Intermittent transitions between multiple dynamical states are characteristic nonlinear phenomena in dynamical systems. It is important to understand a mechanism of an onset of intermittency in mathematical models, because it often replicates an observable phenomenon in physical world. Among several types of intermittency, we focus on crisis-induced intermittency in this tutorial.

In coupled chaotic maps which are widely used in modeling networks of dynamical elements, coexistence of multiple attractors is not uncommon. As a system parameter is varied, such a system often exhibits a sudden crisis inducing intermittent behaviors. We demonstrate that a crisis can be understood by a contact between attractors and a fractal basin boundary as well as an emergence of a snap-back repeller in two coupled chaotic maps. The scenario for the crisis is also illustrated with qualitative changes in basin structure and quantitative changes of fractal dimension of the basin boundary.

The approaches to the crisis is successfully applied to an analysis of a global bifurcation inducing itinerant memory dynamics in a chaotic neural network. In addition to the viewpoints for crisis-induced intermittency, this tutorial provides several basic concepts such as invertible and non-invertible maps, smooth and fractal basin boundaries, fractal dimension, and basin bifurcations in discrete-time dynamical systems.