The Order of Bifurcation Points in Fourth Order Conservative Systems via Braids

Abstract

In second order Lagrangian systems bifurcation branches of periodic solutions preserve certain topological invariants. These invariants are based on the observation that periodic orbits of a second order Lagrangian lie on 3-dimensional (noncompact) energy manifolds and the periodic orbits may have various linking and knotting properties. The main ingredients defining the topological invariants are the discretization of second order Lagrangian systems that satisfy the twist property and the theory of discrete braid invariants developed in [R. W. Ghrist, J. B. Van den Berg, and R. C. Vandervorst, Invent. Math., 152 (2003), pp. 369–432]. In the first part of this paper we recall the essential theory of braid invariants, and in the second part this theory is applied to second order Lagrangian systems and in particular to the Swift–Hohenberg equation. We show that the invariants yield forcing relations on bifurcation branches. We quantify this principle via an order relation on the topological type of a bifurc...