Chaos

The concept of deterministic chaos is a major revolution in continuum mechanics (Bergé et al., 1986). Its implications may turn out to be equivalent to the impact of quantum mechanics on atomic and molecular physics. Solutions to problems in solid and fluid mechanics have generally been thought to be deterministic. If initial and boundary conditions on a region are specified, then the time evolution of the solution is completely determined. This is in fact the case for linear equations such as the Laplace equation, the heat conduction equation, and the wave equation.

However, the problem of fluid turbulence has remained one of the major unsolved problems in physics. Turbulent flows govern the behavior of the oceans and atmosphere. The appropriate Navier–Stokes equations can be written down, but solutions yielding fully developed turbulence cannot be obtained. It is necessary to treat turbulent flows statistically and to carry out spectral analyses.

The concept of deterministic chaos bridges the gap between stable deterministic solutions to equations and deterministic solutions that are unstable to infinitesimal disturbances. Chaotic solutions must also be treated statistically; they evolve in time with exponential sensitivity to initial conditions. A deterministic solution is defined to be chaotic if two solutions that initially differ by a small amount diverge exponentially as they evolve in time. The evolving solutions are predictable only in a statistical sense. A necessary condition that a solution be chaotic is that the governing equations be nonlinear.