Articles you may be interested inAn observation of quadratic algebra, dual family of nonlinear coherent states and their non-classical properties, in the generalized isotonic oscillator A simple and unified approach to identify integrable nonlinear oscillators and systems Bifurcation in the entrained state of a nonlinear oscillator under sinusoidal perturbation with the 2nd harmonics AIP Conf.In this paper we point out the existence of a remarkable nonlocal transformation between the damped harmonic oscillator and a modified Emden-type nonlinear oscillator equation with linear forcing, ẍ + ␣xẋ + x 3 + ␥x = 0, which preserves the form of the time independent integral, conservative Hamiltonian, and the equation of motion. Generalizing this transformation we prove the existence of nonstandard conservative Hamiltonian structure for a general class of damped nonlinear oscillators including Liénard-type systems. Further, using the above Hamiltonian structure for a specific example, namely, the generalized modified Emden equation ẍ + ␣x q ẋ + x 2q+1 = 0, where ␣, , and q are arbitrary parameters, the general solution is obtained through appropriate canonical transformations. We also present the conservative Hamiltonian structure of the damped Mathews-Lakshmanan oscillator equation. The associated Lagrangian description for all the above systems is also briefly discussed.
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Existence of amplitude independent frequencies of oscillation is an unusual property for a nonlinear oscillator. We find that a class of N coupled nonlinear Liénard type oscillators exhibit this interesting property. We show that a specific subset can be explicitly solved from which we demonstrate the existence of periodic and quasiperiodic solutions. Another set of N-coupled nonlinear oscillators, possessing the amplitude independent nature of frequencies, is almost integrable in the sense that the system can be reduced to a single nonautonomous first order scalar differential equation which can be easily integrated numerically .
We present a system of N-coupled Liénard-type nonlinear oscillators which is completely integrable and possesses N time-independent and N time-dependent explicit integrals. In a special case, it becomes maximally superintegrable and admits (2N − 1) time-independent integrals. The results are illustrated for the N = 2 and arbitrary number cases. General explicit periodic (with frequency independent of amplitude) and quasi-periodic solutions as well as decaying-type/frontlike solutions are presented, depending on the signs and magnitudes of the system parameters. Though the system is of a nonlinear damped type, our investigations show that it possesses a Hamiltonian structure and that under a contact transformation it is transformable to a system of uncoupled harmonic oscillators.
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