THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS

doi:10.1007/s10473-024-0319-4

Abstract

Abstract: This paper deals with the problem of limit cycles for the whirling pendulum equation $\dot{x}=y,\ \dot{y}=\sin x(\cos x-r)$ under piecewise smooth perturbations of polynomials of $\cos x$, $\sin x$ and $y$ of degree $n$ with the switching line $x=0$. The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained using the Picard-Fuchs equations, which the generating functions of the associated first order Melnikov functions satisfy. Furthermore, the exact bound of a special case is given using the Chebyshev system. At the end, some numerical simulations are given to illustrate the existence of limit cycles.

Key words: whirling pendulum, limit cycle, Melnikov function, Picard-Fuchs equation, Chebyshev system

CLC Number:

34C08

Jihua yang. THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS[J].Acta mathematica scientia,Series B, 2024, 44(3): 1115-1144.

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