Difference equations and discrete dynamical systems

✍ Scribed by Linda J. S. Allen, Bernd Aulbach, Saber Elaydi, Robert Sacker

• Publisher: WS
• Year: 2005
• Tongue: English
• Leaves: 336
• Category: Library

✦ Synopsis

Difference Equations or Discrete Dynamical Systems is a diverse field which impacts almost every branch of pure and applied mathematics. Not surprisingly, the techniques that are developed vary just as broadly. No more so is this variety reflected than at the prestigious annual International Conference on Difference Equations and Applications. Organized under the auspices of the International Society of Difference Equations, the Conferences have an international attendance and a wide coverage of topics. The contributions from the conference collected in this volume invite the mathematical community to see a variety of problems and applications with one ingredient in common, the Discrete Dynamical System. Readers may also keep abreast of the many novel techniques and developments in the field. The special emphasis of the meeting was on mathematical biology and accordingly about half of the articles are in the related areas of mathematical ecology and mathematical medicine.

✦ Table of Contents

Contents......Page 8
Preface......Page 6
1 Introduction......Page 12
2 A Beverton-Holt competition model......Page 13
3 Concluding remarks......Page 18
A Dynamic Analysis of the Bush Fiscal Policy......Page 22
2. The Basic DADS Growth Model......Page 23
4. Steady State Growth......Page 26
5. Fiscal Policy: Long Run Effects......Page 28
6 A Nonlinear Complication......Page 31
1. INTRODUCTION......Page 38
2. THE GENERAL CASE......Page 39
3. SOME SPECIAL CASES......Page 41
1 Introduction......Page 46
2 Main Result......Page 50
4 Examples......Page 54
5 One Difference Equation of Emden-Fowler Type......Page 55
1. INTRODUCTION......Page 62
2. DYNAMIC CONSISTENCY AND NSFD RULES......Page 64
3. DIFFUSION EQUATION......Page 70
4. DAMPED WAVE EQUATION......Page 72
5. CANCER DYNAMICS......Page 75
6. DISCUSSION......Page 78
1 Introduction......Page 82
2 Background and Definitions......Page 83
3 Theorems......Page 85
4 Simple Models of Population Growth......Page 88
6 Conclusion......Page 95
1 Introduction......Page 98
2 Model for population of mosquitoes......Page 100
3 Global asymptotic stability......Page 101
5 The model is not robust......Page 104
6 Conclusions......Page 108
Global Behavior of Solutions of a Nonlinear Second-Order Nonautonomous Difference Equation......Page 112
1 Introduction and Preliminaries......Page 113
2 Permanence......Page 117
3 Extreme Stability......Page 121
4 Applications......Page 124
1 Introduction......Page 132
2 Reduction to the limiting system......Page 135
3 Average competition......Page 136
4 Numerical simulation......Page 139
5 Conclusions......Page 143
1 Introduction......Page 146
2 The information measures of a discrete random variable......Page 147
3 The classical orthogonal polynomials of a discrete variable......Page 149
4 Information measures of the classical orthogonal polynomials of a discrete variable......Page 151
5 Conclusions......Page 162
1. PRELIMINARIES......Page 166
2. SUFFICIENT CRITERIA FOR AN EXPONENTIAL DICHOTOMY......Page 169
3. TRANSFORMATION OF DIFFERENCE EQUATIONS......Page 172
4. INVARIANT FIBER BUNDLES......Page 173
5. EXAMPLES......Page 176
1 Introduction......Page 182
2 Notations and Preliminary Results......Page 184
3 Main Result......Page 186
4 Extensions and Remarks......Page 191
1 Introduction......Page 196
2 Exact Schemes......Page 198
3 The Non-standard Finite Difference Schemes......Page 200
4 Test Examples and Numerical Results......Page 204
5 Discussion......Page 205
1 INTRODUCTION......Page 210
2 AUTOREGRESSIVE MODELS (CLASSIC)......Page 211
3 GENERALIZED AUTOREGRESSIVE MODELS......Page 212
On Xn+1=BnXn+YnXn-1/An+BnXn with Period-Two Coefficients......Page 218
1 Introduction and Preliminaries......Page 219
2 Background......Page 222
3 The Case r0 r1 < 1......Page 223
4 The Case r0 = r1 = 1......Page 224
5 The Case r0 r1 > 1......Page 225
1. INTRODUCTION......Page 228
2. GLOBAL STABILITY OF PERIODIC ORBITS: AUTONOMOUS DIFFERENCE EQUATIONS......Page 229
3. NON-AUTONOMOUS DIFFERENCE EQUATIONS......Page 232
4. MONOTONE NON-AUTONOMOUS DIFFERENCE EQUATIONS......Page 234
5. MULTIPLE ATTRACTORS IN NONAUTONOMOUS DIFFERENCE EQUATIONS......Page 236
6. OPEN QUESTIONS: PERIODICALLY FORGED NONLINEAR SYSTEMS WITH DELAY......Page 237
7. CONCLUSION......Page 239
1 The general setting......Page 244
2 How to identify integrable equations?......Page 248
3 The singularity confinement criterion in practice......Page 249
4 Complexity and integrability......Page 252
5 Conclusions......Page 255
1. Introduction.......Page 258
2. A linear equation......Page 260
3. Remarks on several delays......Page 265
4. Linear Neutral Functional Differential Equations......Page 268
5. Applications to control......Page 270
6. Bifurcation from equilibrium of nonlinear equations......Page 277
7. The delayed logistic equation.......Page 279
Solvability of the Discrete LQR-Problem under Minimal Assumptions......Page 284
1. THE PROBLEM......Page 285
2. SOLUTION TO (LQR) PROBLEM......Page 288
1 INTRODUCTION......Page 294
2 A STAGE STRUCTURED COMPETITION MODEL......Page 298
3 CONCLUSIONS......Page 307
2 Theorem on the stability under constantly acting perturbations......Page 314
3 The main theorem......Page 315
4 Averaging on an infinite interval......Page 319
5 Averaging on an infinite interval of systems with the right-hand side that vanishes over time......Page 320
6 Dynamics of selection of genetic population in a varying environment......Page 322
1. Motivation and preliminaries......Page 324
2. Symbolic dynamics. Topological entropy and chaos......Page 328
3. Final considerations......Page 333