Chaos in Dissipative Systems

Abstract

A crucial distinction exists between the dissipative systems and conservative Hamiltonian systems. By Liouville’s Theorem, the solution flow for a conservative Hamiltonian system preserves volumes in phase space. Dissipative systems, by contrast, usually give rise to solution flows which contract volumes in phase space. This volume contraction gives rise to a bounded set called an attractor in the phase space, which ultimately contains the solution flow, once the transients associated with initial conditions have died away. Indeed, dissipative systems of dimensions equal to and greater than three can have bounded trajectories, which may be attracted not by a fixed point nor by a periodic/quasi-periodic orbit, but by an object of complicated infinitely many-layered structure called a strange attractor. The trajectories on this attractor diverge continually from each other locally, but remain bounded globally. Besides, the evolution on the attractor is essentially aperiodic. Strange attractors are sometimes modeled by fractals which are geometric objects that are very suited for characterizing roughness and illustrating the difference between the mathematical properties of continuity and differentiability. Adoption of fractal geometry therefore releases one from bondage to smooth surfaces and smooth curves and enables one to come to terms with nature, for as Mandelbrot (The Fractal Geometry of Nature, Freeman, 1983) put it—“clouds are not spheres, mountains are not (cones), and bark is not smooth, nor does lightning travel in a straight line”. (Indeed, the fractal structure of a natural object is closely intertwined with its functionality (like the light-catching capacity of trees or the electrical connectivity of neurons).) Further, fractals are self-similar objects which have the same shape at all scales.