LIMIT CYCLES IN 3D LOTKA–VOLTERRA SYSTEMS APPEARING AFTER PERTURBATION OF HOPF CENTER

2008 ◽  
Vol 18 (12) ◽  
pp. 3647-3656 ◽  
Author(s):  
Ł. J. GOŁASZEWSKI ◽  
P. SŁAWIŃSKI ◽  
H. ŻOŁADEK
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Invariant Torus ◽  
Small Amplitude ◽  
Parameter Family ◽  
Volterra Systems ◽  
Parameter Perturbations ◽  
Two Parameter

We study the system ẋ = x(y+2z+(15/2η2)u), ẏ = y(x-2z-(7/2η2)u), ż = -z(x+y+(4/η2)u), u = x+y+z-1, and its two-parameter perturbations. We show that before perturbation there exists a one-parameter family of periodic solutions obtained via a nondegenarate Hopf bifurcation and after perturbation there remains at most one limit cycle of small amplitude and bounded period. Moreover, we found that a secondary Hopf bifurcation to an invariant torus occurs after the perturbation.

scholarly journals ON PERIODIC SOLUTIONS OF LIÉNARD TYPE EQUATIONS

2013 ◽  
Vol 18 (5) ◽  
pp. 708-716 ◽  
Author(s):  
Svetlana Atslega ◽  
Felix Sadyrbaev
Keyword(s):  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Type Equation ◽  
Convex Shape ◽  
Period Annulus ◽  
Conservative Equation ◽  
Liénard Type Equation

The Liénard type equation x'' + f(x, x')x' + g(x) = 0 (i) is considered. We claim that if the associated conservative equation x'' + g(x) = 0 has period annuli then a dissipation f(x, x') exists such that a limit cycle of equation (i) exists in a selected period annulus. Moreover, it is possible to define f(x, x') so that limit cycles appear in all period annuli. Examples are given. A particular example presents two limit cycles of non-convex shape in two disjoint period annuli.

Limit Cycles in a Class of Quartic Kolmogorov Model with Three Positive Equilibrium Points

2015 ◽  
Vol 25 (06) ◽  
pp. 1550080 ◽  
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Qi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Positive Equilibrium ◽  
Bifurcation Problem ◽  
Kolmogorov Model ◽  
Limit Cycle Bifurcation ◽  
Perturbed Model

Limit cycle bifurcation problem of Kolmogorov model is interesting and significant both in theory and applications. In this paper, we will focus on investigating limit cycles for a class of quartic Kolmogorov model with three positive equilibrium points. Perturbed model can bifurcate three small limit cycles near (1, 2) or (2, 1) under a certain condition and can bifurcate one limit cycle near (1, 1). In addition, we have given some examples of simultaneous Hopf bifurcation and the structure of limit cycles bifurcated from three positive equilibrium points. The limit cycle bifurcation problem for Kolmogorov model with several positive equilibrium points are less seen in published references. Our result is good and interesting.

Hopf Bifurcation of Z2-Equivariant Generalized Liénard Systems

2018 ◽  
Vol 28 (06) ◽  
pp. 1850069 ◽  
Author(s):  
Yusen Wu ◽  
Laigang Guo ◽  
Yufu Chen
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Polynomial Function ◽  
Complete Classification ◽  
Normal Form Theory ◽  
Liénard Systems ◽  
Form Theory ◽  
Center Conditions

In this paper, we consider a class of Liénard systems, described by [Formula: see text], with [Formula: see text] symmetry. Particular attention is given to the existence of small-amplitude limit cycles around fine foci when [Formula: see text] is an odd polynomial function and [Formula: see text] is an even function. Using the methods of normal form theory, we found some new and better lower bounds of the maximal number of small-amplitude limit cycles in these systems. Moreover, a complete classification of the center conditions is obtained for such systems.

scholarly journals Limit Cycles in a Model of Olfactory Sensory Neurons

2019 ◽  
Vol 29 (03) ◽  
pp. 1950038 ◽  
Author(s):  
Yonghui Xia ◽  
Mateja Grašič ◽  
Wentao Huang ◽  
Valery G. Romanovski
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Sensory Neurons ◽  
Limit Cycles ◽  
Center Manifold ◽  
Calcium Oscillations ◽  
Olfactory Sensory Neurons ◽  
Generalized Hopf Bifurcation ◽  
Subcritical Hopf Bifurcation ◽  
Unstable Cycle

We propose an approach to study small limit cycle bifurcations on a center manifold in analytic or smooth systems depending on parameters. We then apply it to the investigation of limit cycle bifurcations in a model of calcium oscillations in the cilia of olfactory sensory neurons and show that it can have two limit cycles: a stable cycle appearing after a Bautin (generalized Hopf) bifurcation and an unstable cycle appearing after a subcritical Hopf bifurcation.

Maximum Number of Small Limit Cycles in Some Rational Liénard Systems with Cubic Restoring Terms

2021 ◽  
Vol 31 (12) ◽  
pp. 2150176
Author(s):  
Jiayi Chen ◽  
Yun Tian
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Upper Bound ◽  
Small Amplitude ◽  
Liénard Systems

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].

