Higher Order Bifurcations of Limit Cycles

Abstract

with analytic * j (=)=O(=), have at most two limit cycles that bifurcate for small ={0 from any period annulus of the unperturbed system. This fact agrees with previous results of Petrov, Dangelmayr and Guckenheimer, and Chicone and Iliev, but shows that the result of three limit cycles for the asymmetrically perturbed, exterior Duffing oscillator, recently obtained by Jebrane and Z 4 o*a dek, is incorrect. The proofs follow by deriving an explicit formula for the kth-order Melnikov function, M k (h), and using a Picard Fuchs analysis to show that, in each case, M k (h) has at most two zeros. Moreover, the method developed in this paper for determining the higher-order Melnikov functions also applies to more general perturbations of these systems. Academic Press

Limit cycles in planar piecewise linear differential systems with nonregular separation line

“…More specifically, the method used in this work, see [15], is a generalization to piecewise linear systems of the extension to higher order perturbations, see [21], of the method of Françoise. The main application in [15] is the computation of the Lyapunov constants for piecewise systems and their use in the center-focus problem.…”

A Chebyshev criterion for Abelian integrals

“…The resultant with respect to z between q(x, z) and the numerator of ω 3 (x, z) is r 3 (x) = 64x 16 and by applying Sturm's Theorem we get that p 3 (x) = 0 for all x ∈ 0, √ 2 − 1 . Thus, ω 3 (x, z) = 0 and q(x, z) = 0 have no common roots, and this fact implies that W [ 3 ](x) = 0 for all x ∈ 0, √ 2 − 1 .…”

“…In this example we study the so-called interior Duffing oscillator. Theorem 1.3 in [16] shows that at most two limit cycles bifurcate from either one of the interior period annuli. If we perform a translation to bring the center on the right half-plane to the origin, the Hamiltonian function of the unperturbed system becomes…”

“…From Theorem 2.1 in [16], it follows that the first non-identically zero Melnikov function is a linear combination of I i (h) = γ h x i ydx for i = 0, 1, 2. Thus, Theorem 1.3 in [16] will follow if we prove that I 0 , I 1 , I 2 is an ECT-system.…”

Cyclicity versus Center Problem

“…From this point of view when a is small it can be seen as the perturbation of a global Hamiltonian center. In this situation the algorithm introduced in [9], see also [12,14], allows to compute the first non-zero Melnikov function associated to the system. The positive simple zeros of this function control the limit cycles of the system that tend to the periodic orbits of the linear system.…”