Nonexistence of Limit Cycles for Planar Systems of Liénard-type
نویسندگان
چکیده
A new method for the nonexistence of limit cycles of the planar system including a generalized Liénard-type system is introduced. It is given by constructing a curve with some invariance defined on a half-open interval and has useful powers for the case which the equilibrium point is stable specially. Moreover, it is also applied to systems with several equilibrium points. It shall be shown that our results are used to many examples. AMS Subject Classifications: 34C07, 34C25, 34C26, 34D20.
منابع مشابه
Bifurcation of Limit Cycles in a Class of Liénard Systems with a Cusp and Nilpotent Saddle
In this paper the asymptotic expansion of first-order Melnikov function of a heteroclinic loop connecting a cusp and a nilpotent saddle both of order one for a planar near-Hamiltonian system are given. Next, we consider the bifurcation of limit cycles of a class of hyper-elliptic Liénard system with this kind of heteroclinic loop. It is shown that this system can undergo Poincarè bifurcation fr...
متن کاملGlobal analysis of a piecewise linear Liénard-type dynamical system
In this paper, we consider a planar dynamical system with a piecewise linear function containing an arbitrary (but finite) number of dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifurcations of its limit cycles, we prove that such a piecewise linear dynamical system with k dropping sections and 2k + 1 singular points can have at ...
متن کاملThe number of medium amplitude limit cycles of some generalized Liénard systems
We will consider two special families of polynomial perturbations of the linear center. For the resulting perturbed systems, which are generalized Liénard systems, we provide the exact upper bound for the number of limit cycles that bifurcate from the periodic orbits of the linear center.
متن کاملExact Number of Limit Cycles for a Family of Rigid Systems
For a given family of planar differential equations it is a very difficult problem to determine an upper bound for the number of its limit cycles. Even when this upper bound is one it is not always an easy problem to distinguish between the case of zero and one limit cycle. This note mainly deals with this second problem for a family of systems with a homogeneous nonlinear part. While the condi...
متن کاملLimit cycles in generalized Liénard systems
This paper presents some new results which we obtained recently for the study of limit cycles of nonlinear dynamical systems. Particular attention is given to small limit cycles of generalized Liénard systems in the vicinity of the origin. New results for a number of cases of the Liénard systems are presented with the Hilbert number, b H ði; jÞ 1⁄4 b H ðj; iÞ, for j = 4, i = 10,11,12,13; j = 5,...
متن کامل