Arnol’d Tongues, Devil’s Staircase, and Self-Similarity in the Driven Chua’s Circuit

Abstract

Empirical recurrent relations, governing the structure of the devil’s staircase in the driven Chua’s circuit are given, which reflect the self-similar structure in an algebraic form. In particular, it turns out that the same formulas hold for both winding and period numbers, but with different “initial conditions”. Some of the finer details such as period-doubling along with numerous coexistence phenomena within staircases of mode-locked states have been revealed by computing high-resolution bifurcation diagrams.

Resonance Pockets of Hill's Equations with Two-Step Potentials

“…One may find in [3] the historical development of the study for resonance regions of Hill's equations. For resonance tongues of certain nonlinear systems, one can refer to [2,5,7,9,13]. A geometric explanation using singularity theory to the appearance of resonance pockets is given in [3] and has been developed in [2,4].…”

Stable Oscillations and Devil's Staircase in the Van Der Pol Oscillator

“…For E large enough, he found a sequence of subharmonics 1, 1/2, 1/3, 1/4, 1/5, 1/6. More recently Pivka et al [1994] performed a numerical investigation of the driven Chua's circuit, which belongs to the class of van der Pol oscillators. They discussed in detail the period-adding law and the corresponding Devil's-staircase structure that results from varying the driving frequency.…”

Resonant Phase Patterns in a Reaction-Diffusion System

“…Entrainment of oscillating systems to external periodic stimuli has been well studied in one-dimensional maps [1], ordinary differential equation models [2], and experiments [3]. The entrainment of spiral tip trajectories in spatially extended excitable systems has also been studied [4].…”