Stabilization of the coupled pendula chain under parametric PT-symmetric driving force

TL;DR

This paper derives a novel coupled PT-symmetric discrete nonlinear Schrödinger equation for a chain of coupled pendula under parametric driving, analyzing stability and demonstrating how destabilization leads to a dynamic equilibrium with finite oscillations.

Contribution

It introduces a new PT-symmetric model for coupled pendula and analyzes its stability properties near parametric resonance.

Findings

Identifies parameter ranges for stability of the zero equilibrium.

Shows destabilization leads to finite amplitude oscillations.

Numerical simulations confirm theoretical stability and destabilization scenarios.

Abstract

We consider a chain of coupled pendula pairs, where each pendulum is connected to the nearest neighbors in the longitudinal and transverse directions. The common strings in each pair are modulated periodically by an external force. In the limit of small coupling and near the 1:2 parametric resonance, we derive a novel system of coupled PT-symmetric discrete nonlinear Schrodinger equation, which has Hamiltonian symmetry but has no gauge symmetry. By using the conserved energy, we find the parameter range for the linear and nonlinear stability of the zero equilibrium. Numerical experiments illustrate how destabilization of the zero equilibrium takes place when the stability constraints are not satisfied. The central pendulum excites nearest pendula and this process continues until a dynamical equilibrium is reached where each pendulum in the chain oscillates at a finite amplitude.