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Center manifold

About: Center manifold is a research topic. Over the lifetime, 3465 publications have been published within this topic receiving 72338 citations.


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Book
01 Aug 1983
TL;DR: In this article, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.
Abstract: Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.- Hyperbolic Sets, Sympolic Dynamics, and Strange Attractors.- Global Bifurcations.- Local Codimension Two Bifurcations of Flows.- Appendix: Suggestions for Further Reading. Postscript Added at Second Printing. Glossary. References. Index.

12,669 citations

Book
23 Jul 2012
TL;DR: In this paper, a motivated introduction to functions of bounded variation is given, followed by a digression on weak* integration, and then the authors present an example of a retarded functional differential equation.
Abstract: 0 Introduction and preview.- 0.1 An example of a retarded functional differential equation.- 0.2 Solution operators.- 0.3 Synopsis.- 0.4 A few remarks on history.- I Linear autonomous RFDE.- I.1 Prelude: a motivated introduction to functions of bounded variation.- I.2 Linear autonomous RFDE and renewal equations.- I.3 Solving renewal equations by Laplace transformation.- I.4 Estimates for det ?(z) and related quantities.- I.5 Asymptotic behaviour for t ? ?.- I.6 Comments.- II The shift semigroup.- II.1 Introduction.- II.2 The prototype problem.- II.3 The dual space.- II.4 The adjoint shift semigroup.- II.5 The adjoint generator and the sun subspace.- II.6 The prototype system.- II.7 Comments.- III Linear RFDE as bounded perturbations.- III.1 The basic idea, followed by a digression on weak* integration.- III.2 Bounded perturbations in the sun-reflexive case.- III.3 Perturbations with finite dimensional range.- III.4 Back to RFDE.- III.5 Interpretation of the adjoint semigroup.- III.6 Equivalent description of the dynamics.- III.7 Complexification.- III.8 Remarks about the non-sun-reflexive case.- III.9 Comments.- IV Spectral theory.- IV.1 Introduction.- IV.2 Spectral decomposition for eventually compact semigroups.- IV.3 Delay equations.- IV.4 Characteristic matrices, equivalence and Jordan chains.- IV.5 The semigroup action on spectral subspaces for delay equations.- IV.6 Comments.- V Completeness or small solutions?.- V.l Introduction.- V.2 Exponential type calculus.- V.3 Completeness.- V.4 Small solutions.- V.5 Precise estimates for ??(z)-1?.- V.6 Series expansions.- V.7 Lower bounds and the Newton polygon.- V.8 Noncompleteness, series expansions and examples.- V.9 Arbitrary kernels of bounded variation.- V.10 Comments.- VI Inhomogeneous linear systems.- VI.1 Introduction.- VI.2 Decomposition in the variation-of-constants formula.- VI.3 Forcing with finite dimensional range.- VI.4 RFDE.- VI.5 Comments.- VII Semiflows for nonlinear systems.- VII.1 Introduction.- VII.2 Semiflows.- VII.3 Solutions to abstract integral equations.- VII.4 Smoothness.- VII.5 Linearization at a stationary point.- VII.6 Autonomous RFDE.- VII.7 Comments.- VIII Behaviour near a hyperbolic equilibrium.- VIII.1 Introduction.- VIII.2 Spectral decomposition.- VIII.3 Bounded solutions of the inhomogeneous linear equation.- VIII.4 The unstable manifold.- VIII.5 Invariant wedges and instability.- VIII.6 The stable manifold.- VIII.7 Comments.- IX The center manifold.- IX.1 Introduction.- IX.2 Spectral decomposition.- IX.3 Bounded solutions of the inhomogeneous linear equation.- IX.4 Modification of the nonlinearity.- IX.5 A Lipschitz center manifold.- IX.6 Contractions on embedded Banach spaces.- IX.7 The center manifold is of class Ck.- IX.8 Dynamics on and near the center manifold.- IX.9 Parameter dependence.- IX.10 A double eigenvalue at zero.- IX.11 Comments.- X Hopf bifurcation.- X.l Introduction.- X.2 The Hopf bifurcation theorem.- X.3 The direction of bifurcation.- X.4 Comments.- XI Characteristic equations.- XI.1 Introduction: an impressionistic sketch.- XI.2 The region of stability in a parameter plane.- XI.3 Strips.- XI.4 Case studies.- XI.5 Comments.- XII Time-dependent linear systems.- XII.1 Introduction.- XII.2 Evolutionary systems.- XII.3 Time-dependent linear RFDE.- XII.4 Invariance of X?: a counterexample and a sufficient condition.- XII.5 Perturbations with finite dimensional range.- XII.6 Comments.- XIII Floquet Theory.- XIII.1 Introduction.- XIII.2 Preliminaries on periodicity and a stability result.- XIII.3 Floquet multipliers.- XIII.4 Floquet representation on eigenspaces.- XIII.5 Comments.- XIV Periodic orbits.- XIV.1 Introduction.- XIV.2 The Floquet multipliers of a periodic orbit.- XIV.3 Poincare maps.- XIV.4 Poincare maps and Floquet multipliers.- XIV.5 Comments.- XV The prototype equation for delayed negative feedback: periodic solutions.- XV.1 Delayed feedback.- XV.2 Smoothness and oscillation of solutions.- XV.3 Slowly oscillating solutions.- XV.4 The a priori estimate for unstable behaviour.- XV.5 Slowly oscillating solutions which grow away from zero, periodic solutions.- XV.6 Estimates, proof of Theorem 5.5(i) and (iii).- XV.7 The fixed-point index for retracts in Banach spaces, Whyburn'0 Introduction and preview.- 0.1 An example of a retarded functional differential equation.- 0.2 Solution operators.- 0.3 Synopsis.- 0.4 A few remarks on history.