Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2004-08-28
Nonlinear Sciences
Exactly Solvable and Integrable Systems
Submitted to Physical Review E 7 pages, 4 figures
Scientific paper
A Li\'enard type nonlinear oscillator of the form $\ddot{x}+kx\dot{x}+\frac{k^2}{9}x^3+\lambda_1 x=0$, 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 $\lambda_1>0$. 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 $\lambda_1\leq0$ 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.
Chandrasekar V. K.
Lakshmanan Meenakshi
Senthilvelan M.
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