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Higher Order Bifurcations of Limit Cycles

✍ Scribed by I.D. Iliev; L.M. Perko

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
242 KB
Volume
154
Category
Article
ISSN
0022-0396

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✦ Synopsis

This paper shows that asymmetrically perturbed, symmetric Hamiltonian systems of the form

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*aΓ‚ 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.

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