The Persistence of a Slow Manifold with Bifurcation

Kristian Uldall Kristiansen; P. Palmer; M. Robert

doi:10.1137/110820671

The Persistence of a Slow Manifold with Bifurcation

Kristian Uldall Kristiansen, P. Palmer, M. Robert

University of Surrey

Research output: Contribution to journal › Journal article › Research › peer-review

Abstract

his paper considers the persistence of a slow manifold with bifurcation in a slow-fast two degree of freedom Hamiltonian system. In particular, we consider a system with a supercritical pitchfork bifurcation in the fast space which is unfolded by the slow coordinate. The model system is motivated by tethered satellites. It is shown that an almost full measure subset of a neighborhood of the slow manifold's normally elliptic branches persists in an adiabatic sense. We prove this using averaging and a blow-up near the bifurcation.

Original language	English
Journal	S I A M Journal on Applied Dynamical Systems
Volume	11
Issue number	2
Pages (from-to)	661-683
ISSN	1536-0040
DOIs	https://doi.org/10.1137/110820671
Publication status	Published - 2012

Bibliographical note

Read More: http://epubs.siam.org/action/showAbstract?page=661&volume=11&issue=2&journalCode=sjaday

Keywords

Singularly perturbed Hamiltonian systems
Slow manifolds with bifurcations
Pitchfork bifurcations
Blow-up
Tethered satellites

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title = "The Persistence of a Slow Manifold with Bifurcation",

abstract = "his paper considers the persistence of a slow manifold with bifurcation in a slow-fast two degree of freedom Hamiltonian system. In particular, we consider a system with a supercritical pitchfork bifurcation in the fast space which is unfolded by the slow coordinate. The model system is motivated by tethered satellites. It is shown that an almost full measure subset of a neighborhood of the slow manifold's normally elliptic branches persists in an adiabatic sense. We prove this using averaging and a blow-up near the bifurcation.",

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AU - Robert, M.

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N2 - his paper considers the persistence of a slow manifold with bifurcation in a slow-fast two degree of freedom Hamiltonian system. In particular, we consider a system with a supercritical pitchfork bifurcation in the fast space which is unfolded by the slow coordinate. The model system is motivated by tethered satellites. It is shown that an almost full measure subset of a neighborhood of the slow manifold's normally elliptic branches persists in an adiabatic sense. We prove this using averaging and a blow-up near the bifurcation.

AB - his paper considers the persistence of a slow manifold with bifurcation in a slow-fast two degree of freedom Hamiltonian system. In particular, we consider a system with a supercritical pitchfork bifurcation in the fast space which is unfolded by the slow coordinate. The model system is motivated by tethered satellites. It is shown that an almost full measure subset of a neighborhood of the slow manifold's normally elliptic branches persists in an adiabatic sense. We prove this using averaging and a blow-up near the bifurcation.

KW - Singularly perturbed Hamiltonian systems

KW - Slow manifolds with bifurcations

KW - Pitchfork bifurcations

KW - Blow-up

KW - Tethered satellites

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The Persistence of a Slow Manifold with Bifurcation

Abstract

Bibliographical note

Keywords

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