Fractal Patterns in the Parameter Space of Bi-stable Duffing Oscillator

TL;DR

This study uncovers fractal patterns in the parameter space of a bi-stable Duffing oscillator, revealing complex boundary structures and steady-state behaviors, with implications for designing advanced mechanical metamaterials.

Contribution

It is the first to identify and analyze fractal patterns in the parameter space of a bi-stable Duffing oscillator, including boundary fractal dimensions and behavior categorization.

Findings

Fractal boundaries separate intra-well and inter-well behaviors.

Three steady-state types identified: switching, reverting, vacillating.

Fractal patterns also appear in continuous bi-stable systems modeled by single-mode approximation.

Abstract

We study the dissipative bi-stable Duffing oscillator with equal energy wells and observe fractal patterns in the parameter space of driving frequency, forcing amplitude, and damping ratio. Our numerical investigation reveals the Hausdorff fractal dimension of the boundaries that separate the oscillator's intra-well and inter-well behaviors. Furthermore, we categorize the inter-well behaviors as three steady-state types: switching, reverting, and vacillating. While fractal patterns in the phase space are well-known and heavily studied, our results point to a new research direction about fractal patterns in the parameter space. Another implication of this study is that the vibration of a continuous bi-stable system modeled using a single-mode approximation also manifests fractal patterns in the parameter space. In addition, our findings can guide the design of next-generation bi-stable and multi-stable mechanical metamaterials.