Common features of the onset of the persistent chaos in nonlinear oscillators: a phenomenological approach (Q1610729)

This is a comparative computational study that covers three archetypal nonlinear oscillators: the twin-well oscillator (oscillator with two minima and one maximum of the potential energy), driven by an external harmonic force, and two types of oscillators with 'infinite' number of potential energy wells: the pendulum with parametric periodic excitation, and the pendulum driven by an external periodic force. This work is focused on the phenomena that occur right prior to critical system parameters, the parameters at which all regular asymptotic solutions disappear and are replaced by irregular, unpredictable over time and sensitive to initial conditions steady-state chaotic solutions, and reveals common features of the system response properties and of the bifurcational scenarios, prior to the onset of chaos. It is pointed out that, in the three systems considered, the onset of chaos is preceded by two and only two asymmetric periodic attractors, which are simultaneously annihilated via the period-doubling-crisis scenario, and that persistent chaos can be viewed as an irregular combination of the `crossing the potential barrier' and the oscillatory component of motion. This study is performed by numerical methods and is mainly based on the software package \textit{Dynamics} [\textit{H. E. Nusse} and \textit{J. A. Yorke}, Dynamics: numerical explorations (Springer, New York) (1998; Zbl 0895.58001)].