Degree for Gradient Equivariant Maps and Equivariant Conley Index

Gęba, Kazimierz

doi:10.1007/978-1-4612-4126-3_5

Cited by 51 publications

(68 citation statements)

References 19 publications

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“…see [18]. We observe that the matrices B(x 0 ), C(x 0 ) are non-degenerate because the orbit G(x 0 ) is non-degenerate.…”

Section: Definition 33 An Orbit G(xmentioning

confidence: 87%

“…Under this stronger assumption they have proved that there is a connected set of non-stationary periodic solutions of system (1.2) emanating from the stationary solution u 0 ≡ 0. In order to prove this theorem they have applied the degree theory for S 1 -equivariant gradient maps, see [18].…”

Section: Note That Since χ(Ci({0} −∇ H )) = Deg B (∇ H B Nmentioning

confidence: 99%

“…The rest of this section is dedicated to the study of the G-equivariant Conley index, see [6,16,18,34], of the isolated invariant set G(x 0 ) considered as a G-orbit of stationary solutions of the equationẋ(t) = −∇ϕ(x(t)). Note that since the orbit G(x 0 ) is non-degenerate,…”

Section: Definition 33 An Orbit G(xmentioning

confidence: 99%

“…(4.2) from the family T . As a topological tool we will use the equivariant Conley index, see [6,16,18,34] and its infinite-dimensional generalization [24].…”

Section: Proof Of the Symmetric Liapunov Center Theoremmentioning

confidence: 99%

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Symmetric Liapunov center theorem

2017

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In this article, using an infinite-dimensional equivariant Conley index, we prove a generalization of the profitable Liapunov center theorem for symmetric potentials. Consider a system ( * )q = −∇U (q), where U (q) is a -invariant potential and is a compact Lie group acting linearly on R n . If system ( * ) possess a non-degenerate orbit of stationary solutions (q 0 ) with trivial isotropy group, such that there exists at least one positive eigenvalue of the Hessian ∇ 2 U (q 0 ), then in any neighborhood of (q 0 ) there is a non-stationary periodic orbit of solutions of system ( * ).

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“…see [18]. We observe that the matrices B(x 0 ), C(x 0 ) are non-degenerate because the orbit G(x 0 ) is non-degenerate.…”

Section: Definition 33 An Orbit G(xmentioning

confidence: 87%

Section: Note That Since χ(Ci({0} −∇ H )) = Deg B (∇ H B Nmentioning

confidence: 99%

Section: Definition 33 An Orbit G(xmentioning

confidence: 99%

“…(4.2) from the family T . As a topological tool we will use the equivariant Conley index, see [6,16,18,34] and its infinite-dimensional generalization [24].…”

Section: Proof Of the Symmetric Liapunov Center Theoremmentioning

confidence: 99%

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Symmetric Liapunov center theorem

2017

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“…The main reference for this subject are the papers of Gęba [8], Gęba & Rybicki [9] and the papers of Rybicki and his collaborators [10,17,19,20]. The equivariant version of the Conley index is the subject of the papers of Floer [6], Floer & Zehnder [7].…”

Section: Let V Be An Orthogonal Representation Of a Compact Lie Groupmentioning

confidence: 99%

Equivariant Morse equation

Styborski

2012

Open Mathematics

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Global Bifurcations of Solutions of Emden–Fowler-Type Equation −Δu(x)=λf(u(x)) on an Annulus in Rn, n⩾3

Rybicki¹

2002

Journal of Differential Equations

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In this article we study global and symmetry-breaking bifurcations of solutions of Emden-Fowler-type elliptic differential equations. As the main tool we use a topological invariant which is suitable for study of global bifurcations and continuations of critical points of equivariant functionals. Namely, we will apply degree for S 1 -equivariant orthogonal maps defined in Rybicki (1994, Nonlinear Anal TMA 23, 83-102

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Degree for Gradient Equivariant Maps and Equivariant Conley Index

Cited by 51 publications

References 19 publications

Symmetric Liapunov center theorem

Symmetric Liapunov center theorem

Equivariant Morse equation

Global Bifurcations of Solutions of Emden–Fowler-Type Equation −Δu(x)=λf(u(x)) on an Annulus in Rn, n⩾3

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