The Equivariant Spectral Flow and Bifurcation for Strongly Indefinite Functionals with Symmetries
Final Report Abstract
We have considered families of differential equations which depend on a real parameter and whose equations are all solved by the constant function 0. Bifurcation theory deals with the question of the existence of values of the parameter at which other solutions arise from the zero solution. Since the solutions of differential equations can often be obtained as critical points of a functional on a Hilbert space, families of functionals play an important role in bifurcation theory. A well-known theorem of bifurcation theory states that for functionals with finite Morse indices, a parameter value is a bifurcation point if the Morse index changes when passing the parameter value. Often those functionals do not have a finite Morse index (e.g. for Hamiltonian systems) and there have been numerous attempts to construct suitable bifurcation invariants in such cases. Fitzpatrick, Pejsachowicz and Recht presented an impressive solution to this issue in 1999. They showed that the spectral flow, an integer invariant from global analysis, is the correct replacement for the Morse indices if the latter are not finite. Their theorem generalises the results known so far and it has a wide range of applications to differential equations. Differential equations in the natural sciences often have symmetries that can be described by the action of a Lie group. Such symmetries can impose restrictions on the Morse index and, for example, force it to be constant. Therefore, in such cases, the bifurcation cannot be detected using the Morse index. In a celebrated paper in 1990, Smoller and Wasserman showed that this problem can be solved using group representations. In recent years, first steps have been made using representation theory to construct bifurcation invariants for special types of equations with symmetries that do not have a finite Morse index. In this project, an equivariant version of the bifurcation theorem by Fitzpatrick, Pejsachowicz and Recht was proven, which can be used to show the existence of bifurcation points even if the spectral flow loses its applicability due to existing symmetries. The new bifurcation invariant is an equivariant spectral flow that is an element of the representation ring of the underlying Lie group. If the action of the group is trivial, we reobtain the theorem of Fitzpatrick, Pejsachowicz and Recht. If the Morse indices are finite, our theorem reduces to that of Smoller and Wasserman. We have applied our bifurcation theorem to Hamiltonian systems and indefinite elliptic systems with symmetries that could not have been studied using previously known methods.
Publications
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The equivariant spectral flow and bifurcation of periodic solutions of Hamiltonian systems. Nonlinear Analysis, 211, 112475.
Izydorek, Marek; Janczewska, Joanna & Waterstraat, Nils
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The relative cup-length in local Morse cohomology. Topological Methods in Nonlinear Analysis, 1-15.
Rot, Thomas; Starostka, Maciej & Waterstraat, Nils
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Bifurcation for a Class of Indefinite Elliptic Systems by Comparison Theory for the Spectral Flow via an Index Theorem. Nonlinear Differential Equations and Applications NoDEA, 32(6).
Janczewska, Joanna; Möckel, Melanie & Waterstraat, Nils
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The equivariant spectral flow and bifurcation for functionals with symmetries: part I. Mathematische Annalen, 393(2), 2187-2226.
Izydorek, Marek; Janczewska, Joanna; Starostka, Maciej & Waterstraat, Nils
