Newhouse sinks in the self-similar bifurcation structure

Abstract

The numerical analyses of the dynamics of periodically driven Toda oscillator suggest the following features. Primary Newhouse orbits (sinks and saddles) are born in sequence when the oscillator proceeds through various subharmonic resonance regions. As the control parameter is swept in the neighboring parameter space of the homoclinic tangency for a primary saddle, first order secondary Newhouse sinks are born around the corresponding primary sink in a series of period n-tupling (n>2) processes. Higher order secondary Newhouse sinks are similarly born, in a recurrent manner, around those first-order secondary sinks, constituting a self-similar bifurcation structure in the parameter space. Each higher (say nth) order secondary Newhouse sink appears and undergoes sequence of period doubling (before being destroyed by crises), within a small subinterval of the control parameter window where the (n-1)th-order secondary Newhouse sink exists. The nth-order secondary Newhouse orbits appear in the basin of the (n-1)th-order secondary Newhouse sink. Thus, the higher-order secondary sinks appear with progressively smaller basins intertwined with the basins of lower-order secondary sinks.