Bounding the number of limit cycles for parametric Liénard systems using symbolic computation methods

2021 ◽  
Vol 96 ◽  
pp. 105716
Author(s):  
Yifan Hu ◽  
Wei Niu ◽  
Bo Huang
Keyword(s):  
Symbolic Computation ◽  
Limit Cycles ◽  
Liénard Systems

scholarly journals Hopf Bifurcation of Limit Cycles in Discontinuous Liénard Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Special Form ◽  
Bifurcation Of Limit Cycles ◽  
Liénard Systems

We consider a class of discontinuous Liénard systems and study the number of limit cycles bifurcated from the origin when parameters vary. We establish a method of studying cyclicity of the system at the origin. As an application, we discuss some discontinuous Liénard systems of special form and study the cyclicity near the origin.

scholarly journals Limit cycles in Liénard systems with saturation

2018 ◽  
Vol 51 (33) ◽  
pp. 127-131
Author(s):  
Thomas Lathuilière ◽  
Giorgio Valmorbida ◽  
Elena Panteley
Keyword(s):  
Limit Cycles ◽  
Liénard Systems

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 BIFURCATION CURVES OF LIMIT CYCLES IN SOME LIÉNARD SYSTEMS

2000 ◽  
Vol 10 (05) ◽  
pp. 971-980 ◽  
Author(s):  
R. LÓPEZ-RUIZ ◽  
J. L. LÓPEZ
Keyword(s):  
Continuous Function ◽  
Limit Cycles ◽  
Nonlinear Regime ◽  
Liénard Systems

Liénard systems of the form [Formula: see text], with f(x) an even continuous function, are considered. The bifurcation curves of limit cycles are calculated exactly in the weak (ε → 0) and in the strongly (ε → ∞) nonlinear regime in some examples. The number of limit cycles does not increase when ε increases from zero to infinity in all the cases analyzed.

Comment on “Liénard systems, limit cycles, and Melnikov theory”

1999 ◽  
Vol 59 (2) ◽  
pp. 2483-2484 ◽  
Author(s):  
Hector Giacomini ◽  
Sébastien Neukirch
Keyword(s):  
Limit Cycles ◽  
Liénard Systems ◽  
Melnikov Theory

BIFURCATIONS OF LIMIT CYCLES FROM A QUINTIC HAMILTONIAN SYSTEM WITH A FIGURE DOUBLE-FISH

2013 ◽  
Vol 23 (07) ◽  
pp. 1350116 ◽  
Author(s):  
MINGHUI QI ◽  
LIQIN ZHAO
Keyword(s):  
Hamiltonian System ◽  
Limit Cycles ◽  
Upper Bound ◽  
Abelian Integrals ◽  
Liénard Systems

In this paper, we consider Liénard systems of the form [Formula: see text] where 0 < ∣∊∣ ≪ 1 and (α, β, γ) ∈ ℝ3. We prove that the least upper bound of the number of isolated zeros of the related Abelian integrals [Formula: see text] is 4 (counting the multiplicity) and this upper bound is a sharp one.

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