Organization of spatially periodic solutions of the steady Kuramoto–Sivashinsky equation

• Systematically investigated steady periodic solutions of the Kuramoto–Sivashinsky equation.
• Obtained all the short steady periodic solutions and classified with symbolic dynamics.
• The return map is studied and bifurcation of fundamental cycles are explored.

A systematic study of spatially periodic steady solutions of the Kuramoto–Sivashinsky equation (KSe) is undertaken from a dynamical systems’ point of view. A recently devised variational method is employed and one new variant is developed. At fixed system size L=43.5L=43.5, important equilibria are identified and shown to organize the dynamics. The first integral of the steady KSe leads to a 3D dynamical system with an integration constant c  . At a typical value of c=0.40194c=0.40194, four simplest cycles are identified and used as basic building blocks to construct longer cycles. The symbolic dynamics based on trajectory topology are very effective in classifying all short periodic orbits. The probation of the return map on a chosen Poincaré section shows the complexity of the dynamics and the bifurcation of building blocks provides a chart to look for possible cycles at given periods. The current study may be conveniently adapted to the identification and classification of cycles in other nonlinear systems.