scholarly journals A cubic system with twelve small amplitude limit cycles

2005 ◽  
Vol 129 (2) ◽  
pp. 83-98 ◽  
Author(s):  
Yirong Liu ◽  
Wentao Huang
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Cubic System

A Cubic System with Eight Small-Amplitude Limit Cycles

1991 ◽  
Vol 47 (2) ◽  
pp. 163-171 ◽  
Author(s):  
E. M. JAMES ◽  
N. G. LLOYD
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Cubic System

BIFURCATIONS OF LIMIT CYCLES CREATED BY A MULTIPLE NILPOTENT CRITICAL POINT OF PLANAR DYNAMICAL SYSTEMS

2011 ◽  
Vol 21 (02) ◽  
pp. 497-504 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Dynamical Systems ◽  
Critical Point ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Hopf Bifurcations ◽  
Bifurcation Phenomena ◽  
Cubic System ◽  
The Stability ◽  
Nilpotent Critical Point ◽  
Planar Dynamical Systems

Bifurcations of limit cycles created from a multiple critical point of planar dynamical systems are studied. It is different from the usual Hopf bifurcations of limit cycles created from an elementary critical point. This bifurcation phenomena depends on the stability of the multiple critical point and the multiple number of the critical point. As an example, a cubic system which can created four small amplitude limit cycles from the origin (a multiple critical point) is given.

scholarly journals A cubic system with eight small-amplitude limit cycles

1994 ◽  
Vol 7 (4) ◽  
pp. 23-27 ◽  
Author(s):  
Shucheng Ning ◽  
Shilong Ma ◽  
Keng Huat Kwek ◽  
Zhiming Zheng
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Cubic System

Bifurcations of limit cycles in a cubic system with cubic perturbations

2006 ◽  
Vol 176 (1) ◽  
pp. 341-358 ◽  
Author(s):  
Hong Zang ◽  
Tonghua Zhang ◽  
Maoan Han
Keyword(s):  
Limit Cycles ◽  
Cubic System

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.

Small-amplitude limit cycles in polynomial Li�nard systems

1996 ◽  
Vol 3 (2) ◽  
pp. 183-190 ◽  
Author(s):  
Colin J. Christopher ◽  
Noel G. Lloyd
Keyword(s):  
Limit Cycles ◽  
Small Amplitude

scholarly journals Five limit cycles for a simple cubic system

1997 ◽  
Vol 41 ◽  
pp. 199-208 ◽  
Author(s):  
N. G. Lloyd ◽  
J. M. Pearson
Keyword(s):  
Limit Cycles ◽  
Simple Cubic ◽  
Cubic System

The number of limit cycles of certain polynomial differential equations

1984 ◽  
Vol 98 (3-4) ◽  
pp. 215-239 ◽  
Author(s):  
T. R. Blows ◽  
N. G. Lloyd
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Equations ◽  
Image Position ◽  
Cubic Systems

SynopsisTwo-dimensional differential systemsare considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

scholarly journals Analytic Integrability of Two Lopsided Systems

2016 ◽  
Vol 26 (02) ◽  
pp. 1650026
Author(s):  
Feng Li ◽  
Pei Yu ◽  
Yirong Liu
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Integrating Factor ◽  
Analytic Integrability ◽  
Inverse Integrating Factor

In this paper, we present two classes of lopsided systems and discuss their analytic integrability. The analytic integrable conditions are obtained by using the method of inverse integrating factor and theory of rotated vector field. For the first class of systems, we show that there are [Formula: see text] small-amplitude limit cycles enclosing the origin of the systems for [Formula: see text], and ten limit cycles for [Formula: see text]. For the second class of systems, we prove that there exist [Formula: see text] small-amplitude limit cycles around the origin of the systems for [Formula: see text], and nine limit cycles for [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document