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37
A Survey of ContinuousTime Computation Theory
 Advances in Algorithms, Languages, and Complexity
, 1997
"... Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuoustime computation. However, while specialcase algorithms and devices are being developed, relatively little work exists o ..."
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Cited by 29 (6 self)
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Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuoustime computation. However, while specialcase algorithms and devices are being developed, relatively little work exists on the general theory of continuoustime models of computation. In this paper, we survey the existing models and results in this area, and point to some of the open research questions. 1 Introduction After a long period of oblivion, interest in analog computation is again on the rise. The immediate cause for this new wave of activity is surely the success of the neural networks "revolution", which has provided hardware designers with several new numerically based, computationally interesting models that are structurally sufficiently simple to be implemented directly in silicon. (For designs and actual implementations of neural models in VLSI, see e.g. [30, 45]). However, the more fundamental...
Global Attractors in Partial Differential Equations
"... this paper, we present the weakly damped Schrodinger equation, which is a system generated by a dispersive equation with weak damping. ..."
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Cited by 18 (0 self)
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this paper, we present the weakly damped Schrodinger equation, which is a system generated by a dispersive equation with weak damping.
Unbounded growth of energy in nonautonomous Hamiltonian systems
 Nonlinearity
, 1998
"... The result of J. Mather on the existence of trajectories with unbounded energy for time periodic Hamiltonian systems on a torus is generalized to a class of multidimensional Hamiltonian systems with Hamiltonian polynomial in momenta. It is assumed that the leading homogeneous term of the Hamiltonian ..."
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Cited by 16 (3 self)
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The result of J. Mather on the existence of trajectories with unbounded energy for time periodic Hamiltonian systems on a torus is generalized to a class of multidimensional Hamiltonian systems with Hamiltonian polynomial in momenta. It is assumed that the leading homogeneous term of the Hamiltonian is autonomous and the corresponding Hamiltonian system has a hyperbolic invariant torus possessing a transversal homoclinic trajectory. Under certain Melnikov type condition, the existence of trajectories with unbounded energy is proved. Instead of the variational methods of Mather, a geometrical approach based on KAM theory and the PoincaréMelnikov method is used. This makes it possible to study a more general class of Hamiltonian systems, but requires additional smoothness assumptions on the Hamiltonian.
Arnold diffusion in perturbations of analytic integrable Hamiltonian systems
, 1998
"... Given an analytic integrable Hamiltonian with three or more degrees of freedom, we construct, arbitrarily close to it, an analytic perturbation with transition chains whose lengths only depend on the unperturbed Hamiltonian. Then we deduce that the perturbed system has Arnold di usion. We provide th ..."
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Cited by 8 (0 self)
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Given an analytic integrable Hamiltonian with three or more degrees of freedom, we construct, arbitrarily close to it, an analytic perturbation with transition chains whose lengths only depend on the unperturbed Hamiltonian. Then we deduce that the perturbed system has Arnold di usion. We provide the technical details of the tools we use.
Remarks on the definition of hyperbolic tori of Hamiltonian systems
, 2000
"... this paper is to discuss the denition of hyperbolic invariant tori of Hamiltonian systems. Hyperbolic tori were rst studied by Poincare. Arnold [1] named hyperbolic tori as whiskered and used them in his famous example of the phenomenon that is now called \Arnold diusion". KAMtype results on the co ..."
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Cited by 8 (0 self)
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this paper is to discuss the denition of hyperbolic invariant tori of Hamiltonian systems. Hyperbolic tori were rst studied by Poincare. Arnold [1] named hyperbolic tori as whiskered and used them in his famous example of the phenomenon that is now called \Arnold diusion". KAMtype results on the conservation of hyperbolic tori under small perturbations were obtained by Gra [16] and Zehnder [27]. In the recent years the notion of a hyperbolic torus was widely used in literature [25, 22, 7], but, as far as we know, there is no universally accepted denition. Usually a denition of hyperbolic torus is given by presenting some normal form of the Hamiltonian in its neighborhood. The goal of this note is to give an invariant (coordinate independent) denition of a hyperbolic turus and derive some consequences. This paper is mainly of the methodological nature and doesn't contain really new results. However, we hope that it will be useful for clarifying the subject.
