Crisis (dynamical systems)

In dynamical systems, a crisis is a sudden qualitative change in the behavior of a chaotic attractor. Crises typically involve a sudden change in the size or nature of the attractor, or its sudden destruction. They are bifurcations that occur when a chaotic attractor interacts with an unstable fixed point, unstable periodic orbit, or another attractor in the system's phase space. The term "crisis" was coined by Celso Grebogi, Edward Ott, and James A. Yorke in the early 1980s.

There are several types of crises, categorized by the nature of the change in the attractor. The three primary types are:

• Boundary Crisis: A boundary crisis occurs when a chaotic attractor collides with its own basin boundary. At the point of the crisis, the attractor is suddenly destroyed. Trajectories that were previously confined to the attractor will now escape to infinity or some other region of the phase space. Before the crisis, the attractor may exist in a region of parameter space with a chaotic saddle near its edge. At the crisis point, the attractor merges with the chaotic saddle, and trajectories can then "leak" out of the region.

• Interior Crisis: An interior crisis involves a sudden expansion of a chaotic attractor. Before the crisis, the attractor exists but is confined to a relatively small region of phase space. At the crisis, the attractor suddenly expands to occupy a much larger region. This expansion occurs because the attractor collides with an unstable fixed point or periodic orbit that was previously in its interior. An interior crisis can be either a merging crisis, where the chaotic attractor merges with another pre-existing, but previously disconnected, attractor, or a widening crisis, where the chaotic attractor suddenly becomes larger in extent.

• Riddled Basin Crisis: A riddled basin crisis is a special case of a boundary crisis where the basin of attraction for an attractor becomes riddled with holes belonging to the basin of attraction of another attractor or infinity. Before the crisis, most points in the basin of attraction of the original attractor will eventually converge to it. However, interspersed among those points are tiny regions (the "riddles") from which trajectories will eventually escape to another attractor. At the crisis, the riddling becomes so pervasive that the basin of attraction is no longer useful for predicting the long-term behavior of the system. Practically, this means that any small perturbation can cause the system to switch to a different attractor or to escape altogether.

Crises are important because they represent a form of non-smooth bifurcation that can dramatically alter the behavior of a dynamical system. They are found in a wide variety of physical, chemical, and biological systems. Understanding crises is crucial for predicting and controlling the behavior of chaotic systems. Characterizing a crisis typically involves determining the parameter value at which it occurs and analyzing the dynamics before and after the crisis. This may involve calculating Lyapunov exponents, fractal dimensions, and other measures of chaoticity.