Nonlinear dynamics of hidden modes in a system with internal symmetry

TL;DR

This paper investigates how hidden internal modes in a symmetric discrete dynamical system can be excited through nonlinear effects, using Floquet analysis and a novel Fourier series method to understand their stability and energy exchange.

Contribution

It introduces a new Fourier series method for analyzing nonlinear normal modes without explicit quadrature integration, revealing how weak nonlinearity excites hidden modes via parametric resonance.

Findings

Hidden modes can be excited by weak cubic nonlinearity.

A novel Fourier series method improves accuracy in Floquet analysis.

Conditions for energy exchange and mode instability are characterized.

Abstract

We consider a discrete dynamical system with internal degrees of freedom (DOF). Due to the symmetry between the internal DOFs, certain internal modes cannot be excited by external forcing (in a case of linear interactions) and thus are considered "hidden". If such a system is weakly asymmetric, the internal modes remain approximately "hidden" from the external excitation, given that small damping is taken into account. However, already in the case of weak cubic nonlinearity, these hidden modes can be excited, even as the exact symmetry is preserved. This excitation occurs through parametric resonance. Floquet analysis reveals instability patterns for the explored modes. To perform this analysis with the required accuracy, we suggest a special method for obtaining the Fourier series of the unperturbed solution for the nonlinear normal mode. This method does not require explicit integration of the arising quadratures. Instead, it employs expansion of the solution at the stage of the implicit quadrature in terms of Chebyshev polynomials. The emerging implicit equations are solved by using a fixed-point iteration scheme. Poincar\'{e} sections help to clarify the correspondence between the loss of stability of the modes and the global structure of the dynamical flow. In particular, the conditions for intensive energy exchange in the system are characterized.