Infinite dimensional dynamical systems

✍ Scribed by John Mallet-Paret; et al (eds.)

• Publisher: Springer
• Year: 2013
• Tongue: English
• Leaves: 495
• Series: Fields Institute communications, v. 64
• Category: Library

✦ Synopsis

This collection covers a wide range of topics of infinite dimensional dynamical systems generated by parabolic and hyperbolic partial differential equations, solitary equations, lattice differential equations, delay differential equations, and stochastic differential equations.Infinite dimensional dynamical systems are generated by equations describing the evolution in time of systems whose status must be depicted in infinite dimensional phase spaces. Studying the long-term behaviors of such systems is important in our understanding of their spatiotemporal pattern formation and global continuation, and has been among the major sources of motivation and applications of new developments in nonlinear analysis and other mathematical theories. The theory of infinite dimensional dynamical systems has also increasingly important applications in the physical, chemical and life sciences.This book collects 19 papers from 48 invited lecturers to the International Conference on Infinite Dimensional Dynamical Systems held at York University, Toronto, in September of 2008. As the conference was dedicated to Professor George Sell from University of Minnesota on the occasion of his 70th birthday, this collection reflects his pioneering work and influence in core areas of dynamical systems, including non-autonomous dynamical systems, skew-product flows, invariant manifolds theory, infinite dimensional dynamical systems, approximation dynamics, and fluid flows. Read more... Preface.- Persistence of Periodic Orbits for Perturbed Dissipative Dynamical Systems (J. Hale, G. Raugel).- Spectral Theory for Forward Nonautonomus Parabolic Equations and Applications (J. Mierczynski, W. Shen).- A Dynamical Systems Approach to Traveling Wave Solutions for Liquid/Vapor Phase Transition (H. Fan, X. Lin).- Instability of Radially-Symmetric Spikes in Systems with Conserved Quantity (A. Pogan, A. Scheel).- Global Hopf Bifurcation Analysis of a Neuron Network Model with Time Delays (M. Li, J. Wei).- Instability of Low Density Supersonic Waves of a Viscous Isentropic Gas Flow Through a Nozel (W. Liu, M. Oh).- A Simple Proof of the Stability of Solitary Waves in the Fermi-Pasta-Ulam Model Near the KdV Limit (A. Hoffman, G. Wayne).- Littlewood Problem for a Singular Subquadratic Potential (X. Li, Y. Yi).- Semiflows for Neutral Equations with State-dependent Delay (H.-O. Walther).- Threshold Dynamics of Scalar Linear Periodic Delay-Differential Equations (Y. Chen, J. Wu).- Differential Equations with Random Delay (T.S. Doan, S. Siegmund).- Beyond Diffusion: Conditional Dispersal in Ecological Models (C. Cosner).- Global Attractor of a Coupled Two-Cell Brusselator Model (Y. You).- Projectors on the Generalized Eigenspaces for Partial Differential Equations with Time Delay (A. Ducrot, P. Magal, S. Ruan).- Global Convergence in Monotone and Uniformly Stable Recurrent Skew-Product Semiflows (Y. Wang, X. Zhao).- The Infinite Hierarchy of Elastic Shell Models: Some Recent Results and a Conjecture (M. Lewicka, R. Pakzad).- Traveling Wavefronts for Lattic Differential Equations with Time Delay and Global Interaction (S. Ma, Z. Zou).- Bifurcation of Limit Cycles from a Non-Hamiltonian Quadratic Integrable System with Homoclinic Loop (Y. Zhao, H. Zhu) Anomalous Diffusion in Polymers: Long-Time Behaviour (D. Vorotnikov)

