scholarly journals Multiple Limit Cycle Bifurcation Surfaces and Global Families of Multiple Limit Cycles

1995 ◽  
Vol 122 (1) ◽  
pp. 89-113 ◽  
Author(s):  
L.M. Perko
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Limit Cycle Bifurcation ◽  
Multiple Limit Cycle

LIMIT CYCLE BIFURCATIONS IN NEAR-HAMILTONIAN SYSTEMS BY PERTURBING A NILPOTENT CENTER

2008 ◽  
Vol 18 (10) ◽  
pp. 3013-3027 ◽  
Author(s):  
MAOAN HAN ◽  
JIAO JIANG ◽  
HUAIPING ZHU
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Unperturbed System ◽  
Bifurcation Theorem ◽  
First Order ◽  
Cubic System ◽  
Limit Cycle Bifurcation ◽  
Almost All ◽  
Nilpotent Center ◽  
Computing Method

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented.

ASYMPTOTIC EXPANSIONS OF MELNIKOV FUNCTIONS AND LIMIT CYCLE BIFURCATIONS

2012 ◽  
Vol 22 (12) ◽  
pp. 1250296 ◽  
Author(s):  
MAOAN HAN
Keyword(s):  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Asymptotic Expansions ◽  
Melnikov Function ◽  
Heteroclinic Loop ◽  
First Order ◽  
Melnikov Functions ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Center

In the study of the perturbation of Hamiltonian systems, the first order Melnikov functions play an important role. By finding its zeros, we can find limit cycles. By analyzing its analytical property, we can find its zeros. The main purpose of this article is to summarize some methods to find its zeros near a Hamiltonian value corresponding to an elementary center, nilpotent center or a homoclinic or heteroclinic loop with hyperbolic saddles or nilpotent critical points through the asymptotic expansions of the Melnikov function at these values. We present a series of results on the limit cycle bifurcation by using the first coefficients of the asymptotic expansions.

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.

Global Dynamics of Memristor Oscillator

2016 ◽  
Vol 26 (12) ◽  
pp. 1650198 ◽  
Author(s):  
Hebai Chen
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Pitchfork Bifurcation ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Generalized Hopf Bifurcation ◽  
Limit Cycle Bifurcation ◽  
Sector Method ◽  
Double Limit ◽  
The Continuum

In this paper, we investigate the global dynamics of a memristor oscillator [Formula: see text] which comes from [Corinto et al., 2011], where [Formula: see text], and [Formula: see text]. Clearly, the case [Formula: see text] is trivial. So far, all results of this oscillator were given only for the case [Formula: see text], where the set of equilibria may change among a singleton, three points and a singular continuum and at most one limit cycle can arise and no limit cycles arise from the continuum. Compared with the case [Formula: see text], this oscillator displays more complicated dynamics for the case when [Formula: see text]. More clearly, one limit cycle may arise from the continuum and at most three limit cycles appear in the case of three equilibria, where generalized pitchfork bifurcation, saddle-node bifurcation, generalized Hopf bifurcation, double limit cycle bifurcation and homoclinic bifurcation may occur. Finally all global phase portraits are given for [Formula: see text] cases on the Poincaré disc, where a generalized normal sector method is applied. Moreover, our partial analytical results are demonstrated by numerical examples.

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.

Bifurcation of Limit Cycles in Small Perturbation of a Class of Liénard Systems

2014 ◽  
Vol 24 (01) ◽  
pp. 1450004 ◽  
Author(s):  
Xianbo Sun ◽  
Hongjian Xi ◽  
Hamid R. Z. Zangeneh ◽  
Rasool Kazemi
Keyword(s):  
Limit Cycle ◽  
Small Perturbation ◽  
Limit Cycles ◽  
Upper Bound ◽  
Bifurcation Of Limit Cycles ◽  
Heteroclinic Loop ◽  
Algebraic Criterion ◽  
Liénard Systems ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Saddle

In this article, we study the limit cycle bifurcation of a Liénard system of type (5,4) with a heteroclinic loop passing through a hyperbolic saddle and a nilpotent saddle. We study the least upper bound of the number of limit cycles bifurcated from the periodic annulus inside the heteroclinic loop by a new algebraic criterion. We also prove at least three limit cycles will bifurcate and six kinds of different distributions of these limit cycles are given. The methods we use and the results we obtain are new.

