Final-state sensitivity characterized by phase transition in singularity spectra of intermingled basins.

Abstract

Final-state sensitivity in a system with intermingled basins is investigated. Under a scaling assumption a dimension of the basins, referred to as the conditional external dimension, is introduced by which the uncertainty exponent is expressed. For an analytically tractable model, which is not a skew product type system, a multifractal analysis on the basin structure is performed. It is shown that the scaling assumption is valid and the conditional external dimension as the left end-point value of the singularity spectrum is determined by the transient motions converging to neither of the chaotic attractors. It is also shown that such transient motions bring about a phase transition in the singularity spectrum.