Multi-pulse evolution and space-time chaos in dissipative systems

We study semilinear parabolic systems on the full space $\R^n$ that admit a family of exponentially decaying pulse-like steady states obtained via translations. The multi-pulse solutions under consideration look like the sum of infinitely many such pulses which are well separated. We prove a global center-manifold reduction theorem for the temporal evolution of such multi-pulse solutions and show that the dynamics of these solutions can be described by an infinite systems of ODEs for the positions of the pulses. As an application of the developed theory, we verify the existence of Sinai-Bunimovich space-time chaos in 1D space-time periodically forced Swift--Hohenberg equation