  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.
