Transient dynamics in strongly nonlinear systems: optimization of initial conditions on the resonant manifold

TL;DR

This paper analyzes the energy exchange dynamics in a coupled oscillator system with impact constraints, using asymptotic methods to predict the critical coupling for the transition from localized to delocalized energy states.

Contribution

It introduces a novel asymptotic approach to analyze the slow-flow in nonlinear oscillators with impact constraints, enabling precise prediction of the delocalization transition.

Findings

Derived an energy-action relation for impact-oscillators.

Predicted the critical coupling value for energy delocalization.

Provided a method to optimize initial conditions on the resonant manifold.

Abstract

We consider a system of two linear and linearly coupled oscillators with ideal impact constraints. Primary resonant energy exchange is investigated by analysis of the slow-flow using the action-angle (AA) formalism. Exact inversion of the action-energy dependence for the linear oscillator with impact constraints is not possible. This difficulty, typical for many models of nonlinear oscillators, is circumvented by matching the asymptotic expansions for the linear and impact limits. The obtained energy-action relation enables the complete analysis of the slow-flow and the accurate description of the critical delocalization transition. The transition from the localization regime to the energy-exchange regime is captured by prediction of the critical coupling value. Accurate prediction of the delocalization transition requires detailed account of the coupling energy with appropriate re-definition and optimization of the limiting phase trajectory on the resonant manifold.