Numerical and experimental observation of Arnol’d resonance webs in an electrical circuit

• Arnol’d resonance webs generated in an electric circuit are investigated.
• Our model uses a piecewise-constant dynamical circuit.
• The explicit solutions are obtained in this circuit.
• The bifurcation structure is precisely studied through Lyapunov analysis.

An extensive bifurcation analysis of partial and complete synchronizations of three-frequency quasi-periodic oscillations generated in an electric circuit is presented. Our model uses two-coupled hysteresis oscillators and a rectangular wave forcing term. The governing equation of the circuit is represented by a piecewise-constant dynamics generating a three-dimensional torus. The Lyapunov exponents are precisely calculated using explicit solutions without numerically solving any implicit equation. By analyzing this extremely simple circuit, we clearly demonstrate that it generates an extremely complex bifurcation structure called Arnol’d resonance web. Inevitably, chaos is observed in the neighborhood of Chenciner bubbles around which regions generating three-dimensional tori emanate. Furthermore, the numerical results are experimentally verified.