V. K. Chandrasekar

A simple and unified approach to identify integrable nonlinear oscillators and systems

In this paper, we consider a generalized second-order nonlinear ordinary differential equation (ODE) of the form x+(k1xq+k2)x+k3x2q+1+k4xq+1+λ1x=0, where ki’s, i=1,2,3,4, λ1, 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∊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 subcases of our results and significantly enlarge the list of integrable equations that exists in the contemporary literature. T...

On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator

Using the modified Prelle-Singer approach, we point out that explicit time independent first integrals can be identified for the damped linear harmonic oscillator in different parameter regimes. Using these constants of motion, an appropriate Lagrangian and Hamiltonian formalism is developed and the resultant canonical equations are shown to lead to the standard dynamical description. Suitable canonical transformations to standard Hamiltonian forms are also obtained. It is also shown that a possible quantum mechanical description can be developed either in the coordinate or momentum representations using the Hamiltonian forms.

On the general solution for the modified Emden-type equation

In this paper, we demonstrate that the modified Emden type equation (MEE),

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

ddot{x}+alpha xdot{x}+beta x^3=0

A unification in the theory of linearization of second-order nonlinear ordinary differential equations

, is integrable either explicitly or by quadrature for any value of

A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators

alpha

Nonstandard conserved Hamiltonian structures in dissipative/damped systems: Nonlinear generalizations of damped harmonic oscillator

and

Extended Prelle-Singer Method and Integrability/Solvability of a Class of Nonlinear nth Order Ordinary Differential Equations

beta

Dynamics of a Completely Integrable N-Coupled Lienard Type Nonlinear Oscillator

. We also prove that the MEE possesses appropriate time-independent Hamiltonian function for the full range of parameters

Interplay of symmetries, null forms, Darboux polynomials, integrating factors and Jacobi multipliers in integrable second-order differential equations

alpha