Self-similar organization of Gavrilov-Silnikov-Newhouse sinks.

The numerical analyses of the Hénon map suggest the following features. As we increase the value of the control parameter around each stable period of the period-1 branch, an infinitely large series of period n-tupled saddle nodes appears in the following sequence (n= ...,5,4,3). The limit of each series is the infinitely large set of homoclinic points, created at the homoclinic tangency for the respective flip saddle (boundary saddle in the case of period 1). These observations are in good agreement with the predictions of Gavrilov, Silnikov, and Robinson. Each newly created sink, referred to as Gavrilov-Silnikov (GS) sink, later constitutes a first-order secondary cascade. The flip (boundary) saddles of these cascades also exhibit homoclinic tangency. Past such tangency, around the respective GS sink, an infinitely large series of period n-tupled saddle nodes (n= ...,5,4,3) seems to appear in a similar manner. The newly created GS sinks later constitute second-order secondary cascades. These phenomena, comprised of the homoclinic tangency of a flip (boundary) saddle, followed by the sequential appearance of an infinitely large sequence of period n-tupled saddle nodes around the respective GS sink, appear to recur in a self-similar manner, creating higher-order and further higher-order GS sinks and the associated secondary cascades. Each secondary cascade survives within a small subinterval of the control parameter window where the respective GS sink from the immediate lower-order secondary cascade exists. These processes appear to continue ad infinitum. Therefore, in the limiting conditions, an infinitely large sequence of sinks may simultaneously coexist in the phase space for an infinitely large number of control parameter values. These observations are in good agreement with the predictions of Newhouse. Thus, the GS sinks may be identified as Gavrilov-Silnikov-Newhouse (GSN) sinks that are organized in a self-similar manner in the phase and parameter space. These features are very similar to those we recently observed in a periodically forced, damped Toda oscillator [B. K. Goswami, Phys. Rev. E 62, 2068 (2000)]. Since, the Hénon map and Toda oscillator are standard models (one from the maps and the other from the oscillators), our observations may provide some strong evidences towards universality in the self-similar organization of GSN sinks in the low-dissipative limit.