Analysis of the multi-phenomenal nonlinear system : Testing Integrability and detecting chaos

Abstract

As a large extension in Hamiltonian form, the system of a PT symmetric dimer of coupled nonlinear oscillators is developed. This system provides an explanation for a number of problems with Hamiltonian dynamics. Integrability is evaluated in the Painlevé sense of the system. The system reported twelve P-cases. First integrals of planar motion are constructed explicitly for each integrable case to show the Liouvillian integrability of the equations of motion. A mixture of numerical approaches is used to test the theoretical conclusions in order to identify the nature of orbits and evaluate the system’s transition from order to chaos. These techniques consist of the Poincaré Section Surface, the maximum Lyapunov Exponent, and the Smaller Alignment Index.