- I Linear autonomous RFDE.- I.1 Prelude: a motivated introduction to functions of bounded variation.- I.2 Linear autonomous RFDE and renewal equations.- I.3 Solving renewal equations by Laplace transformation.- I.4 Estimates for det ?(z) and related quantities.- I.5 Asymptotic behaviour for t ? ?.- I.6 Comments.- II The shift semigroup.- II.1 Introduction.- II.2 The prototype problem.- II.3 The dual space.- II.4 The adjoint shift semigroup.- II.5 The adjoint generator and the sun subspace.- II.6 The prototype system.- II.7 Comments.- III Linear RFDE as bounded perturbations.- III.1 The basic idea, followed by a digression on weak* integration.- III.2 Bounded perturbations in the sun-reflexive case.- III.3 Perturbations with finite dimensional range.- III.4 Back to RFDE.- III.5 Interpretation of the adjoint semigroup.- III.6 Equivalent description of the dynamics.- III.7 Complexification.- III.8 Remarks about the non-sun-reflexive case.- III.9 Comments.- IV Spectral theory.- IV.1 Introduction.- IV.2 Spectral decomposition for eventually compact semigroups.- IV.3 Delay equations.- IV.4 Characteristic matrices, equivalence and Jordan chains.- IV.5 The semigroup action on spectral subspaces for delay equations.- IV.6 Comments.- V Completeness or small solutions?.- V.l Introduction.- V.2 Exponential type calculus.- V.3 Completeness.- V.4 Small solutions.- V.5 Precise estimates for ??(z)-1?.- V.6 Series expansions.- V.7 Lower bounds and the Newton polygon.- V.8 Noncompleteness, series expansions and examples.- V.9 Arbitrary kernels of bounded variation.- V.10 Comments.- VI Inhomogeneous linear systems.- VI.1 Introduction.- VI.2 Decomposition in the variation-of-constants formula.- VI.3 Forcing with finite dimensional range.- VI.4 RFDE.- VI.5 Comments.- VII Semiflows for nonlinear systems.- VII.1 Introduction.- VII.2 Semiflows.- VII.3 Solutions to abstract integral equations.- VII.4 Smoothness.- VII.5 Linearization at a stationary point.- VII.6 Autonomous RFDE.- VII.7 Comments.- VIII Behaviour near a hyperbolic equilibrium.- VIII.1 Introduction.- VIII.2 Spectral decomposition.- VIII.3 Bounded solutions of the inhomogeneous linear equation.- VIII.4 The unstable manifold.- VIII.5 Invariant wedges and instability.- VIII.6 The stable manifold.- VIII.7 Comments.- IX The center manifold.- IX.1 Introduction.- IX.2 Spectral decomposition.- IX.3 Bounded solutions of the inhomogeneous linear equation.- IX.4 Modification of the nonlinearity.- IX.5 A Lipschitz center manifold.- IX.6 Contractions on embedded Banach spaces.- IX.7 The center manifold is of class Ck.- IX.8 Dynamics on and near the center manifold.- IX.9 Parameter dependence.- IX.10 A double eigenvalue at zero.- IX.11 Comments.- X Hopf bifurcation.- X.l Introduction.- X.2 The Hopf bifurcation theorem.- X.3 The direction of bifurcation.- X.4 Comments.- XI Characteristic equations.- XI.1 Introduction: an impressionistic sketch.- XI.2 The region of stability in a parameter plane.- XI.3 Strips.- XI.4 Case studies.- XI.5 Comments.- XII Time-dependent linear systems.- XII.1 Introduction.- XII.2 Evolutionary systems.- XII.3 Time-dependent linear RFDE.- XII.4 Invariance of X?: a counterexample and a sufficient condition.- XII.5 Perturbations with finite dimensional range.- XII.6 Comments.- XIII Floquet Theory.- XIII.1 Introduction.- XIII.2 Preliminaries on periodicity and a stability result.- XIII.3 Floquet multipliers.- XIII.4 Floquet representation on eigenspaces.- XIII.5 Comments.- XIV Periodic orbits.- XIV.1 Introduction.- XIV.2 The Floquet multipliers of a periodic orbit.- XIV.3 Poincare maps.- XIV.4 Poincare maps and Floquet multipliers.- XIV.5 Comments.- XV The prototype equation for delayed negative feedback: periodic solutions.- XV.1 Delayed feedback.- XV.2 Smoothness and oscillation of solutions.- XV.3 Slowly oscillating solutions.- XV.4 The a priori estimate for unstable behaviour.- XV.5 Slowly oscillating solutions which grow away from zero, periodic solutions.- XV.6 Estimates, proof of Theorem 5.5(i) and (iii).- XV.7 The fixed-point index for retracts in Banach spaces, Whyburn's lemma.- XV.8 Proof of Theorem 5.5(ii) and (iv).- XV.9 Comments.- XVI On the global dynamics of nonlinear autonomous differential delay equations.- XVI.1 Negative feedback.- XVI.2 A limiting case.- XVI.3 Chaotic dynamics in case of negative feedback.- XVI.4 Mixed feedback.- XVI.5 Some global results for general autonomous RFDE.- Appendices.- I Bounded variation, measure and integration.- I.1 Functions of bounded variation.- I.2 Abstract integration.- II Introduction to the theory of strongly continuous semigroups of bounded linear operators and their adjoints.- II. 1 Strongly continuous semigroups.- II.2 Interlude: absolute continuity.- II.3 Adjoint semigroups.- II.4 Spectral theory and asymptotic behaviour.- III The operational calculus.- III.1 Vector-valued functions.- III.2 Bounded operators.- III.3 Unbounded operators.- IV Smoothness of the substitution operator.- V Tangent vectors, Banach manifolds and transversality.- V.1 Tangent vectors of subsets of Banach spaces.- V.2 Banach manifolds.- V.3 Submanifolds and transversality.- VI Fixed points of parameterized contractions.- VII Linear age-dependent population growth: elaboration of some of the exercises.- VIII The Hopf bifurcation theorem.- References.- List of symbols.- List of notation.

