Dynamics of heteroclinic tangles with an unbroken saddle connection

Abstract

This paper studies a second-order differential equation with two heteroclinic solutions to two saddle fixed points. When an equation is periodically perturbed, one heteroclinic solution generates tangle while the other remains unbroken. We illustrate chaotic dynamics in the sense of Smale horseshoes and Henon-like attractors with SRB measures. More explicitly, we obtain three different dynamical phenomena, namely the transient heteroclinic tangles containing no physical measures, heteroclinic tangles dominated by sinks representing stable dynamical behavior, and heteroclinic tangles with Henon-like attractors admitting SRB measures representing chaos. We also demonstrate that three types of phenomena repeat periodically as the forcing magnitude goes to zero.