Two-dimensional isochronous nonstandard Hamiltonian systems

TL;DR

This paper introduces a new class of two-dimensional nonstandard Hamiltonian systems that exhibit isochronous and quasi-periodic oscillations, extending the scalar modified Emden equation to a coupled system with Hamiltonian structure.

Contribution

It generalizes the scalar modified Emden equation to two dimensions, identifying a class of Hamiltonian systems with isochronous behavior and PT symmetry.

Findings

The systems admit a Hamiltonian description.

They exhibit periodic and quasi-periodic oscillations.

The systems are derived from a generalized two-dimensional Lifcnard-type equation.

Abstract

We identify a generic class of two dimensional nonstandard Hamiltonian systems which exhibit isochronous behaviour. This class of systems belongs to the two dimensional mixed Li\'enard- type equations and is obtained by generalizing the scalar modified Emden equation (MEE) to two dimensions. We show that the generalized class of equations admits a Hamiltonian description and exhibits periodic and quasi-periodic oscillations for suitable choice of parameters and also $\emph PT$ symmetric property.