Department of Physics

University of Wisconsin - Madison

sprott@physics.wisc.edu

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.

The complete paper is available in
PDF format.

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