Fractal dimensions and ƒ(α) spectrum of chaotic sets near crises

Abstract

We study numerically the behaviour of the generalized dimensions D( q and the ƒ(α) spectrum in dependence on a control parameter in the parametrically driven, damped pendulum. We find a continuous transition of the D( q) of a chaotic attractor near a boundary crisis to those characterizing a chaotic saddle into which the attractor is converted when the crisis occurs. At an interior crisis a chaotic saddle collides with a small chaotic attractor and both chaotic sets merge to a large chaotic attractor. In the vicinity of the crisis-value of the control parameter the D( q) of the large attractor are close to the D( q) of the small attractor for positive q and near to those of the chaotic saddle for non-positive q. Correspondingly, the ƒ(α) spectrum of the large chaotic attractor near the crisis-value is very broad and has a typical phase-transition-like shape.