Evolution Semigroups in Dynamical Systems and Differential Equations

Introduction Semigroups on Banach spaces and evolution semigroups Evolution families and Howland semigroups Characterizations of dichotomy for evolution families Two applications of evolution semigroups Linear skew-product flows and Mather evolution semigroups Characterizations of dichotomy for linear skew-product flows Evolution operators and exact Lyapunov exponents Bibliography List of notations Index.

[1]  The structurally stable linear systems on the half-line are those with exponential dichotomies , 1979 .

[2]  R. Vinograd Exact bounds for exponential dichotomy roughness III. Semistrong dichotomy , 1991 .

[3]  On the stability of differentiability of semigroups , 1995 .

[4]  Yu. I. Lyubich,et al.  Asymptotic stability of linear differential equations in Banach spaces , 1988 .

[5]  N. Lloyd,et al.  ALMOST PERIODIC FUNCTIONS AND DIFFERENTIAL EQUATIONS , 1984 .

[6]  Semi-groupes generalises et equations d’evolution , 1979 .

[7]  S. Abhyankar Algebraic geometry for scientists and engineers , 1990 .

[8]  J. Cronin Fixed points and topological degree in nonlinear analysis , 1995 .

[9]  Weiyao Zeng Exponential dichotomies and transversal homoclinic orbits in degenerate cases , 1995 .

[10]  André Vanderbauwhede,et al.  Center Manifold Theory in Infinite Dimensions , 1992 .

[11]  Y. Yi,et al.  On minimal sets of scalar parabolic equations with skew-product structures , 1995 .

[12]  Aloisio Neves,et al.  On the spectrum of evolution operators generated by hyperbolic systems , 1986 .

[13]  George R. Sell,et al.  Dichotomies for linear evolutionary equations in Banach spaces , 1994 .

[14]  Michael Renardy Spectrally determined growth is generic , 1996 .

[15]  Richard Rebarber Frequency domain methods for proving the uniform stability of vibrating systems , 1993 .

[16]  F. Ledrappier,et al.  A Relativised Variational Principle for Continuous Transformations , 1977 .

[17]  J. J. Schaffer,et al.  LINEAR DIFFERENTIAL EQUATIONS AND FUNCTIONAL ANALYSIS, III. LYAPUNOV'S SECOND METHOD IN THE CASE OF CONDITIONAL STABILITY , 1959 .

[18]  Semigroups and stability of nonautonomous differential equations in Banach spaces , 1994 .

[19]  George R. Sell,et al.  The spectrum of an invariant submanifold , 1980 .

[20]  Yingfei Yi,et al.  Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows , 1998 .

[21]  A note on Anosov diffeomorphisms , 1974 .

[22]  L. Young,et al.  STATISTICAL PROPERTIES OF DYNAMICAL SYSTEMS WITH SOME HYPERBOLICITY , 1998 .

[23]  A short proof for the stability theorem for positive semigroups on $L , 1998 .

[24]  Bert van Keulen,et al.  H-Infinity-Control for Distributed Parameter Systems: A State-Space Approach , 1993 .

[25]  Richard E. Mortensen,et al.  Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam) , 1991, SIAM Rev..

[26]  H. Rodrigues,et al.  Evolution Equations: Dichotomies and the Fredholm Alternative for Bounded Solutions , 1995 .

[27]  Yu. I. Lyubich,et al.  On the spectral mapping theorem for one-parameter groups of operators , 1992 .

[28]  Michael Renardy,et al.  On the linear stability of hyperbolic PDEs and viscoelastic flows , 1994 .

[29]  George Weiss,et al.  Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity , 1994 .

[30]  David Ruelle,et al.  An extension of the theory of Fredholm determinants , 1990 .

[31]  K. Palmer Two linear systems criteria for exponential dichotomy , 1980 .

[32]  M. Pollicott Meromorphic extensions of generalised zeta functions , 1986 .

[33]  R. Nagel,et al.  The Critical Spectrum of a Strongly Continuous Semigroup , 2000 .

[34]  M. Vishik Magnetic field generation by the motion of a highly conducting fluid , 1989 .

