Fractal Basin Boundaries, Long-Lived Chaotic Transients, and Unstable-Unstable Pair Bifurcation

In this paper we consider a new type of bifurcation to chaotic motion. This bifurcation is characterized by the simultaneous appearance of a pair of unstable fixed points or periodic orbits, a, fractal (i.e., nondifferentiable) basin boundary, and a chaotic attractor (also commonly called a strange attractor). Just prior to this type of bifurcation, transient behavior with a chaotic character can take place, and these chaotic transients can be extremely long. The existence of such remarkably long-lived chaotic transients may have important implications for experiments on chaotic systems. To motivate our considerations, we note that the general question of how chaotic attractors arise as a system parameter is varied is of great fundamental interest. According to conventional wisdom, only a small number of distinct types of chaotic attractor onsets are generally seen. Among these are period doubling, ' intermittency, ' and crises. ' In the first two a nonchaotic attracting orbit evolves into a chaotic one. On the other hand, in a crisis a chaotic transient converts into a chaotic attractor. As a concrete example of the latter, say that when p, a parameter of the system, is in the range p & p, a nonchaotic attractor exists. In addition, for p & p, chaotic transients are also observed to occur before orbits settle into the nonchaotic attractor. As p approaches p from above the average duration of a chaotic transient approaches infinity, and past p a chaotic attractor appears by conversion of the chaotic transient. For p &p the chaotic and nonchaotic attractors coexist, each with its own separate basin of attraction. (The basin of attraction of an attractor is the set of initial conditions whose trajectories asymptotically approach that attractor as time increases. ) In general there will be some boundary separating these two basins. The question of how the chaotic attractor is created (as p decreases through p }, or, inversely, destroyed (as p increases through p } has only recently been addressed. ' Generally, the disappearance of the chaotic attractor occurs via a