A geometrical index for the group <i>S</i><sup>1</sup> and some applications to the study of periodic solutions of ordinary differential equations

Consider the Hamiltonian system i = 1,. . . , N.Here, H E C2(R2N, R). In this paper, we investigate the existence of periodic orbits of (HS) on a given energy surface X = { z E WZN ; H ( z) = c} ( c > 0 is a constant). The surface I: is required to verify certain geometric assumptions: B bounds a star-shaped compact region B and u 8 c B c pS for some ellipsoid %'c RZN, 0 < (Y < p. We exhibit a constant S > 0 (depending in an explicit fashion on the lengths of the main axes of Lf and one other geometrical parameter of I) such that if furthermore p2/a2< I + 8, then (HS) has at least N distinct geometric orbits on P. This result is shown to extend and unify several earlier works on this subject (among them works by Weinstein, Rabinowitz, Ekeland-Lasry and Ekeland). In proving this result we construct index theories for an S'-action, from which we derive abstract critical point theorems for S'-invariant functionals. We also derive an estimate for the minimal period of solutions to differential equations.

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“…Our first step will be to recall the geometrical S'-index theory of V. Benci [2] for these classes of subsets. DEFINITION 2.1.…”

Section: Remarksmentioning

confidence: 99%

Existence of multiple periodic orbits on star‐shaped hamiltonian surfaces

Berestycki

Lasry

Mancini

et al. 1985

Comm Pure Appl Math

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“…A G-capacity was introduced by Clapp [6]. An S 1 -equivariant geometrical index was defined by Benci [4], and a G-index for any orthogonal compact Lie group action by Marzantowicz [11]. For the group G = S 1 all these constructions have the normalization property (γ 6) due to a version of the Borsuk-Ulam theorem (see [3] for more information).…”

Section: Existence Of Periodic Solutionsmentioning

confidence: 99%

Periodic solutions of symmetric autonomous Newtonian systems

Marzantowicz

Prieto

Rybicki

2008

Journal of Differential Equations

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In this paper we show the existence and bifurcation of T -periodic solutions of a special form for an autonomous Newtonian system with symmetry. If the phase-space R 2n is equipped with the structure of an orthogonal representation (W, ρ W ) and the potential V : W → R is invariant, then for every such a solution the set of indices of nonvanishing Fourier coefficients is finite and depends on W only. If the potential V depends on the squares of complex coordinates, then for every such a solution T is the minimal period.

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“…Using the geometric S 1 -index by V. Benci [6] one obtains for every k ∈ N a minimax-characterization of critical values c k , k ∈ N, which corresponds to the k-th Fučík eigenvalue, see de Figueiredo-Ruf [16]. In particular, this allows to give a variational characterization of the complete Fučík spectrum.…”

Section: Variational Characterization Of σmentioning

confidence: 99%

On the Fučík spectrum for equations with symmetries

Ruf¹

2011

Nonlinear Elliptic Partial Differential Equations

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Abstract. An overview over some aspects of the Fučiík spectrum are given, in particular in the situation when the problem is invariant under some compact group action. Some recent results concerning the complexity of the Fučiík spectrum are discussed, and some open problems are stated. In the final section, based on the mentioned structure of the Fučiík spectrum, a new multiplicity result for a related equation with asymptotic interference with the spectrum is given.

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A geometrical index for the group S¹ and some applications to the study of periodic solutions of ordinary differential equations

Cited by 79 publications

References 18 publications

Existence of multiple periodic orbits on star‐shaped hamiltonian surfaces

Existence of multiple periodic orbits on star‐shaped hamiltonian surfaces

Periodic solutions of symmetric autonomous Newtonian systems

On the Fučík spectrum for equations with symmetries

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A geometrical index for the group S1 and some applications to the study of periodic solutions of ordinary differential equations

Cited by 79 publications

References 18 publications

Existence of multiple periodic orbits on star‐shaped hamiltonian surfaces

Existence of multiple periodic orbits on star‐shaped hamiltonian surfaces

Periodic solutions of symmetric autonomous Newtonian systems

On the Fučík spectrum for equations with symmetries

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A geometrical index for the group S¹ and some applications to the study of periodic solutions of ordinary differential equations