Figure 1

Multi-bump pulse solutions correspond to intersections between the stable and unstable manifolds of the equilibrium state in the fourth order reversible ODE system. Since the ODE system is conservative, the intersections between these manifolds can be found by inspecting their projections onto a 2D phase plane. The above figure shows a partial picture of the projection of the stable manifold (blue curves) and the projection of the unstable manifold (red curves). For some of the intersections, the corresponding pulse solutions are plotted. The Smale horseshoe occurs in the full picture of the projections of the stable and unstable manifolds. This implies countably many intersections between the stable and unstable manifolds.