The Lorenz system as a gradient-like system

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

Families of periodic solutions for some hamiltonian PDEs

We consider the nonlinear wave equation utt -uxx = ±u3 and the beam equation utt +uxxxx = ±u3 on an interval. Numerical observations indicate that time-periodic solutions for these equations are organized into structures that resemble branches and seem to undergo bifurcations. In addition to describing our observations, we prove the existence of time-periodic solutions for various periods (a set of positive measure in the case of the beam equation) along the main nontrivial "branch." Our proofs are computer-Assisted

Non-radial solutions for some semilinear elliptic equations on the disk

Starting with approximate solutions of the equation −Δu=wu3on the disk, with zero boundary conditions, we prove that there exist true solutions nearby. One of the challenges here lies in the fact that we need simultaneous and accurate control of both the (inverse) Dirichlet Laplacean and nonlinearities. We achieve this with the aid of a computer, using a Banach algebra of real analytic functions, based on Zernike polynomials. Besides proving existence, and symmetry properties, we also determine the Morse index of the solutions

Rigorously computing symmetric stationary states of the Ohta-Kawasaki problem in three dimensions

In this paper we develop a symmetry preserving method for the rigorous computation of stationary states of the Ohta-Kawasaki partial differential equation in three space dimensions. By preserving the relevant symmetries we achieve an enormous reduction in computational cost. This makes it feasible to construct computer-assisted proofs of complex three-dimensional structures. In particular, we provide the first existence proofs for both the double gyroid and body centered cubic packed sphere solutions to this problem

A geometric method for infinite-dimensional chaos : symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line

We propose a general framework for proving that a compact, infinite-dimensional map has an invariant set on which the dynamics is semiconjugated to a subshift of finite type. The method is then applied to certain Poincaré map of the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and with parameter . We give a computer-assisted proof of the existence of symbolic dynamics and countable infinity of periodic orbits with arbitrary large periods