Strange attractor defined

Fig. 4.9 Snapshots of strange attractors defined by the expressions appearing in equation 4,32.

Strange Attractors The motion on strange attractors exhibits many of the properties normally associated with completely random or chaotic behavior, despite being well-defined at all times and fully deterministic. More formally, a strange attractor S is an attractor (meaning that it satisfies properties (i)-(iii) above) that also displays sensitivity to initial conditions. In the case of a one-dimensional map, Xn+i = for example, this means that there exists a <5 > 0 such that for... 

Finally, we define a strange attractor to be an attractor that exhibits sensitive dependence on initial conditions. Strange attractors were originally called strange because they are often fractal sets. Nowadays this geometric property is regarded as less important than the dynamical property of sensitive dependence on initial conditions. The terms chaotic attractor and fractal attractor are used when one wishes to emphasize one or the other of those aspects. 

The method is based on time delays. For instance, define a two-dimensional vector x(r) = (B(t). B(t -I- t)) for some delay T > 0. Then the time series B t) generates a trajectory x(r) in a two-dimensional phase space. Figure 12.4.2 shows the result of this procedure when applied to the data of Figure 12.4.1, using t = 8.8 seconds. The experimental data trace out a strange attractor that looks remarkably like the Rdssler attractor ... 

Classical mechanics provides the least ambiguous statement of the nature of chaotic motion, with chaos also defined through a heirarchy of ideal model systems. We note, at the outset, that isolated molecule dynamics relates to chaotic motion in conservative Hamiltonian systems. This is distinct from chaotic motion in dissipative systems where considerable simplifications result from the reduction in degrees of freedom during evolution10 and where objects such as strange attractors and fractal dimensions play an important role. 

The behaviour illustrated by figs. 4.13 and 4.14 provides an example of final state sensitivity (Grebogi et al., 1983a). The evolution towards one or other final state is unpredictable when the unstable trajectory that defines the boundary of their attraction basins is a strange attractor rather than a simple limit cycle. 

The phase space representation of trajectories computed numerically, as described above, has been introduced in another chapter of this volume. TTie systems considered there are Hamiltonian systems which arise in chemistry in the context of molecular dynamics problems, for example. The difference between Hamiltonian systems and the dissipative ones we are considering in this chapter is that, in the former, a constant of the motion (namely the energy) characterizes the system. A dissipative system, in contrast, is characterized by processes that dissipate rather than conserve energy, pulling the trajectory in toward an attractor (where in refers to the direction in phase space toward the center of the attractor). We have already seen two examples of attractors, the steady state attractor and the limit cycle attractor. These attractors, as well as the strange attractors that arise in the study of chaotic systems, are most easily defined in the context of the phase space in which they exist. 

Whereas attractor in the steady state has zero dimensions and Euclidian dimension of the limit cycle is two, it is not possible to define the Euclidian dimension of the strange attractor. However, using the concept of fractal geometry, it is possible to define the dimension of such an attractor, which is not an integer. 

The distinction is crucial because area-preserving maps cannot have attractors (strange or otherwise). As defined in Section 9.3, an attractor should attract all orbits starting in a sufficiently small open set containing it that requirement is incompatible with area-preservation. 

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