Noise and Transient Chaos
In this chapter, we treat transiently chaotic dynamical systems under the influence of noise, focusing on a number of physical phenomena. Firstly, we will demonstrate that noise can increase the lifetime of transient chaos and induce dynamical interactions among different invariant sets of the system. As a result, the stationary distributions of dynamical variables in a noisy system can be much more extended in the phase space than those in the corresponding deterministic system. Secondly, if the system has a nonchaotic (e.g., periodic) attractor but there is transient chaos due to a coexisting nonattracting chaotic set, noise can cause a trajectory to visit both the original attractor and the chaotic saddle, leading to an extended chaotic attractor. This is the phenomenon of noise-induced chaos, which can arise, for instance, when the dynamical system is in a periodic window. Of particular interest is how the Lyapunov exponent and other ergodic averages scale with the noise strength. Thirdly, if the system has a chaotic attractor, noise can cause trajectories on the attractor to move out of its basin of attraction so that either the attractor is enlarged or the originally attracting motion becomes transient. This is the phenomenon of noise-induced crisis, dynamically due to noise-induced heteroclinic or homoclinic tangencies that cause the attractor to collide with its own basin boundary. An issue of both theoretical and experimental interest is how the average transient lifetime depends on the noise strength.