On repellers in quasi-periodically forced logistic map system

Abstract

We propose a method to identify and to locate "repellers'' in quasi-periodically forced logistic map (QPLM), using a kind of Morse decomposition of nested attracting invariant sets. In order to obtain the invariant sets, we use an auxiliary 1+2-dimensional skew-product map system describing the evolution of a line segment in the phase space of QPLM. With this method, detailed structure of repellers can be visualized, and the emergence of a repeller in QPLM can be detected as an easily observable bifurcation in the auxiliary system. In addition to the method to detect the repellers, we propose a new numerical method for distinguishing a strange non-chaotic attractor (SNA) from a smooth torus attractor, using a correspondence between SNAs in QPLM and attractors with riddled basin in the auxiliary system.