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The limit cycle bifurcations of a whirling pendulum with piecewise smooth perturbations. (English) Zbl 07815387

Summary: 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.

MSC:

34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
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