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Higher order bifurcations of limit cycles. (English) Zbl 0926.34033

Summary: The authors show that asymmetrically perturbed, symmetric Hamiltonian systems of the form \[ \dot x= y,\quad \dot y= \pm(x\pm x^3)+ \lambda_1 y+ \lambda_2 x^2+ \lambda_3 xy+ \lambda_4 x^2y, \] with analytic \(\lambda_j(\varepsilon)= O(\varepsilon)\), have at most two limit cycles that bifurcate for small \(\varepsilon\neq 0\) 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 Żołądek, is incorrect.
The proofs follow by deriving an explicit formula for the \(k\)th-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 for determining higher-order Melnikov functions applies to more general perturbations of these systems. \(\copyright\) Academic Press.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
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