Homoclinic orbits and chaos in a pair of parametrically-driven coupled nonlinear resonators

TL;DR

This paper analyzes the dynamics of coupled nonlinear resonators, revealing the existence of homoclinic and Shilnikov orbits that indicate chaos, using analytical and numerical methods in MEMS and NEMS systems.

Contribution

It provides explicit conditions for the persistence of homoclinic orbits and chaos in parametrically-driven coupled nonlinear resonators through a combination of multiple-scales analysis and Melnikov theory.

Findings

Existence of homoclinic orbits in the slow dynamics.

Persistence of Shilnikov orbits indicating chaos.

Analytical results confirmed by numerical simulations.

Abstract

We study the dynamics of a pair of parametrically-driven coupled nonlinear mechanical resonators of the kind that is typically encountered in applications involving microelectromechanical and nanoelectromechanical systems (MEMS & NEMS). We take advantage of the weak damping that characterizes these systems to perform a multiple-scales analysis and obtain amplitude equations, describing the slow dynamics of the system. This picture allows us to expose the existence of homoclinic orbits in the dynamics of the integrable part of the slow equations of motion. Using a version of the high-dimensional Melnikov approach, developed by Kovacic and Wiggins [Physica D, 57, 185 (1992)], we are able to obtain explicit parameter values for which these orbits persist in the full system, consisting of both Hamiltonian and non-Hamiltonian perturbations, to form so-called Shilnikov orbits, indicating a loss of integrability and the existence of chaos. Our analytical calculations of Shilnikov orbits are confirmed numerically.