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Invariant Manifolds for Analytic Difference Equations

SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2012年第4期11卷 1614-1651页

作者： de la Llave, Rafael Lomeli, Hector E. Georgia Inst Technol Sch Math Atlanta GA 30332 USA Inst Tecnol Autonomo Mexico Dept Math Mexico City 01000 DF Mexico

We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, and smooth dependence on parameters and study several singular limits, even if the difference equations do not define a dynamical system. This method also leads to efficient algorithms that we present with their implementations. The manifolds that we consider include not only the classical strong stable and unstable manifolds but also manifolds associated with nonresonant spaces. When the difference equations are the Euler-Lagrange equations of a discrete variational problem, we have sharper results. Note that, if the Legendre condition fails, the Euler-Lagrange equations cannot be treated as a dynamical system. If the Legendre condition becomes singular, the dynamical system may be singular while the difference equation remains regular. We present numerical applications to several examples in the physics literature: the Frenkel-Kontorova model with long-range interactions and the Heisenberg model of spin chains with a perturbation. We also present extensions to finite differentiable difference equations.

关键词： invariant manifold stable manifold unstable manifold difference equations Euler-Lagrange parameterization method Frenkel-Kontorova Heisenberg model functional equation recurrence

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Computer Assisted Existence Proofs of Lyapunov Orbits at L₂ and Transversal Intersections of Invariant Manifolds in the Jupiter-Sun PCR3BP

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2012年第4期11卷 1723-1753页

作者： Capinski, Maciej J. AGH Univ Sci & Technol Fac Appl Math PL-30059 Krakow Poland

We present a computer assisted proof of existence of a family of Lyapunov orbits which stretches from L-2 (the collinear libration point between the primaries) up to half the distance to the smaller primary in the Jupiter-Sun planar circular restricted three body problem. We then focus on a small family of Lyapunov orbits with energies close to comet Oterma and show that their associated invariant manifolds intersect transversally. Our computer assisted proof provides explicit bounds on the location and on the angle of intersection.

关键词： invariant manifolds restricted three body problem cone conditions parameterization method computer assisted proofs

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RIGOROUS NUMERICS FOR SYMMETRIC CONNECTING ORBITS: EVEN HOMOCLINICS OF THE GRAY-SCOTT EQUATION

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SIAM JOURNAL ON MATHEMATICAL ANALYSIS 2011年第4期43卷 1557-1594页

作者： van den Berg, Jan Bouwe Mireles-James, Jason D. Lessard, Jean-Philippe Mischaikow, Konstantin Vrije Univ Amsterdam Dept Math NL-1081 HV Amsterdam Netherlands Rutgers State Univ Dept Math Piscataway NJ 08854 USA BCAM Basque Ctr Appl Math Derio 48160 Bizkaia Spain Rutgers State Univ BioMaPS Piscataway NJ 08854 USA

In this paper we propose a rigorous numerical technique for the computation of symmetric connecting orbits for ordinary differential equations. The idea is to solve a projected boundary value problem (BVP) in a function space via a fixed point argument. The formulation of the projected BVP involves a high order parameterization of the invariant manifolds at the steady states. Using this parameterization, one can obtain explicit exponential asymptotic bounds for the coefficients of the expansion of the manifolds. Combining these bounds with piecewise linear approximations, one can construct a contraction in a function space whose unique fixed point corresponds to the wanted connecting orbit. We have implemented the method to demonstrate its effectiveness, and we have used it to prove the existence of a family of even homoclinic orbits for the Gray-Scott equation.

关键词： computer assisted proof validated continuation invariant manifolds parameterization method projected boundary conditions radii polynomials

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The parameterization method in optimal control problems and differential-algebraic equations

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2006年第2期185卷 377-390页

作者： Gorbunov, VK Lutoshkin, IV Ulyanovsk State Univ Ulyanovsk 432970 Russia

The paper is devoted to the explanation of the numerical parameterization method (PM) for optimal control (OC) problems with intermediate phase constraint and to its circurnstantiation for classical calculus of variation (CV) problems that arise in connection with singular ODEs or DAEs, especially in cases of their essential degeneracy. The PM is based on a finite parameterization of control functions and on derivation of the problem functional with respect to control parameters. The first and the second derivatives are calculated with the help of adjoint vector and matrix impulses. Results of the solution to one phase constrained OC and two degenerate CV problems, connected with singular DAEs nonreducible to the normal form, are presented. (c) 2005 Elsevier B.V. All rights reserved.

关键词： optimal control phase restrictions calculus of variations differential-algebraic equations parameterization method degenerate problems

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The parameterization method in optimal control problems and differential-algebraic equations

The parameterization method in optimal control problems and ...

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International Workshop on the Technological Aspects of Mathematics

作者： Gorbunov, VK Lutoshkin, IV Ulyanovsk State Univ Ulyanovsk 432970 Russia

关键词： optimal control phase restrictions calculus of variations differential-algebraic equations parameterization method degenerate problems

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A Computational and Geometric Approach to Phase Resetting Curves and Surfaces

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2009年第3期8卷 1005-1042页

作者： Guillamon, Antoni Huguet, Gemma Univ Politecn Cataluna Dept Matemat Aplicada 1 E-08028 Barcelona Catalonia Spain Ctr Recerca Matemat E-08193 Barcelona Catalonia Spain

This work arises from the purpose of applying new tools in dynamical systems to time problems in biological systems. The main aim of this paper is to develop a numerical method to perform the effective computation of the phase advancement when we stimulate an oscillator which has not yet reached the asymptotic state (a limit cycle). That is, we want to extend the computation of the phase resetting curves (PRCs) (the classical tool to compute the phase advancement) to a neighborhood of the limit cycle, obtaining what we call the phase resetting surfaces (PRSs). To achieve this goal we first perform a careful study of the theoretical grounds (the parameterization method for invariant manifolds and another approach using Lie symmetries), which allows us to describe the isochronous sections of the limit cycle and, from them, to obtain the PRSs. In order to make this theoretical framework applicable, we use the numerical algorithms of the parameterization method and other semianalytical tools to extend invariant manifolds;as a result, we design a numerical scheme to compute both the isochrons and the PRSs of a given oscillator. Finally, to illustrate this algorithm, we apply it to some well-known biological models and we include a discussion on different biological and numerical aspects suggested by these examples.

