Stability and bifurcation theory for non-autonomous differential equations : Cetraro, Italy 2011

✍ Scribed by Anna Capietto; Peter E Kloeden; J Mawhin; Sylvia Novo; Rafael Ortega; All authors

• Publisher: Springer
• Year: 2013
• Tongue: English
• Leaves: 314
• Series: Lecture notes in mathematics (Springer-Verlag), 2065.; Lecture notes in mathematics (Springer-Verlag), C.I.M.E foundation subseries
• Category: Library

✦ Synopsis

The Maslov index and global bifurcation for nonlinear boundary value problems / Alberto Boscaggin, Anna Capietto, and Walter Dambrosio -- Discrete-time nonautonomous dynamical systems / P.E. Kloeden, C. Pötzsche, and M. Rasmussen -- Resonance problems for some non-autonomous ordinary differential equations / Jean Mawhin -- Non-autonomous functional differential equations and applications / Sylvia Novo and Rafael Obaya -- Twist mappings with non-periodic angles /Markus Kunze and Rafael Ortega

✦ Table of Contents

Cover......Page 1
Stability and Bifurcation Theory for Non-Autonomous Differential Equations......Page 4
Preface......Page 6
Contents......Page 8
Twist Mappings with Non-Periodic Angles......Page 11
2 The Maslov Index......Page 17
3 The Number of Moments of Verticality......Page 22
4 The Phase-Angles and the Number of Moments of Verticality......Page 26
5 Some Related Notions......Page 31
6 Nonlinear First Order Systems in R2N......Page 34
7 Nonlinear Dirac-Type Systems in the Half-Line......Page 37
References......Page 42
1 Introduction......Page 46
2 Autonomous Difference Equations......Page 47
2.1 Autonomous Semidynamical Systems......Page 49
2.2 Lyapunov Functions for Autonomous Attractors......Page 50
3 Nonautonomous Difference Equations......Page 53
3.1 Processes......Page 54
3.2 Skew-Product Systems......Page 55
3.2.1 Definition......Page 56
3.2.2 Examples......Page 57
3.2.3 Skew-Product Systems as Autonomous Semidynamical Systems......Page 59
4 Nonautonomous Invariant Sets and Attractors of Processes
......Page 60
4.1 Nonautonomous Invariant Sets......Page 61
4.2 Forwards and Pullback Convergence......Page 62
4.3 Forwards and Pullback Attractors......Page 63
4.4 Existence of Pullback Attractors......Page 64
4.5 Limitations of Pullback Attractors......Page 69
5.1 Existence of Pullback Attractors......Page 71
5.2 Comparison of Nonautonomous Attractors......Page 74
5.3 Limitations of Pullback Attractors Revisited......Page 76
5.4 Local Pullback Attractors......Page 78
6 Lyapunov Functions for Pullback Attractors
......Page 79
6.1 Existence of a Pullback Absorbing Neighbourhood System......Page 80
6.2 Necessary and Sufficient Conditions......Page 82
6.2.1 Comments on Theorem 6.4......Page 87
6.2.2 Rate of Pullback Convergence......Page 88
7 Bifurcations......Page 89
7.1 Hyperbolicity and Simple Examples......Page 90
7.2 Attractor Bifurcation......Page 96
7.3 Solution Bifurcation......Page 98
8 Random Dynamical Systems......Page 102
8.1 Random Difference Equations......Page 103
8.2 Random Attractors......Page 104
8.3 Random Markov Chains......Page 106
8.4 Approximating Invariant Measures......Page 108
References......Page 110
1.1 Introduction......Page 113
1.2 Notations......Page 115
1.3 Classes of Homeomorphisms......Page 117
1.4 A Nonlinear Projector......Page 119
2.2.1 Equivalent Fixed Point Problem......Page 122
2.2.2 Existence Result for Singular φ......Page 123
2.3.1 Equivalent Fixed Point Problem for Classical or Singular φ......Page 124
2.3.2 Existence Theorem for Singular φ......Page 127
2.3.3 Forced Planar Polynomial Systems with Singular φ......Page 128
2.3.4 Asymptotic Sign Conditions for Scalar Problems with Singular φ......Page 130
