TY - JOUR

T1 - The order of bifurcation points in fourt horder conservative systems via braids

AU - Van Den Berg, Jan Bouwe

AU - Kramar, Miroslav

AU - Vandervorst, Robert C.

PY - 2011

Y1 - 2011

N2 - In second order Lagrangian systems bifurcati on 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 bifurcation branch. The order will then determine the forcing relation. It is shown that certain braid classes force infinitely many solution curves.

AB - In second order Lagrangian systems bifurcati on 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 bifurcation branch. The order will then determine the forcing relation. It is shown that certain braid classes force infinitely many solution curves.

KW - Conley index

KW - Fourth order equations

KW - Parabolic recurrence relations

KW - Relative braid classes

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U2 - 10.1137/100796558

DO - 10.1137/100796558

M3 - Article

AN - SCOPUS:80051485142

VL - 10

SP - 510

EP - 550

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 2

ER -