✦ Table of Contents

Cover......Page 1
Infinite Dimensional
Dynamical Systems......Page 4
Anomalous Diffusion in Polymers: Long-Time Behaviour......Page 6
Contents......Page 8
1 Introduction......Page 10
2 An Integral Equation Method......Page 15
2.1 An Abstract Result......Page 17
2.1.1 Step 1......Page 29
2.1.2 Step 2......Page 34
2.2 Comparison with the Modified Poincaré Method......Page 42
2.3 Some Examples of Applications......Page 47
2.3.1 Systems of Weakly Damped Wave Equations with Positive Damping......Page 48
2.3.2 Systems of Weakly Damped Wave Equations with Variable Non-negative Damping......Page 51
2.3.3 A System of Damped Wave Equations in a Thin Domain......Page 52
3.1 The Case of Ordinary Differential Equations......Page 55
3.2 The Infinite-Dimensional Case......Page 59
References......Page 63
1 Introduction......Page 65
2.1 Assumptions......Page 68
2.2 Weak Solutions: Definition......Page 73
2.3 Weak Solutions: Basic Properties......Page 75
3.1 Definition......Page 78
3.2 Fundamental Properties......Page 79
3.3 Monotonicity and Continuity with Respect to Zero Order Terms......Page 81
4 Exponential Separation and Equivalent Definition......Page 83
4.1 Definitions and Characterizations......Page 84
4.2 The Classical Case: An Example......Page 89
5.1 Continuity with Respect to the Coefficients......Page 92
5.2 Time Averaging......Page 94
5.3 Space-Averaging......Page 99
6 Applications to Nonlinear Equations of Kolmogorov Type......Page 100
References......Page 105
1 Introduction......Page 108
2.1 Eigenvalues and Eigenvectors at Equilibrium Points......Page 115
2.2 Existence of Liquefaction Waves for λ-=0, λ+=1......Page 117
2.3 Existence of Liquefaction Waves for λ-=0, 0<λ+<1, p+=pe......Page 121
References......Page 123
1 Introduction......Page 125
2 Families of Spikes......Page 128
3 Essential Spectrum......Page 131
4 Tracing the Point Spectrum......Page 132
4.1 Homotopy......Page 133
4.3 The Kernel of Lrad......Page 134
4.4 Fredholm Properties of Lrad on Exponentially Weighted Spaces......Page 135
4.5 Asymptotics of Eigenfunctions......Page 136
4.6 The Eigenvalue Problem Near Zero......Page 137
4.7 Proof of Theorem 1.1......Page 143
References......Page 146
1 Introduction......Page 147
2 Stability and Local Hopf Bifurcation......Page 148
3 Direction and Stability of the Local Hopf Bifurcation at 0......Page 152
4 Global Existence of Periodic Solutions......Page 163
References......Page 173
1 Introduction......Page 175
2 Steady-State Problem......Page 178
3 The Eigenvalue Problem and a Center Manifold Reduction......Page 180
4 The Limiting Eigenvalue Problem with =0......Page 183
5 Instability of Low Density Steady-States......Page 186
References......Page 188
1 Introduction......Page 190
References......Page 197
1 Introduction......Page 198
2 Action-Angle Variables......Page 200
3 New Action-Angle Variables......Page 204
4 Proof of the Main Result......Page 207
References......Page 214
1 Introduction......Page 215
2 Preliminaries......Page 220
3 The Example......Page 227
4 The Semiflow on a Subset of C1......Page 229
5 A Submanifold in C2......Page 238
6 A Semiflow on a Subset of the Manifold X2C2......Page 239
7 A Variational Equation......Page 244
8 A Differentiability Property of Solution Operators......Page 251
9 More on the Variational Equation......Page 257
10 Points of Continuity of Derivatives......Page 262
11 Appendix on Inhomogeneous Linear Delay Differential Equations......Page 267
References......Page 270
1 Introduction......Page 272
2 Preliminaries......Page 274
3 Threshold Dynamics in the Case Where a>0......Page 276
4 Discussion for the Case Where a<0......Page 279
References......Page 281
1 Introduction......Page 282
2 Differential Equations with Random Delay......Page 283
3.1 Integrability......Page 291
3.2 Kuratowski Measure......Page 294
3.3 Multiplicative Ergodic Theorem......Page 298
4 Differential Equations with Bounded Delay......Page 303
References......Page 306
1 Introduction......Page 308
2.1 General Background......Page 309
2.2 Evolution of Dispersal......Page 310
3.1 Systems with Simple Diffusion......Page 311
3.2 Models with Taxis up Environmental Gradients......Page 312
4.1 Definition and Connections to Dispersal Models......Page 315
4.2 Results on an Approximately Ideal Free Model......Page 316
4.3 Remarks......Page 318
References......Page 319
1 Introduction......Page 320
2 Absorbing Property......Page 325
3 κ-Contracting Property for the (v, z) Components......Page 332
4 κ-Contracting Property for the (u, w) Components......Page 341
5 Finite Dimensionality of the Global Attractor......Page 347
References......Page 351
1 Introduction......Page 354
2 Preliminaries......Page 357
3 Some Results on Integrated Solutions and Spectra......Page 359
4 Projectors on the Eigenspaces......Page 369
5 Projector for a Simple Eigenvalue......Page 378
6 Comments on Semilinear Problems and Examples......Page 380
6.1 A Reaction-Diffusion Model with Delay (B is Densely Defined)......Page 383
6.2 An Age-Structured Model with Delay (B is Non-Densely Defined)......Page 386
References......Page 389
1 Introduction......Page 392
2 Preliminaries......Page 394
3 Global Convergence......Page 396
4 Applications......Page 401
References......Page 406
1 Introduction......Page 408
2.1 Three Dimensional Nonlinear Elasticity and the Limiting Lower Dimensional Theories......Page 409
2.2 -Convergence......Page 410
2.3 A Glance at Previously known Results.......Page 411
3 The Kirchhoff Theory for Shells: β=2 and Arbitrary S......Page 412
4 The von-Kármán Theory for Shells: β=4 and Arbitrary S......Page 413
5 The Linear Theory for Shells: bold0mu mumu ββequationββββ>4 and Arbitrary S; β=4 and Approximately Robust S......Page 414
6 Intermediate Theories for Plates and Convex Shells: β(2,4)......Page 415
7 A Conjecture on the Infinite Hierarchy of Shell Models......Page 417
References......Page 420
1 Introduction......Page 423
2 Operational Equations for Profile of Traveling Waves......Page 426
3 Properties of the Operators L and G......Page 428
4 Proof of the Main Theorem......Page 435
5 Applications......Page 438
References......Page 443
1 Introduction......Page 445
2 Reduction of the System and Bifurcation Function......Page 447
3 Picard–Fuchs System and Some Preliminary Results......Page 451
4 Estimate for the Number of Zeros of I(h)......Page 460
5 Proof of Theorem 1.1......Page 469
References......Page 478
1 Introduction......Page 480
2 Function Spaces and Related Notations......Page 482
3 Basic Properties of the Boundary Value Problem......Page 484
4 Semigroups, Semiflows, Trajectory Attractors and Global Attractors......Page 490
5 Attractors for the Polymeric Diffusion Problem......Page 493
References......Page 494