Isochronous Liénard-type nonlinear oscillators of arbitrary dimensions

Tiwari, Avinash; Devi, A. Durga; Pradeep, Ronak; Chandrasekar, V. K.

doi:10.1007/s12043-015-1104-7

Cited by 10 publications

(3 citation statements)

References 43 publications

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“…But it remains an interesting curiosity to provide an explanation for why the analytic representations of both linear and nonlinear coupled system exhibit identical behavior. Moreover, one may also like to extend our method to construct Lagrangians for isochronous Liénard-type oscillators of different dimensions [49,50].…”

Section: Discussionmentioning

confidence: 99%

On the analytic representation of Newtonian systems

Talukdar,

Chatterjee,

Ali

2020

Preprint

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We show that the theory of self-adjoint differential equations can be used to provide a satisfactory solution of the inverse variational problem in classical mechanics. A Newtonian equation when transformed to the self-adjoint form allows one to find an appropriate Lagrangian representation (direct analytic representation) for it. On the other hand, the same Newtonian equation in conjunction with its adjoint provides a basis to construct a different Lagrangian representation (indirect analytic representation) for the system. We obtain the time-dependent Lagrangian of the damped Harmonic oscillator from the self-adjoint form of the equation of motion and at the same time identify the adjoint of the equation with the so called Bateman image equation with a view to construct a time-independent indirect Lagrangian representation. We provide a number of case studies to demonstrate the usefulness of the approach derived by us. We also present similar results for a number of nonlinear differential equations by using an integral representation of the Lagrangian function and make some useful comments.

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Section: Discussionmentioning

confidence: 99%

On the analytic representation of Newtonian systems

Talukdar,

Chatterjee,

Ali

2020

Preprint

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“…One finds that certain Liénard type oscillators are solvable and also admit isochronous oscillations, whose frequencies are independent of their amplitudes [13]. Lie-group techniques, local and nonlocal transformations, etc., are employed to understand/generate new isochronous systems [14][15][16]. In another aspect, a systematic procedure has been developed to construct non-standard forms of Lagrangian/ Hamiltonian for specific nonlinear systems [17,18].…”

Section: Y T X F X Y T Y F X Y T X Y G X Y T X G X Y T Y H X Y T mentioning

confidence: 99%

“…It is observed that the nonlinear systems of the form (1), admitting eight parameter Lie point symmetry elements are linearizable under coordinate transformations if the functions g(x) and f (x) of equation (1) are related by the condition [16]…”

Section: Liénard Type-i Oscillators: Isochronous Oscillationsmentioning

confidence: 99%

Liénard type nonlinear oscillators and quantum solvability

2024

Phys. Scr.

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Liénard-type nonlinear oscillators with linear and nonlinear damping terms exhibit diverse dynamical behavior in both the classical and quantum regimes. In this paper, we consider examples of various one-dimensional Liénard type I and type II oscillators. The associated Euler-Lagrange equations are divided into groups based on the characteristics of the damping and forcing terms. The Liénard type-I oscillators often display localized solutions, isochronous and non-isochronous oscillations and are also precisely solvable in quantum mechanics in general, where the ordering parameters play an important role. These include Mathews-Lakshmanan and Higgs oscillators. However, the classical solutions of some of the nonlinear oscillators are expressed in terms of elliptic functions and have been found to be quasi-exactly solvable in the quantum region. The three-dimensional generalizations of these classical systems add more degrees of freedom, which show complex dynamics. Their quantum equivalents are also explored in this article. The isotonic generalizations of the non-isochronous nonlinear oscillators have also been solved both classically and quantum mechanically to advance the studies. The modiﬁed Emden equation categorized as Liénard type-II exhibits isochronous oscillations at the classical level. This property makes it a valuable tool for studying the underlying nonlinear dynamics. The study on the quantum counterpart of the system provides a deeper understanding of the behavior in the quantum realm as a typical PT-symmetric system.

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Complex bursting dynamics in a Rayleigh–Liénard oscillator

Wang,

Qian

2024

Nonlinear Dyn

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Isochronous Liénard-type nonlinear oscillators of arbitrary dimensions

Cited by 10 publications

References 43 publications

On the analytic representation of Newtonian systems

On the analytic representation of Newtonian systems

Liénard type nonlinear oscillators and quantum solvability

Complex bursting dynamics in a Rayleigh–Liénard oscillator

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