x′ = -y + g1(x) + f1(x)y and y′ = x + g2(x) + f2(x)y, where fi(x) and gi(x) for i = 1, 2 are polynomials of variable x with any given degree. By the averaging theory of first-order for discontinuous differential systems, we provide the criteria on the maximum number of medium amplitude limit cycles for the discontinuous generalized Liénard polynomial differential systems. The upper bound for the number of medium amplitude limit cycles can be attained by specific examples.

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Creation of Limit Cycles in Piecewise Smooth Vector Fields Tangent to Nested Tori

Qualitative Theory of Dynamical Systems ◽

10.1007/s12346-021-00491-9 ◽

2021 ◽

Vol 20 (2) ◽

Author(s):

Tiago Carvalho ◽

Luiz Fernando Gonçalves

Keyword(s):

Limit Cycles ◽

Vector Fields ◽

Piecewise Smooth ◽

Smooth Vector ◽

Piecewise Smooth Vector Fields

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Cyclicity of a Family of Generic Reversible Quadratic Systems with One Center

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419500354 ◽

2019 ◽

Vol 29 (03) ◽

pp. 1950035 ◽

Cited By ~ 1

Author(s):

Jihua Wang ◽

Yanfei Dai

Keyword(s):

Limit Cycles ◽

Upper Bound ◽

Vector Fields ◽

Parameter Family ◽

Reversible Systems ◽

Abelian Integrals ◽

Quadratic Systems ◽

Quadratic Vector Fields ◽

Period Annulus ◽

Straight Line

This paper is concerned with the quadratic perturbations from one parameter family of generic reversible quadratic vector fields having a simple center and an invariant straight line. It is shown that the system can generate at least two limit cycles. As the parameter is rational, we propose a procedure for finding the upper bound to cyclicity of period annulus based on the Chebyshev criterion for Abelian integrals together with one rationalizing transformation. To illustrate our approach, we determine the cyclicity of three representative reversible systems. Our results may be viewed as a contribution to proving the conjecture on cyclicity proposed by Iliev [1998].

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