[35]  David Ruelle,et al.  Spectral properties of a class of operators associated with maps in one dimension , 1991, Ergodic Theory and Dynamical Systems.

[36]  R. Swanson The spectrum of vector bundle flows with invariant subbundles , 1981 .

[37]  Jerzy Zabczyk,et al.  Mathematical control theory - an introduction , 1992, Systems & Control: Foundations & Applications.

[38]  G. S. Litvinchuk,et al.  Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift , 2000 .

[39]  R. Sacker The splitting index for linear differential systems , 1979 .

[40]  George R. Sell,et al.  Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions , 1993 .

[41]  R. Mañé Persistent manifolds are normally hyperbolic , 1978 .

[42]  G. Sell,et al.  Inertial manifolds for reaction diffusion equations in higher space dimensions , 1988 .

[43]  Richardo Mañé,et al.  Lyapounov exponents and stable manifolds for compact transformations , 1983 .

[44]  George R. Sell,et al.  A Spectral Theory for Linear Differential Systems , 1978 .

[45]  Kenneth J. Palmer,et al.  Exponential dichotomies and Fredholm operators , 1988 .

[46]  G. Lumer,et al.  Local Operator Methods and Time Dependent Parabolic Equations on Non-Cylindrical Domains , 1998 .

[47]  A Generalized Integral Manifold Theorem , 1993 .

[48]  D. D. Sokolov,et al.  Kinematic dynamo in random flow , 1985 .

[49]  N. Minh On the proof of characterizations of the exponential dichotomy , 1999 .

[50]  H. H. Rugh On the asymptotic form and the reality of spectra of Perron-Frobenius operators , 1994 .

[51]  R. Rebarber,et al.  Conditions for the equivalence of internal and external stability for distributed parameter systems , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[52]  L. T. Magalhaes The spectrum of invariant sets for dissipative semiflows , 1987 .

[53]  Roland Schnaubelt,et al.  The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions , 1996 .

[54]  G. Pedersen C-Algebras and Their Automorphism Groups , 1979 .

[55]  H. H. Rugh Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems , 1996, Ergodic Theory and Dynamical Systems.

[56]  Joel H. Shapiro,et al.  The essential norm of a composition operator , 1987 .

[57]  P. Walters Relative pressure, relative equilibrium states, compensation functions and many-to-one codes between subshifts , 1986 .

[58]  Asymptotic behavior of ₀-semigroups in Banach spaces , 1996 .

[59]  Xiao-Biao Lin Exponential Dichotomies in Intermediate Spaces with Applications to a Diffusively Perturbed Predator-Prey Model , 1994 .

[60]  R. Schnaubelt,et al.  A Spectral Characterization of Exponentially Dichotomic and Hyperbolic Evolution Families , 1994 .

[61]  R. Saeks,et al.  The Arveson frequency response and systems theory , 1985 .

[62]  D. Ruelle Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval , 1994 .

[63]  M. Vishik Spectrum of small oscillations of an ideal fluid and Lyapunov exponents , 1996 .

[64]  George Weiss,et al.  Representation of shift-invariant operators onL2 byH∞ transfer functions: An elementary proof, a generalization toLp, and a counterexample forL∞ , 1991, Math. Control. Signals Syst..

[65]  Hal L. Smith,et al.  Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions , 1995 .

[66]  Luciano Pandolfi A Lyapunov theorem for semigroups of operators , 1990 .

[67]  L. Magalhães Persistence and smoothness of hyperbolic invariant manifolds for functional differential equations , 1987 .

[68]  Weinian Zhang The Fredholm Alternative and Exponential Dichotomies for Parabolic Equations , 1995 .

[69]  G. Lumer Equations de diffusion dans des domaines (x, t) non-cylindriques et semi-groupes "espace-temps" , 1989 .

[70]  M. Megan,et al.  Exponential dichotomy of evolution operators in Banach spaces , 1992 .

[71]  David Ruelle,et al.  Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics , 1978 .

[72]  P. Thieullen Fibres dynamiques asymptotiquement compacts exposants de Lyapounov. Entropie. Dimension , 1987 .

