Oscillatory solutions of fourth order conservative systems via the Conley index

In this paper we investigate periodic solutions of second order Lagrangian systems which oscillate around equilibrium points of center type. The main ingredients are the discretization of second order Lagrangian systems that satisfy the twist property and the theory of discrete braid invariants developed by Ghrist et al. (2003) [5]. The problem with applying this topological theory directly is that the braid types in our analysis are so-called improper. This implies that the braid invariants do not entirely depend on the topology: the relevant braid classes are non-isolating neighborhoods of the flow, so that their Conley index is not universal. In first part of this paper we develop the theory of the braid invariant for improper braid classes and in the second part this theory is applied to second order Lagrangian system and in particular to the Swift–Hohenberg equation.