scholarly journals Uniqueness of Limit Cycles for a Class of Cubic Systems with Two Invariant Straight Lines

2010 ◽  
Vol 2010 ◽  
pp. 1-17
Author(s):  
Xiangdong Xie ◽  
Fengde Chen ◽  
Qingyi Zhan
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Uniqueness Of Limit Cycles ◽  
Cubic System ◽  
Straight Lines ◽  
Cubic Systems

A class of cubic systems with two invariant straight linesdx/dt=y(1-x2),  dy/dt=-x+δy+nx2+mxy+ly2+bxy2.is studied. It is obtained that the focal quantities ofO(0,0)are,W0=δ; ifW0=0, thenW1=m(n+l); ifW0=W1=0, thenW2=−nm(b+1); ifW0=W1=W2=0, thenOis a center, and it has been proved that the above mentioned cubic system has at most one limit cycle surrounding weak focalO(0,0). This paper also aims to solve the remaining issues in the work of Zheng and Xie (2009).

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.

Bifurcations in a Cubic System with a Degenerate Saddle Point

2014 ◽  
Vol 24 (11) ◽  
pp. 1450144
Author(s):  
Desheng Shang ◽  
Yaoming Zhang
Keyword(s):  
Perturbation Theory ◽  
Qualitative Analysis ◽  
Saddle Point ◽  
Limit Cycles ◽  
Blow Up ◽  
Cubic System ◽  
Cubic Systems

Bifurcations in a cubic system with a degenerate saddle point are investigated using the technique of blow-up, the method of planar perturbation theory and qualitative analysis. It has been found that after appropriate perturbations, at least 12 limit cycles can bifurcate from a degenerate saddle point in a type of cubic systems.

scholarly journals Uniqueness of limit cycles in polynomial systems with algebraic invariants

1994 ◽  
Vol 49 (1) ◽  
pp. 7-20 ◽  
Author(s):  
André Zegeling ◽  
Robert E. Kooij
Keyword(s):  
Simple Proof ◽  
Limit Cycles ◽  
Real Line ◽  
Vector Fields ◽  
Polynomial Systems ◽  
Quadratic Systems ◽  
Codimension Two ◽  
Algebraic Invariants ◽  
Uniqueness Of Limit Cycles ◽  
Cubic Systems

The uniqueness of limit cycles is proved for quadratic systems with an invariant parabola and for cubic systems with four real line invariants. Also a new, simple proof is given of the uniqueness of limit cycles occurring in unfoldings of certain vector fields with codimension two singularities.

Double Bifurcation of Nilpotent Focus

2015 ◽  
Vol 25 (03) ◽  
pp. 1550036 ◽  
Author(s):  
Yirong Liu ◽  
Feng Li
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
Weak Focus ◽  
Bifurcation Phenomenon ◽  
Cubic System ◽  
Cubic Systems ◽  
Planar Dynamical Systems

In this paper, an interesting bifurcation phenomenon is investigated — a 3-multiple nilpotent focus of the planar dynamical systems could be broken into two element focuses and an element saddle, and the limit cycles could bifurcate out from two element focuses. As an example, a class of cubic systems with 3-multiple nilpotent focus O(0, 0) is investigated, we prove that nine limit cycles with the scheme 7 ⊃ (1 ∪ 1) could bifurcate out from the origin when the origin is a weak focus of order 8. At the end of this paper, the double bifurcations of a class of Z2 equivalent cubic system with 3-multiple nilpotent focus or center O(0, 0) are investigated.

Small Amplitude Limit Cycles for Cubic Systems

1993 ◽  
Vol 36 (1) ◽  
pp. 54-63 ◽  
Author(s):  
V. Guíñez ◽  
E. Sáez ◽  
I. Szántó
Keyword(s):  
Periodic Orbit ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Small Amplitude ◽  
The Other ◽  
Simultaneous Generation ◽  
The Family ◽  
Planar Systems ◽  
Image Position ◽  
Cubic Systems

AbstractIn this article we study the simultaneous generation of limit cycles out of singular points and infinity for the family of cubic planar systemsWith a suitable choice of parameters, the origin and four other singularities are foci and infinity is a periodic orbit. We prove that it is possible to obtain the following configuration of limit cycles: two small amplitude limit cycles out of the origin, a small amplitude limit cycle out of each of the other four foci, and a large amplitude limit cycle out of infinity. We also obtain other configurations with fewer limit cycles.

Bifurcation of Nonhyperbolic Limit Cycles in Piecewise Smooth Planar Systems with Finitely Many Zones

2017 ◽  
Vol 27 (10) ◽  
pp. 1750162 ◽  
Author(s):  
Yurong Li ◽  
Liping Yuan ◽  
Zhengdong Du
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Bifurcation Of Limit Cycles ◽  
Planar System ◽  
Planar Systems ◽  
The Stability ◽  
Straight Lines ◽  
First Time

Like for smooth systems, it is very important to discuss the stability and bifurcation of limit cycles in a piecewise smooth planar system. Most of the previous works focus only on hyperbolic limit cycles. Few works have considered nonhyperbolic limit cycles. In fact, to date, no concrete examples of piecewise smooth planar system with nonhyperbolic limit cycles have been given in literature. In this paper, we consider for the first time the bifurcation of nonhyperbolic limit cycles in piecewise smooth planar systems with discontinuities on finitely many straight lines intersecting at the origin. We present a method of Melnikov type to derive two quantities which can be used to determine the stability and the number of limit cycles that can bifurcate from a nonhyperbolic limit cycle of a piecewise smooth planar system. As applications, we present two examples of piecewise smooth systems with two and three zones respectively whose unperturbed system has a nonhyperbolic limit cycle.

scholarly journals Limit Cycles of Planar Piecewise Differential Systems with Linear Hamiltonian Saddles

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1128
Author(s):  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycle ◽  
Differential System ◽  
Limit Cycles ◽  
The Other ◽  
Differential Systems ◽  
Other Hand ◽  
Straight Line ◽  
Line Of Discontinuity ◽  
Straight Lines

We provide the maximum number of limit cycles for continuous and discontinuous planar piecewise differential systems formed by linear Hamiltonian saddles and separated either by one or two parallel straight lines. We show that when these piecewise differential systems are either continuous or discontinuous and are separated by one straight line, or are continuous and are separated by two parallel straight lines, they do not have limit cycles. On the other hand, when these systems are discontinuous and separated by two parallel straight lines, we prove that the maximum number of limit cycles that they can have is one and that this maximum is reached by providing an example of such a system with one limit cycle. When the line of discontinuity of the piecewise differential system is formed by one straight line, the symmetry of the problem allows to take this straight line without loss of generality as the line x=0. Similarly, when the line of discontinuity of the piecewise differential system is formed by two parallel straight lines due to the symmetry of the problem, we can assume without loss of generality that these two straight lines are x=±1.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

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