FolJowing the formulation of the laws of mechanics by Newton, Lagrange
sought to clarify and emphasize their geometrical character. Poincare
and Liapunov successfuIJy developed analytical mechanics further along
these lines. In this approach, one represents the evolution of all
possible states (positions and momenta) by the flow in phase space, or
more efficiently, by mappings on manifolds with a symplectic geometry,
and tries to understand qualitative features of this problem, rather
than solving it explicitly. One important outcome of this line of
inquiry is the discovery that vastly different physical systems can
actually be abstracted to a few universal forms, like Mandelbrot's
fractal and Smale's horse-shoe map, even though the underlying processes
are not completely understood. This, of course, implies that much of the
observed diversity is only apparent and arises from different ways of
looking at the same system. Thus, modern nonlinear dynamics 1 is very
much akin to classical thermodynamics in that the ideas and results
appear to be applicable to vastly different physical systems. Chaos
theory, which occupies a central place in modem nonlinear dynamics,
refers to a deterministic development with chaotic outcome. Computers
have contributed considerably to progress in chaos theory via impressive
complex graphics. However, this approach lacks organization and
therefore does not afford complete insight into the underlying complex
dynamical behavior. This dynamical behavior mandates concepts and
methods from such areas of mathematics and physics as nonlinear
differential equations, bifurcation theory, Hamiltonian dynamics, number
theory, topology, fractals, and others.
