DYNAMICAL SYSTEMS

By R. GIRIO

CONTENTS ONE-DIMENSIONAL MAPS ONE-DIMENSIONAL MAPS COBWEB PLOT: GRAPHICAL REPRESENTATION OF AN ORBIT STABILITY OF FIXED POINTS PERIODIC POINTS THE FAMILY OF LOGISTIC MAPS THE LOGISTIC MAP G(X) 4X(1 X) SENSITIVE DEPENDENCE ON INITIAL CONDITIONS ITINERARIES CHALLENGE 1: PERIOD THREE IMPLIES CHAOS EXERCISES LAB VISIT 1: BOOM, BUST, AND CHAOS IN THE BEETLE CENSUS C O N T E N T S TWO-DIMENSIONAL MAPS MATHEMATICAL MODELS SINKS, SOURCES, AND SADDLES LINEAR MAPS COORDINATE CHANGES NONLINEAR MAPS AND THE JACOBIAN MATRIX STABLE AND UNSTABLE MANIFOLDS MATRIX TIMES CIRCLE EQUALS ELLIPSE CHALLENGE 2: COUNTING THE PERIODIC ORBITS OF LINEAR MAPS ON A TORUS LAB VISIT 2: IS THE SOLAR SYSTEM STABLE? CHAOS LYAPUNOV EXPONENTS CHAOTIC ORBITS CONJUGACY AND THE LOGISTIC MAP TRANSITION GRAPHS AND FIXED POINTS BASINS OF ATTRACTION CHALLENGE 3: SHARKOVSKII’S THEOREM LAB VISIT 3: PERIODICITY AND CHAOS IN A CHEMICAL REACTION FRACTALS CANTOR SETS PROBABILISTIC CONSTRUCTIONS OF FRACTALS FRACTALS FROM DETERMINISTIC SYSTEMS FRACTAL BASIN BOUNDARIES FRACTAL DIMENSION COMPUTING THE BOX-COUNTING DIMENSION CORRELATION DIMENSION CHALLENGE 4: FRACTAL BASIN BOUNDARIES AND THE UNCERTAINTY EXPONENT LAB VISIT 4: FRACTAL DIMENSION IN EXPERIMENTS CHAOS IN TWO-DIMENSIONAL MAPS NUMERICAL CALCULATION OF LYAPUNOV EXPONENTS LYAPUNOV DIMENSION A TWO-DIMENSIONAL FIXED-POINT THEOREM MARKOV PARTITIONS THE HORSESHOE MAP C O N T E N T S CHALLENGE 5: COMPUTER CALCULATIONS AND SHADOWING LAB VISIT 5: CHAOS IN SIMPLE MECHANICAL DEVICES CHAOTIC ATTRACTORS FORWARD LIMIT SETS CHAOTIC ATTRACTORS CHAOTIC ATTRACTORS OF EXPANDING INTERVAL MAPS MEASURE NATURAL MEASURE INVARIANT MEASURE FOR ONE-DIMENSIONAL MAPS CHALLENGE 6: INVARIANT MEASURE FOR THE LOGISTIC MAP LAB VISIT 6: FRACTAL SCUM DIFFERENTIAL EQUATIONS ONE-DIMENSIONAL LINEAR DIFFERENTIAL EQUATIONS ONE-DIMENSIONAL NONLINEAR DIFFERENTIAL EQUATIONS LINEAR DIFFERENTIAL EQUATIONS IN MORE THAN ONE DIMENSION NONLINEAR SYSTEMS MOTION IN A POTENTIAL FIELD LYAPUNOV FUNCTIONS LOTKA-VOLTERRA MODELS CHALLENGE 7: A LIMIT CYCLE IN THE VAN DER POL SYSTEM LAB VISIT 7: FLY VS FLY PERIODIC ORBITS AND LIMIT SETS LIMIT SETS FOR PLANAR DIFFERENTIAL EQUATIONS PROPERTIES OF -LIMIT SETS PROOF OF THE POINCAR´E-BENDIXSON THEOREM CHALLENGE 8: TWO INCOMMENSURATE FREQUENCIES FORM A TORUS LAB VISIT 8: STEADY STATES AND PERIODICITY IN