Morse Theory for Hamiltonian Systems

Abbondandolo, Alberto

doi:10.1201/9781482285741

Cited by 102 publications

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“…In this section, we briefly recall the definition of the Maslov index for paths of Lagrangian subspaces of a symplectic space, where we follow [26] (cf. also [1]). In order to simplify the presentation, we only consider the symplectic space R 2n endowed with the standard scalar product ·, · and the symplectic form…”

Section: The Maslov Index: a Brief Recapitulationmentioning

confidence: 87%

“…In what follows, we use the definition of [7] which applies to any gap-continuous path of (generally) unbounded self-adjoint Fredholm operators on a separable Hilbert space. The differential equations (1) induce such operators A λ , λ ∈ I, on L 2 (R, R 2n ) having as domain H 1 (R, R n ), and such that the kernel of A λ is given by the solutions of the corresponding equation (1). We explain below that the spectral flow of the resulting path A is defined, and heuristically, it counts in an oriented way the instants λ ∈ I for which the equations (1) have non-trivial solutions.…”

Section: Introductionmentioning

confidence: 99%

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Spectral flow, crossing forms and homoclinics of Hamiltonian systems

Waterstraat

2015

Proc. London Math. Soc.

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We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions for bifurcation of homoclinic trajectories of one-parameter families of nonautonomous Hamiltonian vector fields.

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Section: The Maslov Index: a Brief Recapitulationmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Spectral flow, crossing forms and homoclinics of Hamiltonian systems

Waterstraat

2015

Proc. London Math. Soc.

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“…Many methods applied to (1) were inspired by the Hamiltonian systems of type where V = R 2n and J = 0 −Id Id 0 is the symplectic matrix, for which the existence of 2πperiodic solutions was intensively studied (see for example [1,3,14,18,19,20,21,22,27,28,39,40,41,43,44,45,46,50]). Similar methods were also developed for the system (1) (see [2,10,30,31,32,33,34,58]).…”

Section: Introductionmentioning

confidence: 99%

“…[23,24,29,49,52,53,54,55]). These authors applied the Γ×SO(2)-equivariant degree to study the existence of multiple 2π-periodic solutions to (1). However, there is a significant difference between the systems (1) and (3): the system (1) is time-reversible, so it leads to a variational problem with Γ × O(2)-symmetries.…”

Section: Introductionmentioning

confidence: 99%

“…These authors applied the Γ×SO(2)-equivariant degree to study the existence of multiple 2π-periodic solutions to (1). However, there is a significant difference between the systems (1) and (3): the system (1) is time-reversible, so it leads to a variational problem with Γ × O(2)-symmetries. As the equivariant gradient degree provides full equivariant topological classification of the critical set for the related to (1) variational functional J : H 1 (S 1 ; V ) → R, it is important to consider the full symmetry group for J .…”

Section: Introductionmentioning

confidence: 99%

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Multiple periodic solutions for Γ-symmetric Newtonian systems

Da̧bkowski

Krawcewicz

et al. 2017

Journal of Differential Equations

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The existence of periodic solutions in Γ-symmetric Newtonian systemsẍ = −∇f (x) can be effectively studied by means of the Γ × O(2)-equivariant gradient degree with values in the Euler ring U (Γ × O(2)). In this paper we show that in the case of Γ being a finite group, the Euler ring U (Γ × O(2)) and the related basic degrees are effectively computable using Euler ring homomorphisms, the Burnside ring A(Γ × O(2)), and the reduced Γ × O(2)-degree with no free parameters. We present several examples of Newtonian systems with various symmetries, for which we show existence of multiple periodic solutions. We also provide exact value of the equivariant topological invariant for those problems.1 for example a difference of the equivariant gradient degrees on large and small ball). Such cases are for example when the system (1) is asymptotically linear or satisfy a Nagumo-type growth condition. Then clearly the existence of non-trivial solutions (i.e. outside the small ball) can be concluded by the fact that ω = 0. However, one can be also interested to predict the existence of multiple 2π-periodic solutions with different types of symmetry. In such a case, the coefficients of ω corresponding to the so-called maximal orbit type can provide the crucial information in order to formulate such results. But the maximality of such obit types implies that it is a generator of the Burnside ring A(G), therefore it can actually be detected by the Γ × O(2)-equivariant degree with no free parameter, which can be much easier computed than the equivariant gradient degree. Similar arguments apply to the system (2), which we can consider as a bifurcation problem with a parameter λ. More precisely, in this case we are looking for critical values λ o of the parameter λ > 0, to which we can associate the Γ × O(2)-equivariant gradient bifurcation invariants ω(λ o ) ∈ U (Γ×O(2)) classifying the bifurcation of 2π-periodic solutions from the zero solution. The existence and multiplicity of such bifurcating branches of 2π-periodic solutions can be described from the information contained in the invariants ω(λ o ). Consequently, all the essential information needed to establish the existence and multiplicity results for the systems (1) and (2) can be extracted from the Γ × O(2)-equivariant degree (with no free parameter) of J which takes values in the Burnside ring A(G). It is clear that the Γ × O(2)-equivariant degree without free parameter can be easily computed (without getting entangled in complicated technical details), has similar properties and provides enough information for analyzing these problems.Nevertheless, let us emphasize that only the equivariant invariants ω ∈ U (G) (without truncation of its coefficients) provide a complete equivariant topological classification for the related solution sets to (1) or (2).To illustrate the usage and the computations of the associated with the systems (1) and (2) equivariant invariants, in section 7 we present several examples of symmetric Newtonian systems, for which the exact values of the a...

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On a formula for the spectral flow and its applications

Benevieri

Piccione

2010

Math. Nachr.

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ABSTRACT. We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi-Riemannian geodesic, and we compute its value in terms of the Maslov index.

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Morse Theory for Hamiltonian Systems

Cited by 102 publications

References 0 publications

Spectral flow, crossing forms and homoclinics of Hamiltonian systems

Spectral flow, crossing forms and homoclinics of Hamiltonian systems

Multiple periodic solutions for Γ-symmetric Newtonian systems

On a formula for the spectral flow and its applications

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