Chaos in the Kuramoto–Sivashinsky equations—a computer-assisted proof

In this paper, we present a new technique of proving the existence of an infinite number of symmetric periodic solutions. This technique combines the covering relations method introduced by Zgliczy ski (J. Differential Equations 171 (2001) 173; Methods in Nonlinear Anal. 8 (1996) 169; Nonlinearity 10 (1997) 243) with fixed set iteration method described in Lamb (Reversing symmetries in dynamical systems, Ph.D. Thesis, Universiteit van Amsterdam, 1994). As an example we present a computer-assisted proof of symbolic dynamics in the Kuramoto–Sivashinsky equations. Moreover, we prove the existence of an infinite number of symmetric and an infinite number of nonsymmetric periodic solutions