Canard Limit Cycles for Piecewise Linear Liénard Systems with Three Zones

2020 ◽  
Vol 30 (15) ◽  
pp. 2050232
Author(s):  
Shimin Li ◽  
Jaume Llibre
Keyword(s):  
Singular Point ◽  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Piecewise Linear ◽  
Middle Zone ◽  
Differential Systems ◽  
Liénard Systems

This paper deals with planar piecewise linear slow–fast Liénard differential systems with three zones separated by two vertical lines. We show the existence and uniqueness of canard limit cycles for systems with a unique singular point located in the middle zone.

scholarly journals Existence and Uniqueness of Limit Cycles in a Class of Planar Differential Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
Khalil I. T. Al-Dosary
Keyword(s):  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Differential Systems ◽  
Uniqueness Of Limit Cycles

This paper is an extension to the recent results presented by M. Sabatini about the existence and uniqueness of limit cycles of a certain class of planar differential systems in order to include other new classes. A concrete example exhibiting the applicability of the result is introduced.

Bifurcation of Limit Cycles for Some Liénard Systems with a Nilpotent Singular Point

2015 ◽  
Vol 25 (05) ◽  
pp. 1550066 ◽  
Author(s):  
Junmin Yang ◽  
Xianbo Sun
Keyword(s):  
Singular Point ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Bifurcation Of Limit Cycles ◽  
Liénard Systems ◽  
Nilpotent Singular Point ◽  
Nilpotent Saddle

In this paper, we first present some general theorems on bifurcation of limit cycles in near-Hamiltonian systems with a nilpotent saddle or a nilpotent cusp. Then we apply the theorems to study the number of limit cycles for some polynomial Liénard systems with a nilpotent saddle or a nilpotent cusp, and obtain some new estimations on the number of limit cycles of these systems.

scholarly journals Limit Cycles Bifurcating from the Periodic Orbits of a Discontinuous Piecewise Linear Differentiable Center with Two Zones

2015 ◽  
Vol 25 (11) ◽  
pp. 1550144 ◽  
Author(s):  
Jaume Llibre ◽  
Douglas D. Novaes ◽  
Marco A. Teixeira
Keyword(s):  
Equilibrium Point ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential

We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0, we have a linear saddle with its equilibrium point living in x > 0, and in x < 0 we have a linear differential center. Let p be the equilibrium point of this linear center, when p lives in x < 0, we say that it is real, and when p lives in x > 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential system formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N ≥ 2 when p is a real or virtual center. Furthermore, when p is a real center we found an example satisfying N ≥ 3.

scholarly journals The non-existence, existence and uniqueness of limit cycles for quadratic polynomial differential systems

2017 ◽  
Vol 149 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Jaume Llibre ◽  
Xiang Zhang
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System ◽  
Uniqueness Of Limit Cycles

AbstractWe provide sufficient conditions for the non-existence, existence and uniqueness of limit cycles surrounding a focus of a quadratic polynomial differential system in the plane.

scholarly journals ON THE NUMBER OF LIMIT CYCLES FOR DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS IN ℝ2n WITH TWO ZONES

2013 ◽  
Vol 23 (02) ◽  
pp. 1350024 ◽  
Author(s):  
JAUME LLIBRE ◽  
FENG RONG
Keyword(s):  
Small Parameter ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Displacement Function ◽  
Averaging Theory ◽  
Differential Systems ◽  
First Order ◽  
Linear Differential Systems ◽  
Order Expansion ◽  
Linear Differential

We study the number of limit cycles of the discontinuous piecewise linear differential systems in ℝ2n with two zones separated by a hyperplane. Our main result shows that at most (8n - 6)n-1 limit cycles can bifurcate up to first-order expansion of the displacement function with respect to a small parameter. For proving this result, we use the averaging theory in a form where the differentiability of the system is not necessary.

scholarly journals LIMIT CYCLES OF DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS

2011 ◽  
Vol 21 (11) ◽  
pp. 3181-3194 ◽  
Author(s):  
PEDRO TONIOL CARDIN ◽  
TIAGO DE CARVALHO ◽  
JAUME LLIBRE
Keyword(s):  
Limit Cycles ◽  
Piecewise Linear ◽  
Displacement Function ◽  
Two Dimensional ◽  
Bifurcation Of Limit Cycles ◽  
Differential Systems ◽  
First Order ◽  
Linear Differential Systems ◽  
Order Expansion ◽  
Linear Differential

We study the bifurcation of limit cycles from the periodic orbits of a two-dimensional (resp. four-dimensional) linear center in ℝn perturbed inside a class of discontinuous piecewise linear differential systems. Our main result shows that at most 1 (resp. 3) limit cycle can bifurcate up to first-order expansion of the displacement function with respect to the small parameter. This upper bound is reached. For proving these results, we use the averaging theory in a form where the differentiability of the system is not needed.

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