Multi-stability in Doubochinski's Pendulum

TL;DR

This paper investigates multistability in Doubochinski's Pendulum, revealing how nonlinear forcing leads to multiple stable periodic solutions, with transitions influenced by symmetry and a proposed control strategy.

Contribution

It introduces the concept of amplitude quantization as self-adaptive subharmonic resonance and analyzes the relationship between symmetry and subharmonic resonance in nonlinear oscillators.

Findings

Multiple discrete periodic solutions identified in Doubochinski's Pendulum.

Transition between multistable modes is irreversible and can be controlled.

Subharmonic resonance frequency relates to the symmetry of the driving force.

Abstract

The widespread phenomena of multistability is a problem involving rich dynamics to be explored. In this paper, we study the multistability of a generalized nonlinear forcing oscillator excited by $f(x)cos \omega t$. We take Doubochinski's Pendulum as an example. The so-called "amplitude quantization", i.e., the multiple discrete periodical solutions, is identified as self-adaptive subharmonic resonance in response to nonlinear feeding. The subharmonic resonance frequency is found related to the symmetry of the driving force: odd subharmonic resonance occurs under even symmetric driving force and vice versa. We solve the multiple periodical solutions and investigate the transition and competition between these multi-stable modes via frequency response curves and Poincare maps. We find the irreversible transition between the multistable modes and propose a multistability control strategy.