Complex Behavior of Simple Systems

Since the seminal work of Lorenz [1963] 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. This paper will review the history of recent attempts to identify the simplest such system and will describe two candidate systems that are simpler than any previously known. They were discovered by a brute-force numerical search for the algebraically simplest chaotic flows. There are reasons to believe that these cases are the simplest examples with quadratic and piecewise linear nonlinearities. The properties of these systems will be described.