THE NUMBER OF LIMIT CYCLES IN A Z2-EQUIVARIANT LIÉNARD SYSTEM

2013 ◽  
Vol 23 (05) ◽  
pp. 1350085 ◽  
Author(s):  
YANQIN XIONG ◽  
HUI ZHONG
Keyword(s):  
Limit Cycle ◽  
Lower Bounds ◽  
Limit Cycles ◽  
Bifurcation Theory ◽  
Heteroclinic Bifurcation ◽  
Liénard System ◽  
Lienard System ◽  
Limit Cycle Bifurcation

In this paper, we consider the problem of limit cycle bifurcation near center points and a Z2-equivariant compound cycle in a polynomial Liénard system. Using the methods of Hopf, homoclinic and heteroclinic bifurcation theory, we found some new and better lower bounds of the maximal number of limit cycles for this system.

Limit Cycle Bifurcations by Perturbing a Piecewise Hamiltonian System with a Double Homoclinic Loop

2016 ◽  
Vol 26 (06) ◽  
pp. 1650103 ◽  
Author(s):  
Yanqin Xiong
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Sufficient Conditions ◽  
Melnikov Function ◽  
Analytic Method ◽  
Piecewise Polynomial ◽  
Bifurcation Problem ◽  
Homoclinic Loop ◽  
Limit Cycle Bifurcation

This paper is concerned with the bifurcation problem of limit cycles by perturbing a piecewise Hamiltonian system with a double homoclinic loop. First, the derivative of the first Melnikov function is provided. Then, we use it, together with the analytic method, to derive the asymptotic expansion of the first Melnikov function near the loop. Meanwhile, we present the first coefficients in the expansion, which can be applied to study the limit cycle bifurcation near the loop. We give sufficient conditions for this system to have [Formula: see text] limit cycles in the neighborhood of the loop. As an application, a piecewise polynomial Liénard system is investigated, finding six limit cycles with the help of the obtained method.

Stability and Limit Cycle Bifurcation for Two Kinds of Generalized Double Homoclinic Loops in Planar Piecewise Smooth Systems

2014 ◽  
Vol 24 (12) ◽  
pp. 1450153
Author(s):  
Feng Liang ◽  
Maoan Han
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Upper Bound ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Homoclinic Loop ◽  
Limit Cycle Bifurcation

In this paper, we present two kinds of generalized double homoclinic loops in planar piecewise smooth systems. For their stability a criterion is provided. Under nondegenerate conditions, we prove that for each case there are at most five limit cycles which can be bifurcated from the generalized double homoclinic loop. Especially, we construct two concrete systems to show that the upper bound can be achieved in both cases.

AN IMPROVED LOWER BOUND ON THE NUMBER OF LIMIT CYCLES BIFURCATING FROM A QUINTIC HAMILTONIAN PLANAR VECTOR FIELD UNDER QUINTIC PERTURBATION

2010 ◽  
Vol 20 (01) ◽  
pp. 63-70 ◽  
Author(s):  
TOMAS JOHNSON ◽  
WARWICK TUCKER
Keyword(s):  
Vector Field ◽  
Hamiltonian System ◽  
Lower Bound ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Hamiltonian Vector Field ◽  
Planar Vector ◽  
Limit Cycle Bifurcation ◽  
The Given ◽  
Planar Hamiltonian System

The limit cycle bifurcation of a Z2 equivariant quintic planar Hamiltonian vector field under Z2 equivariant quintic perturbation is studied. We prove that the given system can have at least 27 limit cycles. This is an improved lower bound on the possible number of limit cycles that can bifurcate from a quintic planar Hamiltonian system under quintic perturbation.

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