All possible chaotic dynamics can be approximated in three dimensions

Abstract

It is known that neither one- nor two-dimensional ordinary differential equations can exhibit chaotic phenomena, while such behaviour is possible in three-dimensional systems. Here it is shown that the solutions of any dissipative equation can be approximated arbitrarily closely by the trajectories of an appropriate three-dimensional system. This class includes the Navier-Stokes equations, showing that the dynamical system which may describe fully developed turbulence can be approximated using just three degrees of freedom.