Hopfsaddlenode bifurcation for fixed points of 3Ddiffeomorphisms: analysis of a resonance ‘bubble’
, 2007
"... ..."
Anosov Flows of Codimension One
, 1995
"... The main goal of this dissertation is to show the existence of global cross sections for certain classes of Anosov flows. Let \Phi be a C codimension one Anosov flow on a compact Riemannian manifold M of dimension greater than three. Verjovsky conjectured that \Phi admits a global cross section ..."
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Cited by 5 (3 self)
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The main goal of this dissertation is to show the existence of global cross sections for certain classes of Anosov flows. Let \Phi be a C codimension one Anosov flow on a compact Riemannian manifold M of dimension greater than three. Verjovsky conjectured that \Phi admits a global cross section and we affirm this conjecture in the following three cases: 1) if the sum, E , of the strong stable and strong unstable bundle of \Phi is Lipschitz; 2) if E is `Holder continuous for all ` ! 1 and \Phi preserves volume; 3) if the center stable distribution of \Phi is of class C for all ` ! 1 and \Phi preserves volume. We note that 1) and 3) generalize the results of Ghys from [Gh3]. For showing 1), we needed to prove a natural generalization of the theorem of Frobenius on integrability of distributions which are only Lipschitz. We also
jets of diffeomorphisms preserving orbits of vector fields
 Central European Journal of Mathematics, available at arXiv:math/0708.0737
"... Abstract. Let F be a C ∞ vector field defined near the origin O ∈ Rn, F(O) = 0, and (Ft) be its local flow. Denote by Ê(F) the set of germs of orbit preserving diffeomorphisms h: R n → R n at O, and let Êid(F) r, (r ≥ 0), be the identity component of Ê(F) with respect to Cr topology. Then Êid(F) ∞ ..."
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Cited by 5 (5 self)
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Abstract. Let F be a C ∞ vector field defined near the origin O ∈ Rn, F(O) = 0, and (Ft) be its local flow. Denote by Ê(F) the set of germs of orbit preserving diffeomorphisms h: R n → R n at O, and let Êid(F) r, (r ≥ 0), be the identity component of Ê(F) with respect to Cr topology. Then Êid(F) ∞ contains a subset ˆ Sh(F) consisting of maps of the form Fα(x)(x), where α: Rn → R runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then ˆ Sh(F) = Êid(F) 0. In this paper we present a class of examples of vector fields with degenerate singularities at O for which ˆ Sh(F) formally coincides with Êid(F) 1, i.e. on the level of ∞jets at O. We also establish parameter rigidity of linear vector fields and “reduced ” Hamiltonian vector fields of real homogeneous polynomials in two variables. Keywords: orbit preserving diffeomorphism, parameter rigidity, Borel’s theorem. AMSClass: 37C10 1.
Infinite number of homoclinic orbits to hyperbolic invariant tori of Hamiltonian systems
 11] Cresson J., Guillet C., Periodic orbits and Arnold diffusion. Preprint 2001. [12] Delshams A., de la Llave
, 2000
"... A timeperiodic Hamiltonian system on a cotangent bundle of a compact manifold with Hamiltonian strictly convex and superlinear in the momentum is studied. A hyperbolic Diophantine nondegenerate invariant torus N is said to be minimal if it is a Peierls set in the sense of the AubryMather theory. W ..."
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Cited by 4 (1 self)
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A timeperiodic Hamiltonian system on a cotangent bundle of a compact manifold with Hamiltonian strictly convex and superlinear in the momentum is studied. A hyperbolic Diophantine nondegenerate invariant torus N is said to be minimal if it is a Peierls set in the sense of the AubryMather theory. We prove that N has an infinite number of homoclinic orbits. For any family of homoclinic orbits the first and last intersection point with the boundary of a tubular neighborhood U of N define sets in U . If no family of minimal homoclinics defines a noncontractible set in U , we obtain an infinite number of multibump homoclinic, periodic and chaotic orbits. The proof is based on a combination of variational methods of Mather and a generalization of Shilnikov's lemma to invariant tori.