liénard systems
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2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Regilene D. S. Oliveira ◽  
Iván Sánchez-Sánchez ◽  
Joan Torregrosa

AbstractThe present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family.


2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Yassine Bouattia ◽  
Djalil Boudjehem ◽  
Ammar Makhlouf ◽  
Sulima Ahmed Zubair ◽  
Sahar Ahmed Idris

In this paper, we demonstrate using a counterexample for a theorem of the small amplitude limit cycles in some Liénard systems and show that that there will be no solutions unless we add an extra condition. A new condition is derived for some specific Liénard systems where a violation of the small amplitude limit cycles theorem takes place.


2021 ◽  
Vol 31 (12) ◽  
pp. 2150176
Author(s):  
Jiayi Chen ◽  
Yun Tian

In this paper, we obtain an upper bound for the number of small-amplitude limit cycles produced by Hopf bifurcation in one particular type of rational Liénard systems in the form of [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are polynomials in [Formula: see text] with degrees [Formula: see text] and [Formula: see text], respectively. Furthermore, we show that the upper bound presented here is sharp in the case of [Formula: see text].


Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1960
Author(s):  
Federico Zadra ◽  
Alessandro Bravetti ◽  
Marcello Seri

Starting from a contact Hamiltonian description of Liénard systems, we introduce a new family of explicit geometric integrators for these nonlinear dynamical systems. Focusing on the paradigmatic example of the van der Pol oscillator, we demonstrate that these integrators are particularly stable and preserve the qualitative features of the dynamics, even for relatively large values of the time step and in the stiff regime.


2021 ◽  
Vol 31 (10) ◽  
pp. 2150154
Author(s):  
Robert E. Kooij ◽  
André Zegeling

For a family of two-dimensional predator–prey models of Gause type, we investigate the simultaneous occurrence of a center singularity and a limit cycle. The family is characterized by the fact that the functional response is nonanalytical and exhibits group defense. We prove the existence and uniqueness of the limit cycle using a new theorem for Liénard systems. The new theorem gives conditions for the uniqueness of a limit cycle which surrounds a period annulus. The results of this paper provide a mechanism for studying the global behavior of solutions to Gause systems through bifurcation of an integrable system which contains a center and a limit cycle.


2020 ◽  
Vol 30 (15) ◽  
pp. 2050232
Author(s):  
Shimin Li ◽  
Jaume Llibre

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.


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