Complex Behavior of Simple Systems

Since the seminal work of Lorenz [1963J and Rössler [1976], it has been known that complex behavior (chaos) can occur in systems of autonomous ordinary differential equations (ODEs) with as few as three variables and one or two quadratic nonlinearities. Many other simple chaotic systems have been discovered and studied over the years, but it is not known whether the algebraically simplest chaotic flow has been identified. For continuous flows, the Poincaré-Bendixson theorem [Hirsch 1974] implies the necessity of three variables, and chaos requires at least one nonlinearity. With the growing availability of powerful computers, many other examples of chaos have been subsequently discovered in algebraically simple ODEs. Yet the sufficient conditions for chaos in a system of ODEs remain unknown.

Ref: J. C. Sprott, Unifying Themes in Complex Systems IIIB, 3-11 (2006)