A Simple and Unified Approach to Identify Integrable Nonlinear Oscillators and Systems

Abstract

In this paper, we consider a generalized second order nonlinear ordinary differential equation of the form $\ddot{x}+(k_1{x}^q+k_2)\dot{x}+k_3{x}^{2q+1}+k_4{x}^{q+1}+{\lambda }_{1}x=0$, where $k_i$'s, $i=1,2,3,4$, ${\lambda }_1$ and $q$ are arbitrary parameters, which includes several physically important nonlinear oscillators such as the simple harmonic oscillator, anharmonic oscillator, force-free Helmholtz oscillator, force-free Duffing and Duffing-van der Pol oscillators, modified Emden type equation and its hierarchy, generalized Duffing-van der Pol oscillator equation hierarchy and so on and investigate the integrability properties of this rather general equation. We identify several new integrable cases for arbitrary value of the exponent $q,q\in R$. The $q=1$ and $q=2$ cases are analyzed in detail and the results are generalized to arbitrary $q$. Our results show that many classical integrable nonlinear oscillators can be derived as sub-cases of our results and significantly enlarge the list of integrable equations that exist in the contemporary literature. To explore the above underlying results we use the recently introduced generalized extended Prelle-Singer procedure applicable to second order ODEs. As an added advantage of the method we not only identify integrable regimes but also construct integrating factors, integrals of motion and general solutions for the integrable cases, wherever possible, and bring out the mathematical structures associated with each of the integrable cases.

46 pages

References in corpus (1)

• New aspects of integrability of force-free Duffing-van der Pol oscillator and related nonlinear systems

Cited by in corpus (6)

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• A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators
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• Hypergeometric First Integrals of the Duffing and van der Pol Oscillators
• On the linearization of isochronous centre of a modified Emden equation with linear external forcing
• Effect of nonlocal transformations on the linearizability and exact solvability of the nonlinear generalized modified Emden type equations