The Lorenz system as a gradient-like system

Abstract

We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc. do not exist. This condition is based upon the existence of an auxiliary function defined over the state space of the system, in a way analogous to a Lyapunov function for the stability of an equilibrium. For polynomial systems, Lyapunov functions can be found computationally by using sum-of-squares optimisation. We demonstrate this method by finding such an auxiliary function for the Lorenz system. We are able to show that the system is gradient-like for $0\leq\rho\leq12$ when $\sigma=10$ and $\beta=8/3$, significantly extending previous results. The results are rigorously validated by a novel procedure: First, an approximate numerical solution is found using finite-precision floating-point sum-of-squares optimisation. We then prove that there exists an exact solution close to this using interval arithmetic.