关键词： parameterization method Lie symmetries isochrons phase resetting curves numerical computation of invariant objects biological oscillators

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The parameterization method for invariant manifolds III:: Overview and applications

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JOURNAL OF DIFFERENTIAL EQUATIONS 2005年第2期218卷 444-515页

作者： Cabré, X Fontich, E de la Llave, R Univ Texas Dept Math Austin TX 78712 USA ICREA Barcelona 08028 Spain Univ Politecn Catalunya Dept Matemat Aplicada 1 E-08028 Barcelona Spain Univ Barcelona Dept Matemat Aplicada & Anal E-08007 Barcelona Spain

We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We also present several other applications of the method. (c) 2005 Elsevier Inc. All rights reserved.

关键词： parameterization method invariant manifolds slow manifolds resonances

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Iterative techniques for computing the linearized manifolds of quasiperiodic tori

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CHAOS 2006年第2期16卷 23129-1-23129页

作者： Wysham, Derin B. Meiss, James D. Univ Colorado Boulder CO 80309 USA

We develop an iterative technique for computing the unstable and stable eigenfunctions of the invariant tori of diffeomorphisms. Using the approach of Jorba [Nonlinearity 14, 943 (2001)], the linearized equations are rewritten as a generalized eigenvalue problem. Casting the system in this light allows us to take advantage of the speed of eigenvalue solvers and create an efficient method for finding the first-order approximations to the invariant manifolds of the torus. We present a numerical scheme based on the power method that can be used to determine the behavior normal to such tori, and give some examples of the application of the method. We confirm the qualitative conclusions of the Melnikov calculations of Lomeli and Meiss [Nonlinearity 16, 1573 (2003)] for a volume-preserving mapping.

关键词： VOLUME-PRESERVING MAPS INVARIANT-MANIFOLDS parameterization method SCHRODINGER-OPERATORS DIFFERENTIAL SYSTEMS UNSTABLE MANIFOLDS INTERSECTIONS REDUCIBILITY COMPUTATION SPECTRUM

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The obstruction criterion for non-existence of invariant circles and renormalization

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NONLINEARITY 2006年第8期19卷 1907-1937页

作者： de la Llave, Rafael Olvera, Arturo Univ Texas Dept Math Austin TX 78712 USA Univ Nacl Autonoma Mexico IIMAS Dept Matemat & Mecan Mexico City 01000 DF Mexico

The goal of this paper is to show that the renormalization group and the obstruction criterion can work together. We formulate a conjecture which supplements the standard renormalization scenario for the breakdown of golden circles in twist maps. We show rigorously that if the conjecture was true then the following hold. (1) The stable manifold of the non-trivial fixed point would be part of the boundary between the existence of smooth invariant tori and hyperbolic orbits with golden mean rotation number. In particular, the boundary of the set of twist maps with a torus with a golden mean rotation number would include a smooth submanifold in the space of analytic mappings. Moreover, if the conjecture was true, in the domain of universality (i.e. a small neighbourhood of the non-trivial fixed point), we would have the following (2),(3),(4). (2) The obstruction criterion for non-existence of golden mean invariant circles (Olvera and Simo 1987 Physica D 26 181-92) is sharp. That is, for maps in the university class there is either a golden invariant circle or the condition in Olvera and Simo for non-existence of golden circles applies. (3) The criterion of Greene (1979 J. Math. Phys. 20 1183-201) for existence of invariant circles if and only if there the residues of approximating orbits are finite would be valid. That is, for maps in the universality class there would be a smooth invariant circle if and only if the residue of periodic orbits approximating the circle goes to zero. (4) If there is no invariant circle, there are uniformly hyperbolic sets with golden mean rotation number. We also provide numerical evidence which suggests that the conjecture is true. We derive several scaling relations for observables related to the obstruction criterion and verify them.

关键词： AREA-PRESERVING MAPS PERIODIC-ORBITS parameterization method HAMILTONIANFLOWS TWIST MAPS MANIFOLDS TORI FREQUENCY EXISTENCE BEHAVIOR

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method of linearizing the 3d L₃/L₂ white line ratio as a function of magnetic moment

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ULTRAMICROSCOPY 2001年第1期88卷 1-16页

作者： Pease, DM Fasihuddin, A Daniel, M Budnick, JI Univ Connecticut Dept Phys Storrs CT 06269 USA

We have developed a parameterization method which linearizes the relationship between local magnetic moment and the 3d L-3/L-2 "white line" ratio as observed in electron energy loss spectroscopy or X-ray near edge absorption spectroscopy. We first establish that the parameterization linearizes an existing theoretical result for ratio versus moment. We then test our method on data sets for which a white line ratio has been previously published by other authors, who have studied a series of compounds using a consistent deconvolution procedure. Finally, we apply our linearization method to the observed ratios of a series of 3d transition metals, and to the Cr L edges for a AuxCr((1-x)) alloy. In addition we obtain, for the first time, experimental results on the Au L-3 and L-2 edge white lines of this alloy system. These results are consistent with a model in which the large local moment in this system is not limited to Cr dopants, but extends into the gold matrix. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： parameterization method 3d L-3/L-2 line ratio magnetic moment

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