2.3.5 Another Equivalent Fixed Point Problem......Page 134
2.3.7 Weak Asymptotic Sign Conditions for Singular φ......Page 135
2.3.8 A Localized Sign Condition and Periodic Nonlinearitiesfor Singular φ......Page 137
2.3.9 Lower and Upper Solutions for Singular φ......Page 139
2.3.10 Periodic Nonlinearity for Singular or Bounded φ......Page 143
2.3.11 Ambrosetti–Prodi Problem: Coercive Restoring Force and Singular φ......Page 145
2.3.12 Ambrosetti–Prodi Problem: Bounded Restoring Force and Singular φ......Page 148
2.3.13 Singular Restoring Forces and Singular φ......Page 151
2.4.1 Equivalent Fixed Point Problem......Page 155
2.4.2 Existence Theorems......Page 156
3.2.1 The Functional......Page 157
3.2.2 Critical Points and Solutions of Differential Systems......Page 159
3.3.1 A Sufficient Condition for Minimization......Page 163
3.3.2 Periodic Nonlinearities......Page 164
3.3.3 Asymptotically Positive Potential......Page 165
3.3.4 Nonlinearities with Power Growth Restriction......Page 166
3.3.5 Convex Potentials......Page 168
3.4.1 Palais–Smale Condition......Page 169
3.4.2 Bounded Nonlinearities with Anti-coercive Potential......Page 170
3.5.1 Introduction and Hypotheses......Page 172
3.5.2 Existence of Three Periodic Solutions......Page 175
3.6.1 The Action Functional......Page 179
3.6.2 Existence of Two BV Solutions......Page 181
4.1 Introduction......Page 182
4.2.1 Equivalent Hamiltonian System......Page 184
4.2.2 The Hamiltonian Action Functional......Page 185
4.3.2 Existence of Multiple Periodic Solutions......Page 187
References......Page 189
1 Introduction......Page 195
2.1 Flows Over Compact Metric Spaces......Page 199
2.2 Almost Periodic and Almost Automorphic Dynamics......Page 202
2.3 Some Important ODEs Examples......Page 205
2.3.1 Existence of an Almost Automorphic but Non-almost Periodic Minimal Set......Page 207
2.3.2 An Omega-Limit Set Which Contains Two Minimal Sets......Page 208
2.3.3 Existence of Non-uniquely Ergodic Minimal Sets......Page 209
2.3.4 Bifurcation in the Scalar, Coercive and Concave Case......Page 211
2.4 Ordered Banach Spaces: Monotone Skew-Product Semiflows......Page 213
3 Non-autonomous FDEs with Finite Delay......Page 216
3.1 Cooperative and Irreducible Systems of Finite Delay Equations......Page 220
3.2 Semicontinuous Equilibria and Almost Automorphic Extensions......Page 226
3.3.1 Monotone and Concave Semiflows......Page 230
3.3.2 Monotone and Sublinear Semiflows......Page 232
3.4 A Non-autonomous Cyclic Feedback System......Page 235
3.4.1 The Concave Case......Page 236
3.4.2 The Sublinear Case......Page 237
3.5 Cellular Neural Networks......Page 239
4.1 Stability and Extensibility Results for Omega-Limit Sets......Page 247
4.2 FDEs with Infinite Delay......Page 250
4.2.1 Example: Linear Cellular Neural Networks with Infinite Delay......Page 257
4.3 Monotone FDEs with Infinite Delay......Page 259
4.4 Compartmental Systems......Page 265
References......Page 269
1 Introduction......Page 274
2 Symplectic Maps in the Plane and in the Cylinder......Page 279
3 The Twist Condition and the Generating Function......Page 283
4.1 The Frenkel–Kontorowa Model......Page 286
4.2 A General Framework......Page 287
5 Existence of Complete Orbits......Page 290
6 The Action Functional of a Newtonian Equation......Page 295
7 Impact Problems and Generating Functions......Page 299
7.1 The Dirichlet Problem......Page 300
7.2 The Condition of Elastic Bouncing......Page 302
7.3 A Bouncing Ball......Page 303
8 Comments and Bibliographical Remarks......Page 305
References......Page 308
List of Participants......Page 310