  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.
