Crisis (dynamical systems)

Crisis in dynamical systems theory is a sudden discontinuous change in the structure of a chaotic attractor caused by its collision with an unstable periodic orbit or saddle point. Coined by Celso Grebogi, Edward Ott, and James Yorke in 1982, the term describes three distinct types of catastrophic bifurcation: boundary crisis (destruction of the attractor as it hits a basin of attraction boundary), interior crisis (sudden expansion of the attractor as it collides with an unstable orbit inside its basin), and attractor merging crisis (two separate chaotic attractors fuse into one).

Crisis is a route to chaos as well as a route from chaos. In a boundary crisis, a chaotic attractor and its basin vanish together; trajectories that once settled into aperiodic motion now escape to a distant attractor or diverge to infinity. The phenomenon is the dynamical analog of a phase transition in statistical mechanics: a small parameter change produces a qualitative reorganization of the system's global behavior. Following a crisis, the system often exhibits crisis-induced intermittency — episodes of nearly periodic behavior punctuated by chaotic bursts as trajectories repeatedly approach and rebound from the destroyed attractor's remnants.