Proceedings of the Sixth International Conference on Difference Equations, Augsburg, Germany, 2001 : new progress in difference equations

Proceedings of the Sixth International Conference on Difference Equations, Augsburg, Germany 2001: New Progress in Difference Equations Contents Preface Contributors Part I: Opening Lecture Chapter 1: Difference Equations and Boundary Value Problems 1 Introduction 2 Reduction of BVP to difference equations: some examples 2.1 Simplest examples 2.2 Continuity via discreteness: “continuous lattice” of uncoupled oscillators 2.3 More examples 2.4 Which properties of maps specify the behavior of solutions of DE and BVP? 3 Difference equations and boundary value problems as dynamical systems 3.1 Long-time behavior of solutions 3.2 Self-similarity of solutions 3.3 Self-stochasticity of solutions 3.4 “Universalities” for infinite-dimensional dynamical systems 4 Conclusion: ideal turbulence References Part II: Of General Interest Chapter 2: “Real” Analysis Is a Degenerate Case of Discrete Analysis The ICDEA Conferences: An Asymptotically Stable Recurrence Discrete Analysis: Yet Another Cinderella Story Don’t Worry, the Continuous Heritage is not a Total Waste My Perhaps Not So Foolish ‘April Fool’s Jokes’ Towards a FINITE (and hence RIGOROUS) Foundation of Mathematics The True Derivative My First Love: DISCRETE Analytic Functions Continuous Analysis is a DEGENERATE (not LIMITING) case of Discrete Symbolic Analysis Neo-Pythagoreanism or: Anaxogoras deserved to be drowned Interface with Numerics: Interval Arithmetics Blessed Are The Deltaifference Equations for They Shall Inherit Math Philosophical Conclusion Chapter 3: On the Discrete Nature of Physical Laws 1 Introduction 2 The dynamics of the electron 2.1 The chronon 2.2 Overlapping with numerical analysis 3 The Verhulst equation 4 Solitons and the discrete wave equation 4.1 The complete discrete wave equation References Part III: Discrete Dynamical Systems Chapter 4: Linear Self-Assemblies: Equilibria, Entropy and Convergence Rates 1 Introduction 2 Linear self-assemblies 2.1 Total entropy density 3 n-linear polymerization 3.1 Dynamics of irreversible polymerization 3.2 Dynamics of reversible polymerization 3.3 Uniqueness of the n-linear equilibrium References Chapter 5: Synchronization in a Discrete Circular Network 1 Introduction 2 Synchronization criteria for… 3 Synchronization criteria for… 4 A special case References Chapter 6: Bifurcation of Periodic Points in Reversible Diffeomorphisms 1 Introduction 2 Preliminaries 3 Orbit space formulation and reduction 4 Normal form of reversible diffeomorphisms 5 Stability of bifurcating periodic points 6 Application 6.1 The primary branch 6.2 Period-doubling bifurcations 6.3 Subharmonic bifurcation with… References Chapter 7: Evolution of the Global Behavior of a Class of Difference Equations 1 Introduction 2 Associated diffeomorphisms and fixed points 3 Dispersal to infinity References Chapter 8: A Survey of Exponential Dynamics 1 Introduction 2 Exponential Dynamics 3 Cantor Bouquets Properties 4 Indecomposable Continua 4.1 Topological Preliminaries 4.2 Construction of… 4.3 Dynamics on… 5 Final remarks References Chapter 9: Farey’s Rule for Bifurcations of Periodic Trajectories in a Class of Multi-Valued Interval Maps 1 Properties of a class of two-sheeted one-dimensional maps 1.1 Types of periodic trajectories 1.2 Properties of the bifurcation diagram 2 Properties of solutions of difference equation 3 Appendix References Chapter 10: Combinatorics of Angle-Doubling: Translation Principles for the Mandelbrot set 1 Introduction 2 More on translation-equivalence 3 Proof of Theorem 1.4 References Chapter 11: The Inflation and Perturbation of Nonautonomous Difference Equations and Their Pullback Attractors 1 Introduction 2 The autonomous case 3 Nonautonomous difference equations 3.1 Skew-product formalism 3.2 Pullback attractors for difference cocycles 4 Inflated difference cocycles and pullback attractors 4.1 Inflated pullback attractors 5 Perturbation of pullback attractors 5.1 Perturbation of a shadowing driving system References Chapter 12: A Short Introduction to Asynchronous Systems 1 Introduction 2 Asynchronous systems 3 More formal look 4 Short historical survey 5 First set of problems 6 Chazan-Miranker theorem 7 Complexity issues 8 Robustness of stability 9 Finiteness conjecture 10 Concluding remarks References Chapter 13: A Local-Global Stability Principle for Discrete Systems and and Difference Equations 1 Introduction 2 A general local-global stability principle in metric spaces 3 Cone mappings 4 A local-global stability principle for discrete systems 5 A lemma on roots of certain polynomials 6 Examples A. Population dynamics B. Cobb-Douglas function C. Multidimensional Pielou function D. Nonlinear Fibonacci function (a) Common Fibonacci and linear generalization (b) Fibonacci with exponents in [0, 1] (c) Fibonacci with exponents in [-1, +1] (d) Mixture of linear and inverse Fibonacci (e) Inverse Fibonacci (f) Some special cases References Chapter 14: Stability Implications of Bendixson Conditions for Difference Equations 1 Differential equations in… 2 Difference equations in… 3 Other Bendixson conditions