On the existence of connecting orbits for critical values of the energy

Fusco, Giorgio; Gronchi, Giovanni F.; Novaga, Matteo

doi:10.1016/j.jde.2017.08.067

Cited by 8 publications

(7 citation statements)

References 11 publications

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“…The first existence proofs of a heteroclinic connection in the vector case were given by Rabinowitz [15] and by Sternberg [18,17]. For more recent developments on the heteroclinic connection problem we refer to [6,3,5,14,19,8,9,4]. The aforementioned works provide sufficient conditions for the existence of heteroclinic orbits, in various settings.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Nondegeneracy of heteroclinic orbits for a class of potentials on the plane

Jendrej¹,

Smyrnelis

2022

Applied Mathematics Letters

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In the scalar case, the nondegeneracy of heteroclinic orbits is a well-known property, commonly used in problems involving nonlinear elliptic, parabolic or hyperbolic P.D.E. On the other hand, Schatzman proved that in the vector case this assumption is generic, in the sense that for any potential W : R m → R, m ≥ 2, there exists an arbitrary small perturbation of W , such that for the new potential minimal heteroclinic orbits are nondegenerate. However, to the best of our knowledge, nontrivial explicit examples of such potentials are not available. In this paper, we prove the nondegeneracy of heteroclinic orbits for potentials W : R 2 → [0, ∞) that can be written as W (z) = |f (z)| 2 , with f : C → C a holomorphic function.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Nondegeneracy of heteroclinic orbits for a class of potentials on the plane

Jendrej¹,

Smyrnelis

2022

Applied Mathematics Letters

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“…In particular, such systems admit infinitely many periodic orbits and homoclinics to the Lyapunoff orbit near the critical energy level containing a saddle-center equilibrium. Motivated by the Allen-Cahn equation, Fusco-Gronchi-Novaga [12,13] use variational methods to study the existence of periodic motions of kinetic plus potential Hamiltonians.…”

Section: Applicationsmentioning

confidence: 99%

On the multiplicity of periodic orbits and homoclinics near critical energy levels of Hamiltonian systems in $\mathbb {R}^4$

Paulo

Salomão²

2019

Trans. Amer. Math. Soc.

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We study two-degree-of-freedom Hamiltonian systems. Let us assume that the zero energy level of a real-analytic Hamiltonian function H : R 4 → R contains a saddle-center equilibrium point lying in a strictly convex sphere-like singular subset S 0 ⊂ H −1 (0). From previous work [8] we know that for any small energy E > 0, the energy level H −1 (E) contains a closed 3-ball S E in a neighborhood of S 0 admitting a singular foliation called 2 − 3 foliation. One of the binding orbits of this singular foliation is the Lyapunoff orbit P 2,E contained in the center manifold of the saddle-center. The other binding orbit lies in the interior of S E and spans a one parameter family of disks transverse to the Hamiltonian vector field. In this article we show that the 2 − 3 foliation forces the existence of infinitely many periodic orbits and infinitely many homoclinics to P 2,E in S E . Moreover, if the branches of the stable and unstable manifolds of P 2,E inside S E do not coincide then the Hamiltonian flow on S E has positive topological entropy. We also present applications of these results to some classical Hamiltonian systems.

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“…Remark 1. The assumptions on W imply (see for example [14], [22] and [12]) the existence of a Lipschitz continuous map u H : R → R m that satisfies lim t→±∞ u(t) = a ± ,…”

Section: The Finite Dimensional Casementioning

confidence: 99%

“…Then Theorem 5.5 in [1] or Corollary 1.5 in [12] yields the existence of a brake orbit u δ that oscillates between Γ − and Γ + and whose period T δ diverges to +∞ as δ → 0 + . Even though the condition (1.3) can be relaxed by allowing Γ ± to contain hyperbolic critical points of W [12], the extension of this approach to the infinite dimensional setting requires new ideas to overcome the difficulties related to the formulation of a condition analogous to To avoid these pathologies the idea is to minimize on a set of T -periodic maps. But we can not expect that u δ is a minimizer in the class of maps of period T = T δ .…”

Section: Introductionmentioning

confidence: 99%

On the existence of heteroclinic connections

Fusco

Gronchi

Novaga

2017

São Paulo J. Math. Sci.

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We consider a potential W : R m → R with two different global minima a − , a + and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian systemhas a family of T -periodic solutions u T which, along a sequence T j → +∞, converges locally to a heteroclinic solution that connects a − to a + . We then focus on the elliptic systemthat we interpret as an infinite dimensional analogous of (0.1), where x plays the role of time and W is replaced by the action functional J R (u) = R ( 1 2 |u y | 2 + W (u))dy. We assume that J R has two different global minimizersū − ,ū + : R → R m in the set of maps that connect a − to a + . We work in a symmetric context and prove, via a minimization procedure, that (0.2) has a family of solutions u L : R 2 → R m , which is L-periodic in x, converges to a ± as y → ±∞ and, along a sequence L j → +∞, converges locally to a heteroclinic solution that connectsū − toū + .

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On the existence of connecting orbits for critical values of the energy

Cited by 8 publications

References 11 publications

Nondegeneracy of heteroclinic orbits for a class of potentials on the plane

Nondegeneracy of heteroclinic orbits for a class of potentials on the plane

On the multiplicity of periodic orbits and homoclinics near critical energy levels of Hamiltonian systems in $\mathbb {R}^4$

On the existence of heteroclinic connections

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