Limit cycles in the equation of whirling pendulum with autonomous perturbation. (English) Zbl 1060.34504

Lichardová, Hana

Appl. Math., Praha 44, No. 4, 271-288 (1999).

The paper deals with a two-parameter Hamiltonian system with an autonomous perturbation which models a whirling pendulum. Using the Melnikov function method, the author proves the existence and uniqueness of a limit cycle of the system in a certain region of a two-dimensional space of parameters. After a short description of local bifurcations of the system, the global analysis of the phase portrait is given by means of the study of properties of the Melnikov function whose zeros correspond to limit cycles. The needed properties of the Melnikov function the author gets either by a direct study of corresponding elliptic integrals and by the Li and Zhang criterion, which allows one to examine the monotonicity of the ratio of two Abelian integrals without calculating them.

Reviewer: Irena Rachůnková (Olomouc)

MSC:

34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23	Bifurcation theory for ordinary differential equations
37G15	Bifurcations of limit cycles and periodic orbits in dynamical systems
70K05	Phase plane analysis, limit cycles for nonlinear problems in mechanics

Keywords:

whirling pendulum; Hamiltonian system; autonomous perturbation; Melnikov function; limit cycle; homoclinic orbit; elliptic integral

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