References Chapter 15: Optimal Topological Chaos in Dynamic Economies 1 Introduction 2 Mathematical note 2.1 Periodic orbit and topological entropy 2.2 Sharkovskii’s Theorem 3 Model 4 Interior topological chaos 5 Boundary topological chaos 6 Proof of Theorem 1 References Chapter 16: Dynamics of the Tangent Map 1 Introduction 2 Types of bifurcations 3 Combinatorial trees References Chapter 17: Thresholds, Mode Switching and Complex Dynamics 1 Introduction 2 Polymodal systems 3 Ejector cycles and an application References Chapter 18: Computation of Nonautonomous Invariant Manifolds 1 Introduction 2 Preparations 3 Box algorithm 4 Main result References Part IV: Dynamic Equations on Time Scales Chapter 19: Exponential Functions and Laplace Transforms for Alpha Derivatives 1 Introduction 2 Alpha derivatives, exponentials, and Laplace transforms 3 An example References Chapter 20: Integration on Measure Chains 1 Introduction 2 Basic fact about measure chains 3 The Cauchy integral 4 The Riemann integral 5 The Cauchy-Riemann integral 6 Measure and integral on measure chains 7 Conclusion: The best is (almost) for free References Chapter 21: Asymptotic Formulae for Dynamic Equations on Time Scales with a Functional Perturbation 1 Introduction 2 Basic notions in time scales 3 Main result 4 Some lemmas and proofs References Chapter 22: Oscillation of a Matrix Dynamic Equation on a Time Scale 1 Introduction 2 Preliminaries 3 Main results 4 Examples References Chapter 23: Continuous Dependence in Time Scale Dynamics 1 Dynamical equations on time scales 2 Metrics for dynamical equations on different time scales 3 Uniqueness … continuous dependence 4 Towards a Gronwall Lemma on pairs of time scales References Chapter 24: On the Riemann Integration on Time Scales 1 Introduction 2 The Riemann delta and nabla integrals 3 Properties of the Riemann integrals 4 Fundamental Theorem of Calculus References Chapter 25: Cauchy Functions and Taylor’s Formula for Time Scales… 1 Introduction 2 Preliminaries 3 Taylor monomials 4 Taylor’s Theorem References Chapter 26: Embedding a Class of Time Scale Dynamics into O.D.E Dynamics with Impulse Effect 1 Introduction 2 Preliminaries 2.1 Dynamical equations on time scales 2.2 Differential equations with impulse effect 3 Some examples 4 Embedding results for o.d.e.’s with impulse effect and dynamic equations 5 Summary and further work References Chapter 27: An Oscillation Criterion for a Dynamic Sturm-Liouville Equation 1 Introduction 2 Main result 3 Sketch of the proof References Chapter 28: Two Perturbation Results for Semi-Linear Dynamic Equations on Measure Chains 1 Introduction and preliminaries 2 Perturbation results References Part V: Miscellaneous on Difference Equations Chapter 29: Conjugate Singular and Nonsingular Discrete Boundary Value Problems 1 Introduction 2 Nonsingular positone two-point problems 3 Nonsingular and singular positone two-point problems 4 Semipositone two-point problems 5 Nonsingular multi-point problems References Chapter 30: Asymptotic Solutions of a Discrete Schrödinger Equation Arising from a Dirac Equation with Random Mass 1 Introduction 2 Derivation of the model equation 2.1 Effective non-random Hamiltonian 2.2 Diagonalization of the supersymmetric Hamiltonian 3 Analysis of the Schrödinger equation 3.1 The Birkhoff-Adams Theorem 3.2 The Schrödinger equation with Omega=0 3.3 The full Schrödinger equation 4 Conclusion References Chapter 31: Existence of Bounded Solutions of Discrete Delayed Equations 1 Introduction 2 Formulation of problem 3 Preliminaries 4 Results Acknowledgment References Chapter 32: Difference phi-Laplacian Periodic Boundary Value Problems: Existence and Localization of Solutions 1 Introduction 2 Existence results 3 Discontinuous and functional dependence References Chapter 33: Asymptotic Behavior of Solutions of… 1 Introduction and preliminaries 2 Periodic solutions of equation (1) 3 Existence and asymptotic behavior of solutions of equation (1) 4 An invariant interval for solutions of eq.(1) References Chapter 34: Limit Behavior for Quasilinear Difference Equations 1 Introduction 2 Necessary conditions 3 Main results Concluding remarks Acknowledgment References Chapter 35: Difference Equations in the Qualitative Theory of Delay Differential Equations 1 Introduction 2 Preliminaries 3 The asymptotic behavior of solutions References Chapter 36: Properties of a Class of Numbers Related to the Fibonacci, Lucas and Pell Numbers 1 Introduction 2 Preliminary results 4 Applications References Chapter 37: Oscillation Theory of a Class of Higher Order Sturm-Liouville Difference Equations 1 Introduction 2 Preliminaries 3 General (non)oscillation criteria 4 Two-terms Sturm-Liouville equations 5 The exact value of the oscillation constant References Chapter 38: A Transformation for the Riccati Difference Operator 1 Introduction 2 Preliminaries 3 Proof of the main theorem References Chapter 39: On the Dynamics of… 1 Introduction and preliminaries 2 Local stability, identities, and invariant and attracting intervals 2.1 Local stability analysis 2.2 Identities and invariant intervals 2.3 Attracting Intervals 3 Global stability when p>q 4 Global stability when p=q 5 Global stability when p