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

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

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].

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.

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