Bifurcation results for critical points of families of functionals

Порталури, Алессандро; Waterstraat, Nils

doi:10.57262/die/1391091370

Cited by 1 publication

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References 14 publications

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“…There exist general statements about basic bifurcations in the critical points of functional problem, typically under technical assumptions which allow Lyapunov-Schmidt reduction to a finite dimensional problem, see e.g. [27] or [28]. Note that even in the case d = 1, i.e.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation of solutions to Hamiltonian boundary value problems

McLachlan

Offen

2018

Nonlinearity

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A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for secondorder systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples.

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Section: Introductionmentioning

confidence: 99%