An attracting set A in a trapping region D is defined as a nonempty closed set formed from some open neighborhood,
We mentioned before that for a dissipative bouncing ball system the trapping region is contracting, so the open neighborhood typically consists of a collection of smaller and smaller regions as it approaches the attracting set.
An attractor is an attempt to define the asymptotic solution of a dynamical system . It is that part of the solution that is left after the ``transient'' part is thrown out. Consider Figure 1.14, which shows the approach of several phase space trajectories of the bouncing ball system toward a period one cycle.
Figure 1.14: A periodic attractor and its transient.
The orbits appear to consist of two parts: the transient--the initial part of the orbit that is spiraling toward a closed curve--and the attractor--the closed periodic orbit itself.
In the previous section we saw examples of several different types of attractors. For small table amplitudes, the ball comes to rest on the table. For these equilibrium solutions the attractor consists of a single point in the phase space of the table's reference frame. At higher table amplitudes periodic orbits can exist, in which case the attractor is a closed curve in phase space. In a dissipative system this closed curve representing a periodic motion is also known as a limit cycle. At still higher table amplitudes, a more complicated set called a strange attractor can appear. The phase space plot of a strange attractor is a complicated curve that never quite closes. After a long time, this curve appears to sketch out a surface. Each type of attractor--a point, closed curve, or strange attractor (something between a curve and a surface)--represents a different type of motion for the system--equilibrium, periodic, or chaotic.
Except for the equilibrium solutions, each of the attractors just described in the phase space has its corresponding representation in the impact map. In general, the representation of the attractor in the impact map is a geometric object of one dimension less than in phase space. For instance, a periodic orbit is a closed curve in phase space, and this same period n orbit consists of a collection of n points in the impact map. The impact map for a chaotic orbit consists of an infinite collection of points.
For a nonlinear system, many attractors can coexist. This naturally raises the question as to which orbits and collections of initial conditions go to which attractors. For a given attractor, the domain of attraction , or basin of attraction , is the collection of all those initial conditions whose orbits approach and always remain near that attractor. That is, it is the collection of all orbits that are ``captured'' by an attractor. Like the attractors themselves, the basins of attraction can be simple or complex [7].
Figure 1.15 shows a diagram of basins of attraction in the bouncing ball system. The phase space is dominated by the black regions, which indicate initial conditions that eventually become sticking solutions. The white sinusoidal regions at the bottom of Figure 1.15 show unphysical initial conditions--phases and velocities that the ball cannot obtain. The gray regions represent initial conditions that approach a period one orbit. (See Plate 1 for a color diagram of basins of attraction in the bouncing ball system.)
Figure 1.15: Basins of attraction in the bouncing ball system. (Generated by the Bouncing Ball program.)
