NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS IN TRANSPORT PROCESSES

By JAMES PEREZ

CONTENTS INTRODUCTION LINEAR SYSTEMS DEFINITION OF A LINEAR SYSTEM EXPONENTIAL OF A LINEAR TRANSFORMATION SOLUTION OF THE INITIAL VALUE PROBLEM FOR LINEAR HOMOGENEOUS SYSTEMS COMPUTATION OF THE EXPONENTIAL OF A MATRIX ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS EXERCISES EXISTENCE THEORY THE INITIAL VALUE PROBLEM THE CAUCHY-PEANO EXISTENCE THEOREM THE PICARD EXISTENCE THEOREM EXTENSION OF SOLUTIONS CONTINUOUS DEPENDENCE ON INITIAL CONDITIONS FLOW OF NONAUTONOMOUS SYSTEMS FLOW OF AUTONOMOUS SYSTEMS GLOBAL SOLUTIONS STABILITY LIAPUNOV STABILITY NONAUTONOMOUS LINEAR SYSTEMS FUNDAMENTAL MATRICES FLOQUET THEORY STABILITY OF LINEAR PERIODIC SYSTEMS PARAMETRIC RESONANCE – THE MATHIEU EQUATION EXISTENCE OF PERIODIC SOLUTIONS RESULTS FROM FUNCTIONAL ANALYSIS OPERATORS ON BANACH SPACE FRÉCHET DIFFERENTIATION THE CONTRACTION MAPPING PRINCIPLE IN BANACH SPACE THE IMPLICIT FUNCTION THEOREM IN BANACH SPACE THE LIAPUNOV–SCHMIDT METHOD DEPENDENCE ON INITIAL CONDITIONS AND PARAMETERS SMOOTH DEPENDENCE ON INITIAL CONDITIONS CONTINUOUS DEPENDENCE ON PARAMETERS LINEARIZATION AND INVARIANT MANIFOLDS AUTONOMOUS FLOW AT REGULAR POINTS THE HARTMAN-GROBMAN THEOREM INVARIANT MANIFOLDS PERIODIC SOLUTIONS EXISTENCE OF PERIODIC SOLUTIONS IN RN: NONCRITICAL CASE STABILITY OF PERIODIC SOLUTIONS TO NONAUTONOMOUS PERIODIC SYSTEMS STABLE MANIFOLD THEOREM FOR NONAUTONOMOUS PERIODIC SYSTEMS STABILITY OF PERIODIC SOLUTIONS TO AUTONOMOUS SYSTEMS EXISTENCE OF PERIODIC SOLUTIONS IN RN: CRITICAL CASE THE POINCARÉ-BENDIXSON THEOREM CENTER MANIFOLDS AND BIFURCATION THEORY THE CENTER MANIFOLD THEOREM THE CENTER MANIFOLD AS AN ATTRACTOR CO-DIMENSION ONE BIFURCATIONS POINCARÉ NORMAL FORMS THE HOPF BIFURCATION HOPF BIFURCATION VIA LIAPUNOV–SCHMIDT THE BIRKHOFF SMALE HOMOCLINIC THEOREM HOMOCLINIC SOLUTIONS OF NEWTON’S EQUATION THE LINEARIZED OPERATOR PERIODIC PERTURBATIONS OF NEWTON’S EQUATION EXISTENCE OF A TRANSVERSE HOMOCLINIC POINT CHAOTIC DYNAMICS