DYNAMICS REPORTED reports on recent developments in dynamical systems. Dynamical systems of course originated from ordinary differential equations. Today, dynamical systems cover a much larger area, including dynamical processes described by functional and integral equations, by partial and stochastic differential equations, etc. Dynamical systems have involved remarkably in recent years. A wealth of new phenomena, new ideas and new techniques are proving to be of considerable interest to scientists in rather different fields. It is not surprising that thousands of publications on the theory itself and on its various applications are appearing DYNAMICS REPORTED presents carefully written articles on major subjects in dynamical systems and their applications, addressed not only to specialists but also to a broader range of readers including graduate students. Topics are advanced, while detailed exposition of ideas, restriction to typical results - rather than the most general one- and, last but not least, lucid proofs help to gain the utmost degree of clarity. It is hoped, that DYNAMICS REPORTED will be useful for those entering the field and will stimulate an exchange of ideas among those working in dynamical systems Summer 1991 Christopher K. R. T Jones Drs Kirchgraber Hans-Otto Walther Managing Editors Table of Contents Limit Relative Category and Critical Point Theory G. Fournier, D. Lupo, M. Ramos, M. Willem 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Relative Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Relative Cupiength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4. Limit Relative Category . . . . . . . . . . . . . . . . . . . . . . . '" . . . . " . . . . . . . . . . . . . . . . 10 5. The Deformation Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6. Critical Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7. Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reihe
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Für höhere Schule und Studium
Für Beruf und Forschung
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Höhe: 23.5 cm
Breite: 15.5 cm
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ISBN-13
978-3-540-56727-1 (9783540567271)
DOI
10.1007/978-3-642-78234-3
Schweitzer Klassifikation
Limit Relative Category and Critical Point Theory.- 1. Introduction.- 2. Relative Category.- 3. Relative Cuplength.- 4. Limit Relative Category.- 5. The Deformation Lemma.- 6. Critical Point Theorems.- 7. Some Applications.- 8. A Perturbation Theorem.- References.- Coexistence of Infinitely Many Stable Solutions to Reaction Diffusion Systems in the Singular Limit.- 1. Introduction and singular limit slow dynamics.- 2. Intuitive Approach to the Stability of Multi-layered Solutions.- 2.1. Slow Manifold for Mono-layered Solution.- 2.2. Local Slow Manifold for Double-layered Solution.- 2.3. Formal linearization.- 3. The SLEP Method for the Stability of Normal N-layered Solutions.- 3.1. Normal N-layered Solution.- 3.2. Asymptotic Behaviors of Critical Eigenvalues and Eigenfunctions of Le,a.- 3.3. Derivation of the SLEP System.- 3.4. Eigenvalue problem for the SLEP matrix.- 4. Recovery Process of Stability.- 4.1. Instability of Multi-layered Solutions for the Shadow System.- 4.2. Recovery of Stability (From the shadow system to the full system).- 5. Concluding Remarks.- Appendix A.- Appendix B.- Appendix C.- References.- Recent advances in regularity of second-order hyperbolic mixed problems, and applications.- 1. Introduction.- I: Regularity Theory.- 2. Regularity under Dirichlet boundary conditions.- 3. Regularity under Neumann Boundary Conditions.- 4. Cosine/Sine (Semigroup) Representation Formulae of the Solutions.- 4.1. Dirichlet Case.- 4.2. Neumann Case.- II: Applications.- 5. Well-posedness of Semi-linear Wave Equations with Neumann Boundary Conditions.- 5.1. Statement of Local Well-posedness.- 5.2. Proof of Theorem 5.1.- 5.3. Proof of Theorem 5.2.- 6. Local Exponential Stability of Damped Wave Equations with Semi-linear Boundary Conditions.- 6.1. Orientation.- 6.2. Dirichlet Boundary Conditions: Theorem 6.1.- 6.3. Dirichlet Boundary Conditions: Proof of Theorem 6.1.- 6.4. Neumann Boundary Conditions. Theorem 6.2.- 6.5. Neumann Boundary Conditions. Proof of Theorem 6.2.- 7. Exact Controllability of Semi-linear Hyperbolic Problems.- 7.1. Semi-linear Wave Equation of Neumann Type. Exact Controllability.- 7.2. Semi-linear Wave Equation of Dirichlet Type. Exact Controllability.- 7.3. An Exact Controllability Result for Abstract Semi-linear Systems. Theorem 7.4.- 7.4. Application of Theorem 7.4 to the Semi-linear Wave Problem (7.12) with Dirichlet Control.- 8. Riccati Operator Equations and Hyperbolic Mixed Problems.- 8.1. Introduction; the Case where B and 1Z are Bounded.- 8.2. The Unbounded Case: Main statements for T < oo.- 8.2.1. The Unbounded Case for T < oo. The Differential Riccati Equation. Theorems 8.1, 8.2.- 8.3. Application of Theorems 8.1 and 8.2: The DRE for Boundary Control and Boundary Observation for Hyperbolic Mixed Problems of Neumann Type.- 8.4. The Unbounded Case for T = oo. The Algebraic Riccati Equation. Theorem 8.4.- 8.5. Application of Theorem 8.4: The ARE for Boundary Control for Hyperbolic Mixed Problems of Dirichlet Type.- References.