1,000 citations

Book ChapterDOI
Mikio Sato1
TL;DR: In this paper, the authors interpreted the time evolution of a solution as the dynamical motion of a point on a Grassmann manifold, and a generic solution corresponds to a generic point whose orbit (in the infinitely many time variables) is dense in the manifold, whereas degenerate solutions corresponding to points bound on those closed submanifolds that are stable under the time evolve describe the solutions to various specialized equations, such as KdV, Boussinesq, nonlinear Schrodinger, and sine-Gordon.
Abstract: Publisher Summary Soliton Equations as Dynamical Systems on Infinite Dimensional Grassmann Manifold The totality of the solutions of the Kadomtsev– Petviashvili equation as well as of its multicomponent generalization forms an infinite dimensional Grassmann manifold. In this picture, the time evolution of a solution is interpreted as the dynamical motion of a point on this manifold. A generic solution corresponds to a generic point whose orbit (in the infinitely many time variables) is dense in the manifold, whereas degenerate solutions corresponding to points bound on those closed submanifolds that are stable under the time evolution describe the solutions to various specialized equations, such as KdV, Boussinesq, nonlinear Schrodinger, and sine-Gordon.

835 citations

Journal ArticleDOI
TL;DR: In this paper, a methodology is presented which extends to non-linear systems the concept of normal modes of motion which is well developed for linear systems and demonstrates how an approximate nonlinear version of superposition can be employed to reconstruct the overall motion from the individual nonlinear modal dynamics.

568 citations

Proceedings Article
01 Jan 2002
TL;DR: A nonlinear mapping from a high-dimensional sample space to a low-dimensional vector space is constructed, effectively recovering a Cartesian coordinate system for the manifold from which the data is sampled, and is pseudo-invertible.
Abstract: We construct a nonlinear mapping from a high-dimensional sample space to a low-dimensional vector space, effectively recovering a Cartesian coordinate system for the manifold from which the data is sampled. The mapping preserves local geometric relations in the manifold and is pseudo-invertible. We show how to estimate the intrinsic dimensionality of the manifold from samples, decompose the sample data into locally linear low-dimensional patches, merge these patches into a single low-dimensional coordinate system, and compute forward and reverse mappings between the sample and coordinate spaces. The objective functions are convex and their solutions are given in closed form.

489 citations


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No. of papers in the topic in previous years
YearPapers
202391
2022181
2021127
2020114
2019127
2018125