An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems

2018 ◽  
Vol 28 (06) ◽  
pp. 1850078 ◽  
Author(s):  
Pei Yu ◽  
Maoan Han ◽  
Jibin Li
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Correct Answer ◽  
Homogeneous Polynomial ◽  
Bifurcation Theory ◽  
Polynomial Systems ◽  
Homogeneous Polynomials ◽  
Appl Math

In the two articles in Appl. Math. Comput., J. Giné [2012a, 2012b] studied the number of small limit cycles bifurcating from the origin of the system: [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are homogeneous polynomials of degree [Formula: see text]. It is shown that the maximal number of the small limit cycles, denoted by [Formula: see text], satisfies [Formula: see text] for [Formula: see text]; and [Formula: see text], [Formula: see text]. It seems that the correct answer for their case [Formula: see text] should be [Formula: see text]. In this paper, we apply Hopf bifurcation theory and normal form computation, and perturb the isolated, nondegenerate center (the origin) to prove that [Formula: see text] for [Formula: see text]; and [Formula: see text] for [Formula: see text], which improve Giné’s results with one more limit cycle for each case.

A BIPARAMETRIC BIFURCATION IN 3D CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH TWO ZONES: APPLICATION TO CHUA'S CIRCUIT

2007 ◽  
Vol 17 (02) ◽  
pp. 445-457 ◽  
Author(s):  
E. FREIRE ◽  
E. PONCE ◽  
J. ROS
Keyword(s):  
Hopf Bifurcation ◽  
Linear Systems ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Continuous Systems ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Piecewise Linear Systems ◽  
Limit Cycle Bifurcation

In this paper, a possible degeneration of the focus-center-limit cycle bifurcation for piecewise smooth continuous systems is analyzed. The case of continuous piecewise linear systems with two zones is considered, and the coexistence of two limit cycles for certain values of parameters is justified. Finally, the Chua's circuit is shown to exhibit the analyzed bifurcation. The obtained bifurcation set in the parameter plane is similar to the degenerate Hopf bifurcation for differentiable systems.

scholarly journals Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fang Wu ◽  
Lihong Huang ◽  
Jiafu Wang
Keyword(s):  
Periodic Solutions ◽  
Displacement Function ◽  
Piecewise Smooth ◽  
Parameter Family ◽  
Piecewise Smooth System ◽  
Closed Orbits ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Smooth System ◽  
Two Parameter

<p style='text-indent:20px;'>In this paper, we consider the nonsmooth bifurcation around a class of critical crossing cycles, which are codimension-2 closed orbits composed of tangency singularities and regular orbits, for a two-parameter family of planar piecewise smooth system with two zones. By the construction of suitable displacement function (equivalently, Poincar<inline-formula><tex-math id="M1">\begin{document}$ {\rm\acute{e}} $\end{document}</tex-math></inline-formula> map), the stability and the existence of periodic solutions under the variation of the parameters inside this system are characterized. More precisely, we obtain some parameter regions on the existence of crossing cycles and sliding cycles near those loops. As applications, several examples are given to illustrate our main conclusions.</p>

ANDRONOV–HOPF BIFURCATION OF HIGHER CODIMENSIONS IN A LIÉNARD SYSTEM

2012 ◽  
Vol 22 (11) ◽  
pp. 1250271 ◽  
Author(s):  
ALEXANDER GRIN ◽  
KLAUS R. SCHNEIDER
Keyword(s):  
Hopf Bifurcation ◽  
Parameter Space ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Control Parameter ◽  
Unique Equilibrium ◽  
Liénard System ◽  
Lienard System

We study a polynomial Liénard system depending on three parameters a, b, c and exhibiting the following properties: (i) The origin is the unique equilibrium for all parameters. (ii) If a crosses zero, then the origin changes its stability, and Andronov–Hopf bifurcation arises. We consider a as control parameter and investigate the dependence of Andronov–Hopf bifurcation on the "unfolding" parameters b and c. We establish and describe analytically the existence of surfaces and curves located near the origin in the parameter space connected with the existence of small-amplitude limit cycles of multiplicity two and three (existence of degenerate Andronov–Hopf bifurcation).

HOPF BIFURCATION OF LIÉNARD SYSTEMS BY PERTURBING A NILPOTENT CENTER

2012 ◽  
Vol 22 (08) ◽  
pp. 1250203 ◽  
Author(s):  
JING SU ◽  
JUNMIN YANG ◽  
MAOAN HAN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Nonlinear Oscillators ◽  
Melnikov Function ◽  
Liénard System ◽  
First Order ◽  
Liénard Systems ◽  
Lienard System ◽  
Nilpotent Center

As we know, Liénard system is an important model of nonlinear oscillators, which has been widely studied. In this paper, we study the Hopf bifurcation of an analytic Liénard system by perturbing a nilpotent center. We develop an efficient method to compute the coefficients bl appearing in the expansion of the first order Melnikov function by finding a set of equivalent quantities B2l+1 which are able to calculate directly and can be used to study the number of small-amplitude limit cycles of the system. As an application, we investigate some polynomial Liénard systems, obtaining a lower bound of the maximal number of limit cycles near a nilpotent center.

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