2. Crisis-induced intermittency

The family of two coupled chaotic maps introduced in the previous section exhibits intermittent behaviors as the system parameter a is varied (b is fixed at 1.9). The onset of intermittency is induced by a global bifurcation called a crisis [2] or an explosion [14]. Due to the crisis, two distant hyperchaotic attractors out of the diagonal merge into a larger chaotic attractor without a direct contact of them. This section describes the changes in the dynamics with typical orbital motions before and after the crisis.

(1) Coexistence state before the crisis

There coexist two symmetrical chaotic attractors out of the diagonal. An orbit asymptotes to one of them, but it is difficult to predict the final state from a given initial condition. Even two close initial points can finally settle in different attractors. This implies a complicated basin structure in the phase plane.

(2) Invariant manifolds at the crisis

Before the crisis, unstable invariant manifolds of the repelling fixed point P on the diagonal departs in the opposite direction and leave towards the respective attractor. Once each manifold enters the attracting region, it moves around in the region forever. At the crisis, the unstable manifolds can escape from the attracting regions and return to the vicinity of the repeller P, respectively, as shown in Fig. 6. Hence, P is a snap-back repeller [11]. The first emergence of the snap-back repeller at the crisis corresponds to a homoclinic bifurcation in a continuous dynamical system.

Fig.6: Unstable manifolds of the snap-back repeller P on the diagonal.

(3) Intermittency immediately after the crisis

After the crisis, the unstable manifolds make infinitely many intersections and thereby a typical orbit iterate intermittent transitions between the regions above and below the diagonal. As the parameter a is increased from the crisis point, the lifetime in each region decays exponentially.