Results 1  10
of
67
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. ..."
Abstract

Cited by 42 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 39 (3 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 33 (6 self)
 Add to MetaCart
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...
Hopfsaddlenode bifurcation for fixed points of 3Ddiffeomorphisms: analysis of a resonance ‘bubble’
, 2007
"... ..."
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 ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
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 t ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
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.
Connectivity and design of planar global attractors of Sturm type. III: Small and Platonic examples. Submitted
, 2007
"... Based on a MorseSmale structure we study planar global attractors Af of the scalar reactionadvectiondiffusion equation ut = uxx + f(x, u, ux) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of f, and hyperbolicity of equilibria. We call Af Stur ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
(Show Context)
Based on a MorseSmale structure we study planar global attractors Af of the scalar reactionadvectiondiffusion equation ut = uxx + f(x, u, ux) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of f, and hyperbolicity of equilibria. We call Af Sturm attractor because our results strongly rely on nonlinear nodal properties of Sturm type. The planar Sturm attractor consists of equilibria of Morse index 0, 1, or 2, and their heteroclinic connecting orbits. The unique heteroclinic orbits between adjacent Morse levels define a plane graph Cf which we call the connection graph. Its 1skeleton C1f consists of the unstable manifolds (separatrices) of the index1 Morse saddles. We present two results which completely characterize the connection graphs Cf and their 1skeletons C1f, in purely graph theoretical terms. Connection graphs are characterized by the existence of pairs of Hamiltonian paths with certain chiral restrictions on face passages. Their 1skeletons are characterized by the existence of cyclefree orientations with only one maximum and only one minimum. Such orientations are called bipolar in [FMR95]. In the present paper we show the equivalence of the two characterizations. Moreover we show that connection graphs of Sturm attractors indeed satisfy the required properties. In [FiRo07a] we show, conversely, how to design a planar Sturm attractor with prescribed plane connection graph or 1skeleton of the required properties. In [FiRo07b] we describe all planar Sturm attractors with up to 11 equilibria. We also design planar Sturm attractors with prescribed Platonic 1skeletons. 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 ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
(Show Context)
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.
LORENZ LIKE FLOWS: EXPONENTIAL DECAY OF CORRELATIONS FOR THE POINCARÉ MAP, LOGARITHM LAW, QUANTITATIVE RECURRENCE
, 2009
"... In this paper we prove that the Poincaré map associated to a Lorenz like flow has exponential decay of correlations with respect to Lipschitz observables. This implies that the hitting time associated to the flow satisfies a logarithm law. The hitting time τr(x, x0) is the time needed for the orbit ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
(Show Context)
In this paper we prove that the Poincaré map associated to a Lorenz like flow has exponential decay of correlations with respect to Lipschitz observables. This implies that the hitting time associated to the flow satisfies a logarithm law. The hitting time τr(x, x0) is the time needed for the orbit of a point x to enter for the first time in a ball Br(x0) centered at x0, with small radius r. As the radius of the ball decreases to 0 its asymptotic behavior is a power law whose exponent is related to the local dimension of the SRB measure at x0: for each x0 such that the local dimension dµ(x0) exists, log τr(x, x0) lim = dµ(x0) − 1 r→0 − log r holds for µ almost each x. In a similar way it is possible to consider a quantitative recurrence indicator quantifying the speed of coming back of an orbit to its starting point. Similar
Perturbation Theory of Dynamical Systems
, 2001
"... Please send any comments to berglund@math.ethz.ch Zurich, November 2001 3 4 Contents 1 Introduction and Examples 1 1.1 OneDimensional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Forced Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Si ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Please send any comments to berglund@math.ethz.ch Zurich, November 2001 3 4 Contents 1 Introduction and Examples 1 1.1 OneDimensional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Forced Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Singular Perturbations: The Van der Pol Oscillator . . . . . . . . . . . . . . 10 2 Bifurcations and Unfolding 13 2.1 Invariant Sets of Planar Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 Periodic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.3 The PoincareBendixson Theorem . . . . . . . . . . . . . . . . . . . 21 2.2 Structurally Stable Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 Definition of Structural Stability . . . . . . . . . . . . . . . . . . . . 23 2.2.2 Peixot