Crisis Bifurcation Research Articles

Bistability and torus break-up in a semiconductor laser with phase-conjugate feedback

We present a detailed study of a route to chaos via quasiperiodicity on a torus in a semiconductor laser with phase-conjugate feedback. Highlighting the use of new tools that go far beyond mere simulation, we compute bifurcation diagrams and unstable manifolds of saddle periodic orbits. In this way, we show how a torus breaks up with a final sudden onset of chaos in a crisis bifurcation. We also identify regions of bistability between periodic solutions and other attractors in the system.

Intermittency in families of unimodal maps

We consider intermittency in one-parameter families of unimodal maps, induced by saddle node and boundary crisis bifurcations. In these bifurcations either a periodic orbit or a periodic interval disappears to give rise to chaotic bursts. We prove asymptotic formulae for the frequency with which orbits visit the region previously occupied by the attractor. For this, we extend Pianigiani's results on conditionally invariant measures for the logistic family to more general families.

Onset of resonance in chaotically driven systems

We investigate the onset of chaotic resonance (CR) behavior by studying the dynamics of chaotically driven bistable systems near the crisis bifurcation point. Our analysis reveals the existence and identification of a characteristic natural frequency associated with the dynamics of the system confirming that classical resonance is responsible for CR. We also classify the different routes via which CR can originate. The ability to manipulate the route to CR can be of importance in developing technological applications.

The evolution of crisis: Bifurcation and demise of a snapback repeller

This paper studies the dynamics of members of the two-parameter family of maps x → μx(1 − x v ), emphasizing the evolution from snapback repeller to crisis bifurcations. The example of the square root map v = 1 2 is taken to represent the subfamily where v is fixed and taken from the range 1 2 ≤ v ≤ 1 . A map from such a subfamily is shown to be conjugate with a map with negative Schwarzian derivative. This allows a characterization of crisis as the demise of a snapback repeller on a proper subinterval.