The number of small-amplitude limit cycles of Liénard equations

Abstract

We consider second order differential equations of Liénard type:Such equations have been very widely studied and arise frequently in applications. There is an extensive literature relating to the existence and uniqueness of periodic solutions: the paper of Staude[6] is a comprehensive survey. Our interest is in the number of periodic solutions of such equations.

Variational approach to a class of nonlinear oscillators with several limit cycles

“…Many results on existence and uniqueness have been established, less is known about the number and location of limit cycles which do not satisfy Liénard's conditions [5,6,7]. Lins, de Melo and Pugh [8] conjectured that if F is a polynomial of degree 2n + 1 or 2n + 2 then there can be at most n limit cycles.This conjecture was proved [9] for small ν, that is, for small departures from the hamiltonian case. For ν → 0 the number and position of the limit cycles is given by the real roots of a polynomial obtained from Melnikov's function [5,7].…”

“…where the extremum is taken over all positive functions p(u) that vanish at the end points and φ(u) is the function of p(u) given by (9). If we succeed in finding all the extrema, we have found all limit cycles.…”

Center Problem for Several Differential Equations via Cherkas' Method

“…As to the case iv , with q s 2, we can assume that b and b are 3 2 Ž . nonzero constants, and apply again Theorem 2.6 v with the functions 5 . O x are analytic functions.…”

On the number of limit cycles of a class of polynomial differential systems

“…Blows & Lloyd (1984), Lloyd & Lynch (1988) and Lynch (1995) have used inductive arguments in order to prove the following results:…”