Abstract:
The Duffing model features a linearly damped and periodically driven oscillator, moving in the field of force of the $\phi^4$ double-well potential. The model has been instrumental in theoretical investigations of dynamical properties of systems with parity-conserving symmetry, where a bistable substrate connects two metastable states separated by an energy barrier. Physical systems of interest include nonlinear feedback-controlled mass-spring-damper oscillators, active hysteresis circuits (such as memristors), protein chains prone to hydrogen bond-mediated conformational transitions, centro-symmetric crystals and so on. The present work proposes a model for a linearly damped and periodically driven oscillator, moving in an hyperbolic potential. The hyperbolic potential has two degenerate minima that can be smoothly tuned by varying a real parameter, leaving unaffected the height of the energy barrier. We examine the system dynamics in the absence and presence of damping and forcing. In the regime of free motion the exact analytical solution is obtained, and shown to reproduce a periodic structure consisting of regularly spaced “sech” solitons described by a Jacobi elliptic function. When damping and forcing are taken into consideration, numerical simulations suggest that the system dynamics can transit from periodic to chaotic regimes or vice-versa, via period-doubling or period-halving bifurcations. The Poincaré map of the hyperbolic Duffing model exhibits the well-known characteristic signatures of chaos precursors of the standard Duffing model, which turns out to be a particular physical context of the bistable oscillator model with the hyperbolic double-well potential.
