ORDINARY DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS
CONTENTS BASIC MATERIAL AND ASYMPTOTICS INTRODUCTION EXISTENCE AND UNIQUENESS THE GRONWALL LEMMA CONCEPTS OF ASYMPTOTIC APPROXIMATION NAIVE FORMULATION OF PERTURBATION PROBLEMS REFORMULATION IN THE STANDARD FORM THE STANDARD FORM IN THE QUASILINEAR CASE AVERAGING: THE PERIODIC CASE VAN DER POL EQUATION A LINEAR OSCILLATOR WITH FREQUENCY MODULATION ONE DEGREE OF FREEDOM HAMILTONIAN SYSTEM THE NECESSITY OF RESTRICTING THE INTERVAL OF TIME BOUNDED SOLUTIONS AND A RESTRICTED TIME SCALE OF VALIDITY COUNTER EXAMPLE OF CRUDE AVERAGING TWO PROOFS OF FIRST-ORDER PERIODIC AVERAGING HIGHER-ORDER PERIODIC AVERAGING AND TRADE-O? HIGHER-ORDER PERIODIC AVERAGING ESTIMATES ON LONGER TIME INTERVALS MODI?ED VAN DER POL EQUATION PERIODIC ORBIT OF THE VAN DER POL EQUATION METHODOLOGY OF AVERAGING HANDLING THE AVERAGING PROCESS LIE THEORY FOR MATRICES LIE THEORY FOR AUTONOMOUS VECTOR FIELDS LIE THEORY FOR PERIODIC VECTOR FIELDS SOLVING THE AVERAGED EQUATIONS AVERAGING PERIODIC SYSTEMS WITH SLOW TIME DEPENDENCE PENDULUM WITH SLOWLY VARYING LENGTH UNIQUE AVERAGING AVERAGING AND MULTIPLE TIME SCALE METHODS AVERAGING: THE GENERAL CASE BASIC LEMMAS; THE PERIODIC CASE GENERAL AVERAGING LINEAR OSCILLATOR WITH INCREASING DAMPING SECOND-ORDER AVERAGING EXAMPLE OF SECOND-ORDER AVERAGING ALMOST-PERIODIC VECTOR FIELDS EXAMPLE ATTRACTION EQUATIONS WITH LINEAR ATTRACTION EXAMPLES OF REGULAR PERTURBATIONS WITH ATTRACTION TWO SPECIES A PERTURBATION THEOREM TWO SPECIES, CONTINUED EXAMPLES OF AVERAGING WITH ATTRACTION ANHARMONIC OSCILLATOR WITH LINEAR DAMPING DU?NG’S EQUATION WITH DAMPING AND FORCING THEORY OF AVERAGING WITH ATTRACTION AN ATTRACTOR IN THE ORIGINAL EQUATION CONTRACTING MAPS ATTRACTING LIMIT-CYCLES ADDITIONAL EXAMPLES PERTURBATION OF THE LINEAR TERMS DAMPING ON VARIOUS TIME SCALES PERIODIC AVERAGING AND HYPERBOLICITY COUPLED DU?NG EQUATIONS, AN EXAMPLE REST POINTS AND PERIODIC SOLUTIONS THE REGULAR CASE THE AVERAGING CASE LOCAL CONJUGACY AND SHADOWING EXTENDED ERROR ESTIMATE FOR SOLUTIONS APPROACHING AN ATTRACTOR EXTENSION TO LARGER COMPACT SETS EXTENSIONS AND DEGENERATE CASES AVERAGING OVER ANGLES THE CASE OF CONSTANT FREQUENCIES TOTAL RESONANCES THE CASE OF VARIABLE FREQUENCIES EXAMPLES EINSTEIN PENDULUM NONLINEAR OSCILLATOR OSCILLATOR ATTACHED TO A FLYWHEEL SECONDARY (NOT SECOND ORDER) AVERAGING FORMAL THEORY SLOWLY VARYING FREQUENCY HIGHER ORDER APPROXIMATION IN THE REGULAR CASE 7 10 GENERALIZATION OF THE REGULAR CASE 7 10 1 TWO-BODY PROBLEM WITH VARIABLE MASS PASSAGE THROUGH RESONANCE THE INNER EXPANSION THE OUTER EXPANSION THE COMPOSITE EXPANSION REMARKS ON HIGHER-DIMENSIONAL PROBLEMS THE CASE OF MORE THAN ONE ANGLE EXAMPLE OF RESONANCE LOCKING EXAMPLE OF FORCED PASSAGE THROUGH RESONANCE INNER AND OUTER EXPANSION TWO EXAMPLES THE FORCED MATHEMATICAL PENDULUM AN OSCILLATOR ATTACHED TO A FLY-WHEEL