Finding numerically Newhouse sinks near a homoclinic tangency and investigation of their chaotic transients

Abstract

For Hénon map of nearly classical parameter values, we search numerically for Newhouse sinks. We show how to find successively the Newhouse sinks of higher period, which is the estimation of coordinates of the sinks from power laws of properties of the sinks, and investigate numerically a sequence of sinks of period from 8 to 60 that we obtained. We also show how to verify the existences of obtained sinks by interval arithmetic. The sinks of period from 8 to 14 from among our obtained sinks was verified mathematically.

In the case that we observed, when the sink exists, most orbits converge to it, and the orbit that seems to be Hénon attractor is not an attractor but just a long chaotic transient. The narrowness of the main bands of basins of the sinks causes the long chaotic transients. We also investigate numerically the chaotic transients and their rambling time.