An unusual Liénard type nonlinear oscillator with properties of a linear harmonic oscillator

2004/08/28 by V. K. Chandrasekar, M. Senthilvelan, Chandrasekar, V. K. +3
Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #nlin.SI

paper · pdf · doi:10.48550/arxiv.nlin/0408054

Submitted to Physical Review E 7 pages, 4 figures

arxiv created 2005/04/20 · arxiv updated 2009/12/01

Abstract

A Liénard type nonlinear oscillator of the form x+kxx+(k²)/(9)x³+λ₁ x=0, which may also be considered as a generalized Emden type equation, is shown to possess unusual nonlinear dynamical properties. It is shown to admit explicit nonisolated periodic orbits of conservative Hamiltonian type for λ₁>0. These periodic orbits exhibit the unexpected property that the frequency of oscillations is completely independent of amplitude and continues to remain as that of the linear harmonic oscillator. This is completely contrary to the standard characteristic property of nonlinear oscillators. Interestingly, the system though appears deceptively a dissipative type for λ₁≤0 does admit a conserved Hamiltonian description, where the characteristic decay time is also independent of the amplitude. The results also show that the criterion for conservative Hamiltonian system in terms of divergence of flow function needs to be generalized.

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