AN INTRODUCTION TO DIFFERENCE EQUATIONS
CONTENTS DYNAMICS OF FIRST-ORDER DI?ERENCE EQUATIONS INTRODUCTION LINEAR FIRST-ORDER DI?ERENCE EQUATIONS IMPORTANT SPECIAL CASES EQUILIBRIUM POINTS THE STAIR STEP (COBWEB) DIAGRAMS THE COBWEB THEOREM OF ECONOMICS NUMERICAL SOLUTIONS OF DI?ERENTIAL EQUATIONS EULER’S METHOD A NONSTANDARD SCHEME CRITERION FOR THE ASYMPTOTIC STABILITY OF EQUILIBRIUM POINTS PERIODIC POINTS AND CYCLES THE LOGISTIC EQUATION AND BIFURCATION 2-CYCLES 22-CYCLES THE BIFURCATION DIAGRAM BASIN OF ATTRACTION AND GLOBAL STABILITY (OPTIONAL) LINEAR DI?ERENCE EQUATIONS OF HIGHER ORDER DI?ERENCE CALCULUS THE POWER SHIFT FACTORIAL POLYNOMIALS THE ANTIDI?ERENCE OPERATOR GENERAL THEORY OF LINEAR DI?ERENCE EQUATIONS LINEAR HOMOGENEOUS EQUATIONS WITH CONSTANT COE?CIENTS NONHOMOGENEOUS EQUATIONS: METHODS OF UNDETERMIND COE?CEINTS THE METHOD OF VARIATION OF CONSTANTS (PARAMETERS) LIMITING BEHAVIOR OF SOLUTIONS NONLINEAR EQUATIONS TRANSFORMABLE TO LINEAR EQUATIONS APPLICATIONS PROPAGATION OF ANNUAL PLANTS GAMBLER’S RUIN NATIONAL INCOME THE TRANSMISSION OF INFORMATION SYSTEMS OF LINEAR DI?ERENCE EQUATIONS AUTONOMOUS (TIME-INVARIANT) SYSTEMS THE DISCRETE ANALOGUE OF THE PUTZER ALGORITHM THE DEVELOPMENT OF THE ALGORITHM FOR AN THE BASIC THEORY THE JORDAN FORM: AUTONOMOUS (TIME-INVARIANT) SYSTEMS REVISITED DIAGONALIZABLE MATRICES THE JORDAN FORM BLOCK-DIAGONAL MATRICES LINEAR PERIODIC SYSTEMS MARKOV CHAINS REGULAR MARKOV CHAINS ABSORBING MARKOV CHAINS A TRADE MODEL THE HEAT EQUATION STABILITY THEORY A NORM OF A MATRIX NOTIONS OF STABILITY STABILITY OF LINEAR SYSTEMS NONAUTONOMOUS LINEAR SYSTEMS AUTONOMOUS LINEAR SYSTEMS PHASE SPACE ANALYSIS LIAPUNOV’S DIRECT, OR SECOND, METHOD STABILITY BY LINEAR APPROXIMATION ONE SPECIES WITH TWO AGE CLASSES HOST–PARASITOID SYSTEMS A BUSINESS CYCLE MODEL THE NICHOLSON–BAILEY MODEL THE FLOUR BEETLE CASE STUDY HIGHER-ORDER SCALAR DI?ERENCE EQUATIONS LINEAR SCALAR EQUATIONS SU?CIENT CONDITIONS FOR STABILITY STABILITY VIA LINEARIZATION GLOBAL STABILITY OF NONLINEAR EQUATIONS FLOUR BEETLES A MOSQUITO MODEL THE Z-TRANSFORM METHOD AND VOLTERRA DI?ERENCE EQUATIONS DE?NITIONS AND EXAMPLES PROPERTIES OF THE Z-TRANSFORM THE INVERSE Z-TRANSFORM AND SOLUTIONS OF DI?ERENCE EQUATIONS THE POWER SERIES METHOD THE PARTIAL FRACTIONS METHOD THE INVERSION INTEGRAL METHOD VOLTERRA DI?ERENCE EQUATIONS OF CONVOLUTION TYPE: THE SCALAR CASE EXPLICIT CRITERIA FOR STABILITY OF VOLTERRA EQUATIONS VOLTERRA SYSTEMS A VARIATION OF CONSTANTS FORMULA THE Z-TRANSFORM VERSUS THE LAPLACE TRANSFORM OSCILLATION THEORY THREE-TERM DI?ERENCE EQUATIONS SELF-ADJOINT SECOND-ORDER EQUATIONS NONLINEAR DI?ERENCE EQUATIONS ASYMPTOTIC BEHAVIOR OF DI?ERENCE EQUATIONS TOOLS OF APPROXIMATION POINCAR´E’S THEOREM IN?NITE PRODUCTS AND PERRON’S EXAMPLE ASYMPTOTICALLY DIAGONAL SYSTEMS HIGH-ORDER DI?ERENCE EQUATIONS SECOND-ORDER DI?ERENCE EQUATIONS A GENERALIZATION OF THE POINCAR´E–PERRON THEOREM BIRKHO?’S THEOREM EXTENSIONS OF THE POINCAR´E AND PERRON THEOREMS AN EXTENSION OF PERRON’S SECOND THEOREM POINCAR´E’S THEOREM REVISITED APPLICATIONS TO CONTINUED FRACTIONS AND ORTHOGONAL POLYNOMIALS CONTINUED FRACTIONS: FUNDAMENTAL RECURRENCE FORMULA CONVERGENCE OF CONTINUED FRACTIONS CONTINUED FRACTIONS AND IN?NITE SERIES CLASSICAL ORTHOGONAL POLYNOMIALS THE FUNDAMENTAL RECURRENCE FORMULA FOR ORTHOGONAL POLYNOMIALS MINIMAL SOLUTIONS, CONTINUED FRACTIONS, AND ORTHOGONAL POLYNOMIALS 10 CONTROL THEORY 10 1 1 DISCRETE EQUIVALENTS FOR CONTINUOUS SYSTEMS CONTROLLABILITY 10 2 1 CONTROLLABILITY CANONICAL FORMS OBSERVABILITY 10 3 1 OBSERVABILITY CANONICAL FORMS STABILIZATION BY STATE FEEDBACK (DESIGN VIA POLE PLACEMENT) 10 4 1 STABILIZATION OF NONLINEAR SYSTEMS BY FEEDBACK OBSERVERS 10 5 1 EIGENVALUE SEPARATION THEOREM