Scaling and Critical Exponent for Gap Filling at Crisis in Chaotic Systems

The asymptotic behavior of a physical system depends on its parameters. Mathematically, such a behavior is determined by some asymptotic set in the phase space of the system. The asymptotic set can be either regular (e.g., periodic) or chaotic. When the asymptotic set is chaotic, the system exhibits sensitive dependence on initial conditions. That is, small changes in the initial state of the system can cause large changes in the final state. A chaotic set can be either attracting or nonattracting, the former corresponds to a chaotic attractor and the latter to a chaotic saddle which leads to transient chaos. When the asymptotic set is a chaotic attractor, the state of the system appears random all the time. When the asymptotic set is a nonattracting chaotic saddle, the system's state usually behaves randomly for a finite amount of time but eventually becomes regular. In the phase space, a chaotic saddle is typically a fractal set (Cantor-like set) of points with an infinite number of gaps of all scales in between. As a system parameter changes, qualitative changes in the asymptotic set of the system can occur. These changes can be, for example, from a chaotic attractor to a chaotic saddle, or from a chaotic set to a regular set, or vice versa.

The focus of this talk is on sudden changes in chaotic attractors as a system parameter changes. This is referred to as a "crisis." We will describe sudden enlargement of chaotic attractors, an event called the interior crisis. Before the crisis, the asymptotic set of the system is a small chaotic attractor. An interior crisis is triggered by the collision of a small chaotic attractor with a coexisting nonattracting chaotic saddle. After the crisis, the asymptotic set of the system is a larger chaotic attractor, and the original chaotic saddle is converted into part of the larger chaotic attractor. The gaps in-between various pieces of the chaotic saddle are densely filled after the crisis. This is referred to as "gap filling." We argue that gap filling is caused by the birth of an infinite number of unstable periodic orbits which do not exist before the crisis. As a consequence, we expect the topological entropy, which quantifies the number of periodic orbits of a chaotic set, to grow after the crisis. We give a quantitative scaling theory for the growth of the topological entropy. The theory is confirmed by numerical experiments, and it is expected to be universal, i.e., it holds regardless details of the system.