[73]  J. Willems,et al.  Topological classification and structural stability of linear systems , 1980 .

[74]  A. Manning,et al.  Ergodic theory, symbolic dynamics, and hyperbolic spaces , 1991 .

[75]  Fast dynamo problem for a smooth map on a two-torus , 1993 .

[76]  Jacob Palis,et al.  Fifty problems in dynamical systems , 1975 .

[77]  René T. Rau,et al.  Hyperbolic evolution semigroups on vector valued function spaces , 1994 .

[78]  T. Randolph,et al.  Evolution semigroups and stability of time-varying systems on Banach spaces , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[79]  S. Gils,et al.  Center manifolds and contractions on a scale of Banach spaces , 1987 .

[80]  Hiroki Tanabe,et al.  Equations of evolution , 1979 .

[81]  D. Mayer,et al.  The Ruelle-Araki Transfer Operator in Classical Statistical Mechanics , 1980 .

[82]  P. Walters INVARIANT MEASURES AND EQUILIBRIUM STATES FOR SOME MAPPINGS WHICH EXPAND DISTANCES , 1978 .

[83]  H. K. Moffatt Magnetic Field Generation in Electrically Conducting Fluids , 1978 .

[84]  Auf der Morgenstelle Exponential Bounds and Hyperbolicity of Evolution Families , 1996 .

[85]  W. Parry,et al.  Zeta functions and the periodic orbit structure of hyperbolic dynamics , 1990 .

[86]  W. Ruess,et al.  Asymptotically Almost Periodic Solutions of Evolution Equations in Banach Spaces , 1995 .

[87]  V. I. Oseledec A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[88]  J. Neerven,et al.  Characterization of Exponential Stability of a Semigroup of Operators in Terms of Its Action by Convolution on Vector-Valued Function Spaces overR+ , 1996 .

[89]  V. Wróbel Stability and spectra ofC0-semigroups , 1989 .

[90]  David Ruelle,et al.  The thermodynamic formalism for expanding maps , 1989 .

[91]  C. Corduneanu Evolutionary Integral Equations and Applications (J. Pruss) , 1995, SIAM Rev..

[92]  A. Pazy On the Applicability of Lyapunov’s Theorem in Hilbert Space , 1972 .

[93]  Kenneth J. Palmer,et al.  Exponential Dichotomies, the Shadowing Lemma and Transversal Homoclinic Points , 1988 .

[94]  Vu Quoc Phong,et al.  The Operator EquationAX−XB=C, Admissibility, and Asymptotic Behavior of Differential Equations , 1998 .

[95]  David Ruelle,et al.  Characteristic Exponents and Invariant Manifolds in Hilbert Space , 1982 .

[96]  E. Nordgren Composition operators on hilbert spaces , 1978 .

[97]  James F. Selgrade,et al.  Isolated invariant sets for flows on vector bundles , 1975 .

[98]  Exponential Dichotomy for a Nonautonomous System of Parabolic Equations , 1998 .

[99]  George R. Sell,et al.  Existence of dichotomies and invariant splittings for linear differential systems, II☆ , 1976 .

[100]  George Weiss,et al.  Weak Lp-stability of a linear semigroup on a Hilbert space implies exponential stability , 1988 .

[101]  J. Shapiro Composition Operators: And Classical Function Theory , 1993 .

[102]  J. J. Schaffer,et al.  Linear differential equations and function spaces , 1966 .

[103]  V. Berkovich Spectral Theory and Analytic Geometry over Non-Archimedean Fields , 1990 .

[104]  Naresh K. Sinha,et al.  Robust stability of discrete-time systems , 1999, Int. J. Syst. Sci..

[105]  S. Montgomery-Smith Stability and Dichotomy of Positive Semigroups on $L_p$ , 1994, math/9406211.

[106]  C. Eugene Wayne,et al.  Invariant Manifolds for Parabolic Partial Differential Equations on Unbounded Domains , 1997 .

[107]  R. Mañé Quasi-Anosov diffeomorphisms and hyperbolic manifolds , 1977 .