Chaotic phase similarities and recurrences in a damped-driven Duffing oscillator

We report strong evidence of remarkably close periodic repetitions of the structuring of the parameter space of a damped-driven Duffing oscillator as the amplitude of the drive increases. Families of period-adding cascades and some intricate networks of periodic oscillations embedded in chaotic phases are also found to recur closely as the driving force grows. Such surprising regularities suggest that some hitherto unknown renormalization mechanism may be operating in higher codimension, controlling the alternation of chaos and order in parameter space of certain flows.

Archetypal oscillator for smooth and discontinuous dynamics

We propose an archetypal system to investigate transitions from smooth to discontinuous dynamics. In the smooth regime, the system bears significant similarities to the Duffing oscillator, exhibiting the standard dynamics governed by the hyperbolic structure associated with the stationary state of the double well. At the discontinuous limit, however, there is a substantial departure in the dynamics from the standard one. In particular, the velocity flow suffers a jump in crossing from one well to another, caused by the loss of local hyperbolicity due to the collapse of the stable and unstable manifolds of the stationary state. In the presence of damping and external excitation, the system has coexisting attractors and also a chaotic saddle which becomes a chaotic attractor when a smoothness parameter drops to zero. This attractor can bifurcate to a high-period periodic attractor or a chaotic sea with islands of quasiperiodic attractors depending on the strength of damping.

Indeterminate jumps to resonance from a tangled saddle-node bifurcation

We identify in this paper a new type of bifurcation which yields an unpredictable outcome and therefore seems to be an important new ingredient of nonlinear dissipative dynamics. It arises when the unstable manifold of the saddle of a saddle-node bifurcation is heteroclinically tangled with the inset of a distant saddle which is itself homoclinically tangled so that it forms a fractal basin boundary between two remote attractors. At the saddle-node fold, a slowly evolving system will thus find itself sitting precisely on a fractal basin boundary, and in the presence of even infinitesimal noise we cannot predict to which of the two remote attractors the system will jump. We show here that such an indeterminate tangled saddle-node bifurcation is a common ingredient in the resonance of softening systems.

Birkhoff signature change: a criterion for the instability of chaotic resonance

For periodically forced nonlinear oscillators permitting escape from a potential well a relation is observed between two well-known phenomena, the period-doubling cascade leading to the chaotic escape of the resonant attractor and the complex dynamics associated with the creation of a structurally unstable homoclinic orbit. The particular homoclinic orbit is identified as that created at the initial change of the period one Birkhoff signature of the invariant manifolds of the hilltop saddle. The primary resonant attractor may thus be viewed as the period one simple Newhouse orbit. Significant subharmonic and superharmonic escape events may likewise be associated with nearby Birkhoff signature changes. Significant information about the global dynamics may thus be obtained with little numerical effort by inspection of the signatures of the invariant manifolds of the hilltop saddle.

Steady states in the twin-well potential oscillator: Computer simulations and approximate analytical studies

The paper is focused on the phenomena of various steady-state oscillations exhibited by the twin-well potential system. Regions of existence of different attractors in the system parameter domain are examined and a picture book of different steady states for fixed damping and forcing is presented: 20 different combinations of single or coexisting, small orbit or large orbit, periodic and chaotic attractors are displayed. Computer simulations are followed by an approximate analytical analysis: A study of various forms of instability of periodic solutions gives close form approximate criteria for occurrence of T-periodic small orbit and large orbit oscillations, and for cross-well chaos.