A SQUID NEURON CHAOS IN DIFFERENTIAL EQUATIONS THE LORENZ ATTRACTOR STABILITY IN THE LARGE, INSTABILITY IN THE SMALL THE R¨OSSLER ATTRACTOR CHUA’S CIRCUIT FORCED OSCILLATORS LYAPUNOV EXPONENTS IN FLOWS C O N T E N T S CHALLENGE 9: SYNCHRONIZATION OF CHAOTIC ORBITS LAB VISIT 9: LASERS IN SYNCHRONIZATION STABLE MANIFOLDS AND CRISES 10 1 THE STABLE MANIFOLD THEOREM 10 2 HOMOCLINIC AND HETEROCLINIC POINTS 10 3 CRISES 10 4 PROOF OF THE STABLE MANIFOLD THEOREM 10 5 STABLE AND UNSTABLE MANIFOLDS FOR HIGHER DIMENSIONAL MAPS CHALLENGE 10: THE LAKES OF WADA LAB VISIT 10: THE LEAKY FAUCET: MINOR IRRITATION OR CRISIS? BIFURCATIONS 11 1 SADDLE-NODE AND PERIOD-DOUBLING BIFURCATIONS 11 2 BIFURCATION DIAGRAMS 11 3 CONTINUABILITY 11 4 BIFURCATIONS OF ONE-DIMENSIONAL MAPS 11 5 BIFURCATIONS IN PLANE MAPS: AREA-CONTRACTING CASE 11 6 BIFURCATIONS IN PLANE MAPS: AREA-PRESERVING CASE 11 7 BIFURCATIONS IN DIFFERENTIAL EQUATIONS 11 8 HOPF BIFURCATIONS CHALLENGE 11: HAMILTONIAN SYSTEMS AND THE LYAPUNOV CENTER THEOREM LAB VISIT 11: IRON + SULFURIC ACID ??HOPF BIFURCATION CASCADES 12 1 CASCADES AND 4 669201609 12 2 SCHEMATIC BIFURCATION DIAGRAMS 12 3 GENERIC BIFURCATIONS 12 4 THE CASCADE THEOREM CHALLENGE 12: UNIVERSALITY IN BIFURCATION DIAGRAMS LAB VISIT 12: EXPERIMENTAL CASCADES STATE RECONSTRUCTION FROM DATA 13 1 DELAY PLOTS FROM TIME SERIES 13 2 DELAY COORDINATES 13 3 EMBEDOLOGY CHALLENGE 13: BOX-COUNTING DIMENSION AND INTERSECTION C H A P T E R O N E ONE-DIMENSIONAL MAPS THE FUNCTION F(X) 2X IS A RULE THAT ASSIGNS TO EACH NUMBER X A NUMBER TWICE AS LARGE THIS IS A SIMPLE MATHEMATICAL MODEL WE MIGHT IMAGINE THAT X DENOTES THE POPULATION OF BACTERIA IN A LABORATORY CULTURE AND THAT F(X) DENOTES THE POPULATION ONE HOUR LATER THEN THE RULE EXPRESSES THE FACT THAT THE POPULATION DOUBLES EVERY HOUR IF THE CULTURE HAS AN INITIAL POPULATION OF 10,000 BACTERIA, THEN AFTER ONE HOUR THERE WILL BE F(10,000) 20,000 BACTERIA, AFTER TWO HOURS THERE WILL BE F(F(10,000)) 40,000 BACTERIA, AND SO ON A DYNAMICAL SYSTEM CONSISTS OF A SET OF POSSIBLE STATES, TOGETHER WITH A RULE THAT DETERMINES THE PRESENT STATE IN TERMS OF PAST STATES IN THE PREVIOUS PARAGRAPH, WE DISCUSSED A SIMPLE DYNAMICAL SYSTEM WHOSE STATES ARE POPULATION LEVELS, THAT CHANGE WITH TIME UNDER THE RULE XN F(XN1) 2XN1 HERE THE VARIABLE N STANDS FOR TIME, AND XN DESIGNATES THE POPULATION AT TIME N WE WILL