Continuation and bifurcations of periodic orbits and symbolic dynamics in the Swift-Hohenberg equation

Steady states of the Swift--Hohenberg equation are studied. For the associated four--dimensional ODE we prove that on the energy level $E=0$ two smooth branches of even periodic solutions are created through the saddle-node bifurcation. We also show that these orbits satisfy certain geometric properties, which implies that the system has positive topological entropy for an explicit and wide range of parameter values of the system. The proof is computer-assisted and it uses rigorous computation of bounds on certain Poincar\'e map and its higher order derivatives.