Critical Points of Functions

Arnold, Vladimir I.; Goryunov, V. V.; Lyashko, O. V.; Васильев, Виктор Анатольевич

doi:10.1007/978-3-642-58009-3_1

Cited by 14 publications

(28 citation statements)

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“…This extends Remark 3.5. persistent under small perturbations. For exact notions see [76], for instance. The figure to the right shows the value of the bifurcation test equation for A 4 along the line of cusps parametrised by µ 1 .…”

Section: Application Of Pseudoarclength Continuation To Augmented Dismentioning

confidence: 99%

Detection of high codimensional bifurcations in variational PDEs

2020

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We derive bifurcation test equations for A-series singularities of nonlinear functionals and, based on these equations, we propose a numerical method for detecting high codimensional bifurcations in parameter-dependent PDEs such as parameterdependent semilinear Poisson equations. As an example, we consider a Bratutype problem and show how high codimensional bifurcations such as the swallowtail bifurcation can be found numerically. In particular, our original contributions are (1) the use of the Infinite-Dimensional Splitting Lemma, (2) the unified and simplified treatment of all A-series bifurcations, (3) the presentation in Banach spaces, i.e. our results apply both to the PDE and its (variational) discretization, (4) further simplifications for parameter-dependent semilinear Poisson equations (both continuous and discrete), and (5) the unified treatment of the continuous problem and its discretisation. ‡ corresponding author

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Section: Application Of Pseudoarclength Continuation To Augmented Dismentioning

confidence: 99%

Detection of high codimensional bifurcations in variational PDEs

2020

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“…Classical catastrophe theory considers the bifurcation behaviour of critical points of smooth, parameter dependent functions r µ : R k → R, where µ ∈ R l is a parameter. Stable bifurcations have been classified, see, for instance (Arnold et al, 1998). Figure 1 shows the first five elementary catastrophes.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation preserving discretisations of optimal control problems

Offen,

Ober-Blöbaum

2021

Preprint

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The first order optimality conditions of optimal control problems (OCPs) can be regarded as boundary value problems for Hamiltonian systems. Variational or symplectic discretisation methods are classically known for their excellent long term behaviour. As boundary value problems are posed on intervals of fixed, moderate length, it is not immediately clear whether methods can profit from structure preservation in this context. When parameters are present, solutions can undergo bifurcations, for instance, two solutions can merge and annihilate one another as parameters are varied. We will show that generic bifurcations of an OCP are preserved under discretisation when the OCP is either directly discretised to a discrete OCP (direct method) or translated into a Hamiltonian boundary value problem using first order necessary conditions of optimality which is then solved using a symplectic integrator (indirect method). Moreover, certain bifurcations break when a non-symplectic scheme is used. The general phenomenon is illustrated on the example of a cut locus of an ellipsoid.

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“…Thom's results were extended by Arnold in a series of papers including [4][5][6][7]. Results are summarised in [8,9]. A good introduction to the algebraic framework of catastrophe theory and the results obtained by Thom is [3].…”

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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Critical Points of Functions

Cited by 14 publications

References 0 publications

Detection of high codimensional bifurcations in variational PDEs

Detection of high codimensional bifurcations in variational PDEs

Bifurcation preserving discretisations of optimal control problems

Bifurcation of solutions to Hamiltonian boundary value problems

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