FROM AVERAGING TO NORMAL FORMS CLASSICAL, OR FIRST-LEVEL, NORMAL FORMS DI?ERENTIAL OPERATORS ASSOCIATED WITH A VECTOR FIELD LIE THEORY NORMAL FORM STYLES THE SEMISIMPLE CASE THE NONSEMISIMPLE CASE THE TRANSPOSE OR INNER PRODUCT NORMAL FORM STYLE THE SL2 NORMAL FORM HIGHER LEVEL NORMAL FORMS HAMILTONIAN NORMAL FORM THEORY 10 1 INTRODUCTION 10 1 1 THE HAMILTONIAN FORMALISM 10 1 2 LOCAL EXPANSIONS AND RESCALING 10 1 3 BASIC INGREDIENTS OF THE FLOW 10 2 NORMALIZATION OF HAMILTONIANS AROUND EQUILIBRIA 10 2 1 THE GENERATING FUNCTION NORMAL FORM POLYNOMIALS CANONICAL VARIABLES AT RESONANCE 10 4 PERIODIC SOLUTIONS AND INTEGRALS 10 5 TWO DEGREES OF FREEDOM, GENERAL THEORY 10 5 2 THE LINEAR FLOW 10 5 3 DESCRIPTION OF THE ?1 : ?2-RESONANCE IN NORMAL FORM GENERAL ASPECTS OF THE K : L-RESONANCE, K ?= L 10 6 TWO DEGREES OF FREEDOM, EXAMPLES 10 6 1 THE 1 : 2-RESONANCE 10 6 2 THE SYMMETRIC 1 : 1-RESONANCE 10 6 3 THE 1 : 3-RESONANCE 10 6 4 HIGHER-ORDER RESONANCES 10 7 THREE DEGREES OF FREEDOM, GENERAL THEORY 10 7 1 INTRODUCTION 10 7 2 THE ORDER OF RESONANCE 10 7 3 PERIODIC ORBITS AND INTEGRALS 10 7 4 THE ?1 : ?2 : ?3-RESONANCE 10 7 5 THE KERNEL OF AD(H0) 10 8 THREE DEGREES OF FREEDOM, EXAMPLES 10 8 1 THE 1 : 2 : 1-RESONANCE 10 8 2 INTEGRABILITY OF THE 1 : 2 : 1 NORMAL FORM 10 8 3 THE 1 : 2 : 2-RESONANCE 10 8 4 INTEGRABILITY OF THE 1 : 2 : 2 NORMAL FORM 10 8 5 THE 1 : 2 : 3-RESONANCE 10 8 6 INTEGRABILITY OF THE 1 : 2 : 3 NORMAL FORM 10 8 7 THE 1 : 2 : 4-RESONANCE 10 8 8 INTEGRABILITY OF THE 1 : 2 : 4 NORMAL FORM 10 8 9 SUMMARY OF INTEGRABILITY OF NORMALIZED SYSTEMS 10 8 10GENUINE SECOND-ORDER RESONANCES CLASSICAL (FIRST–LEVEL) NORMAL FORM THEORY 11 1 INTRODUCTION 11 2 LEIBNIZ ALGEBRAS AND REPRESENTATIONS 11 3 COHOMOLOGY 11 4 A MATTER OF STYLE 11 4 1 EXAMPLE: NILPOTENT LINEAR PART IN R2 11 5 INDUCED LINEAR ALGEBRA 11 5 1 THE NILPOTENT CASE 11 5 2 NILPOTENT EXAMPLE REVISITED 11 5 3 THE NONSEMISIMPLE CASE 11 6 THE FORM OF THE NORMAL FORM, THE DESCRIPTION PROBLEM NILPOTENT (CLASSICAL) NORMAL FORM 12 1 INTRODUCTION 12 2 CLASSICAL INVARIANT THEORY 12 3 TRANSVECTANTS 12 4 A REMARK ON GENERATING FUNCTIONS 12 5 THE JACOBSON–MOROZOV LEMMA 12 6 DESCRIPTION OF THE FIRST LEVEL NORMAL FORMS 12 6 1 THE N2 CASE 12 6 2 THE N3 CASE 12 6 3 THE N4 CASE 12 6 4 INTERMEZZO: HOW FREE? 12 6 5 THE N2,2 CASE 12 6 6 THE N5 CASE 12 6 7 THE N2,3 CASE 12 7 DESCRIPTION OF THE FIRST LEVEL NORMAL FORMS 12 7 1 THE N2,2,2 CASE 12 7 2 THE N3,3 CASE 12 7 3 THE N3,4 CASE 12 7 4 CONCLUDING REMARK HIGHER–LEVEL NORMAL FORM THEORY 13 1 INTRODUCTION 13 1 1 SOME STANDARD RESULTS 13 2 ABSTRACT FORMULATION OF NORMAL FORM THEORY 13 3 THE HILBERT–POINCAR´E SERIES OF A SPECTRAL SEQUENCE 13 4 THE ANHARMONIC OSCILLATOR 2R IS INVERTIBLE 2R IS 13 4 3 THE M-ADIC APPROACH 13 5 THE HAMILTONIAN 1 : 2-RESONANCE 13 6 AVERAGING OVER ANGLES 13 7 DE?NITION OF NORMAL FORM 13 8 LINEAR CONVERGENCE, USING THE NEWTON METHOD