All possible chaotic dynamics can be approximated in three dimensions

It is known that neither onenor two-dimensional ordinary differential equations can exhibit chaotic phenomena, while such behaviour is possible in threedimensional systems. Here it is shown that the solutions of any dissipative equation can be approximated arbitrarily closely by the trajectories of an appropriate threedimensional system. This class includes the Navier-Stokes equations, showing that fully developed turbulent flows can be approximated using just three degrees of freedom. AMS classification scheme numbers: 34C35, 34D40, 34D45, 35B35, 35B40, 76F20 ∗ email: [email protected] 1