| Preface to the First Edition List of Symbols 1 Dynamics of First-Order Difference Equations 1.1 Introduction 1.2 Linear First-Order Difference Equations 1.2.1 Important Special Cases 1.3 Equilibrium Points 1.3.1 The Stair Step (Cobweb) Diagrams 1.3.2 The Cobweb Theorem of Economics 1.4 Numerical Solutions of Differential Equations 1.4.1 Euler's Method 1.4.2 A Nonstandard Scheme 1.5 Criterion for the Asymptotic Stability of Equilibrium Points 1.6 Periodic Points and Cycles 1.7 The Logistic Equation and Bifurcation 1.7.1 Equilibrium Points 1.7.2 2-Cycles 1.7.3 22-Cycles 1.7.4 The Bifurcation Diagram 1.8 Basin of Attraction and Global Stability (Optional) 2 Linear Difference Equations of Higher Order 2.1 Difference Calculus 2.1.1 The Power Shift 2.1.2 Factorial Polynomials 2.1.3 The Antidifference Operator 2.2 General Theory of Linear Difference Equations 2.3 Linear Homogeneous Equations with Constant Coefficients 2.4 Nonhomogeneous Equations: Methods of Undetermind Coefficeints 2.4.1 The Method of Variation of Constants (Parameters) 2.5 Limiting Behavior of Solutions 2.6 Nonlinear Equations Transformable to Linear Equations 2.7 Applications 2.7.1 Propagation of Annual Plants 2.7.2 Gambler's Ruin 2.7.3 National Income 2.7.4 The Transmission of Information 3 Systems of Linear Difference Equations 3.1 Autonomous (Time-Invariant) Systems 3.1.1 The Discrete Analogue of the Putzer Algorithm 3.1.2 The Development of the Algorithm for An 3.2 The Basic Theory 3.3 The Jordan Form: Autonomous (Time-Invariant) Systems Revisited 3.3.1 Diagonalizable Matrices 3.3.2 The Jordan Form 3.3.3 Block-Diagonal Matrices 3.4 Linear Periodic Systems 3.5 Applications 3.5.1 Markov Chains 3.5.2 Regular Markov Chains 3.5.3 Absorbing Markov Chains 3.5.4 A Trade Model 3.5.5 The Heat Equation 4 Stability Theory 4.1 A Norm of a Matrix 4.2 Notions of Stability …… 5 Higher-Order Scalar Difference Eqations 6 The Z-Transform Method and volterra Difference Equations 7 Oscillation Theory 8 Asymptotic Behavior of Difference Equations 9 Applications to Contnued Fractions and Orthogonal Polynomials 10 Control Theory A Stability of Nonhyperbolic Fixed Points of Maps on the Real Line B The Vamdermonde Matrix C Stability of Nomdifferentiable D Stahble Manifold and the Hartman-Grobman-Cushing Theorems E The Levin-May Theorem F Classical Orthogonal Polynomials G Identities and Formulas Answers and Hints to Selected Problems Maple Programs References Index |
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