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

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.

References in corpus (10)

• Cooling a nanomechanical resonator with quantum back-action
• Basins of attraction of a nonlinear nanomechanical resonator
• Mass Detection with Nonlinear Nanomechanical Resonator
• Classical to Quantum Transition of a Driven Nonlinear Nanomechanical Resonator
• Intrinsic localized modes in parametrically-driven arrays of nonlinear resonators
• Activation barrier scaling and crossover for noise-induced switching in a micromechanical parametric oscillator
• Paths of fluctuation induced switching
• Classical to quantum transition of a driven nonlinear nanomechanical resonator
• Pattern selection in parametrically-driven arrays of nonlinear resonators
• Driven localized excitations in the acoustic spectrum of small nonlinear macroscopic and microscopic lattices

Cited by in corpus (6)

• Persistent coherent beating in coupled parametric oscillators
• A Passive Phase Noise Cancellation Element
• Theory of coupled parametric oscillators beyond coupled Ising spins
• A Generic Rotating-Frame-Based Approach to Chaos Generation in Nonlinear MEMS NEMS Resonators
• Homoclinic chaos in a pair of parametrically-driven coupled SQUIDs
• Dynamics of a suspended nanowire driven by an ac Josephson current in an inhomogeneous magnetic field