Abstract
We establish sufficient criteria for the existence of a limit cycle in the Liénard system x[over ̇]=y-ɛF(x),y[over ̇]=-x, where F(x) is odd. In their simplest form the criteria lead to the result that, for all finite nonzero ɛ, the amplitude of the limit cycle is less than ρ and 0≤a≤ρ≤u, where F(a)=0 and ∫(0)(u)F(x)dx=0. We take the van der Pol oscillator as a specific example and establish that for all finite, nonzero ɛ, the amplitude of its limit cycle is less than 2.0672, a value whose precision is limited by the capacity of our symbolic computation software package. We show how the criterion for the upper bound can be extended to establish a bound on the amplitude of a limit cycle in systems where F(x) contains both odd and even components. We also show how the criteria can be used to establish bounds for bifurcation sets.
Highlights
Oscillatory phenomena occur widely in nature [1] and they arise in engineered systems [2]
We show that for α = 0 + α ( α > 0) the system given by Eq (30) has a limit cycle α bounded by 0
Let HN∗ (x,y) designate the GN polynomial of degree N in y generated with a nonzero constant of integration k for the integral of dgK,N dx
Summary
Oscillatory phenomena occur widely in nature [1] and they arise in engineered systems [2]. Some of the simplest mathematical descriptions of nonlinear sustained oscillations are based on limit cycles in second-order systems [3]. An area of early research in relation to specific nonlinear systems was the identification of whether limit cycles could occur and, if so, the determination of their amplitude and frequency [4,5,6]. The behavior of coupled systems of nonlinear oscillators has attracted attention, since the development of the Kuramoto model for synchronisation in oscillator ensembles [7]. A recent development in relation to the analysis of oscillatory dynamics which are stable under continuous perturbations includes a generalization of the limit-cycle concept, the “chronotaxic limit cycle” [11]
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