ELEMENTS OF APPLIED BIFURCATION THEORY
CONTENTS INTRODUCTION TO DYNAMICAL SYSTEMS DE?NITION OF A DYNAMICAL SYSTEM STATE SPACE TIME EVOLUTION OPERATOR ORBITS AND PHASE PORTRAITS INVARIANT SETS DE?NITION AND TYPES EXAMPLE 1 9 (SMALE HORSESHOE) STABILITY OF INVARIANT SETS DI?ERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS POINCAR´E MAPS TIME-SHIFT MAPS POINCAR´E MAP AND STABILITY OF CYCLES POINCAR´E MAP FOR PERIODICALLY FORCED SYSTEMS EXERCISES APPENDIX 1: IN?NITE-DIMENSIONAL DYNAMICAL SYSTEMS DE?NED BY REACTION-DI?USION EQUATIONS APPENDIX 2: BIBLIOGRAPHICAL NOTES TOPOLOGICAL EQUIVALENCE, BIFURCATIONS, AND STRUCTURAL STABILITY OF DYNAMICAL SYSTEMS EQUIVALENCE OF DYNAMICAL SYSTEMS TOPOLOGICAL CLASSI?CATION OF GENERIC EQUILIBRIA AND ?XED POINTS HYPERBOLIC EQUILIBRIA IN CONTINUOUS-TIME SYSTEMS HYPERBOLIC ?XED POINTS IN DISCRETE-TIME SYSTEMS HYPERBOLIC LIMIT CYCLES BIFURCATIONS AND BIFURCATION DIAGRAMS TOPOLOGICAL NORMAL FORMS FOR BIFURCATIONS STRUCTURAL STABILITY APPENDIX: BIBLIOGRAPHICAL NOTES ONE-PARAMETER BIFURCATIONS OF EQUILIBRIA IN CONTINUOUS-TIME DYNAMICAL SYSTEMS SIMPLEST BIFURCATION CONDITIONS THE NORMAL FORM OF THE FOLD BIFURCATION GENERIC FOLD BIFURCATION THE NORMAL FORM OF THE HOPF BIFURCATION GENERIC HOPF BIFURCATION APPENDIX 1: PROOF OF LEMMA 3 2 ONE-PARAMETER BIFURCATIONS OF FIXED POINTS IN DISCRETE-TIME DYNAMICAL SYSTEMS THE NORMAL FORM OF THE ?IP BIFURCATION GENERIC ?IP BIFURCATION THE “NORMAL FORM” OF THE NEIMARK-SACKER BIFURCATION GENERIC NEIMARK-SACKER BIFURCATION APPENDIX 1: FEIGENBAUM’S UNIVERSALITY 4 10 APPENDIX 2: PROOF OF LEMMA 4 3 4 11 APPENDIX 3: BIBLIOGRAPHICAL NOTES BIFURCATIONS OF EQUILIBRIA AND PERIODIC ORBITS IN N-DIMENSIONAL DYNAMICAL SYSTEMS CENTER MANIFOLD THEOREMS CENTER MANIFOLDS IN CONTINUOUS-TIME SYSTEMS CENTER MANIFOLDS IN DISCRETE-TIME SYSTEMS CENTER MANIFOLDS IN PARAMETER-DEPENDENT SYSTEMS BIFURCATIONS OF LIMIT CYCLES COMPUTATION OF CENTER MANIFOLDS QUADRATIC APPROXIMATION TO CENTER MANIFOLDS IN EIGENBASIS PROJECTION METHOD FOR CENTER MANIFOLD COMPUTATION APPENDIX 1: HOPF BIFURCATION IN REACTION-DIFFUSION SYSTEMS ON THE INTERVAL WITH DIRICHLET BOUNDARY CONDITIONS BIFURCATIONS OF ORBITS HOMOCLINIC AND HETEROCLINIC TO HYPERBOLIC EQUILIBRIA HOMOCLINIC AND HETEROCLINIC ORBITS ANDRONOV-LEONTOVICH THEOREM HOMOCLINIC BIFURCATIONS IN THREE-DIMENSIONAL SYSTEMS: SHIL’NIKOV THEOREMS HOMOCLINIC BIFURCATIONS IN N-DIMENSIONAL SYSTEMS REGULAR HOMOCLINIC ORBITS: MELNIKOV INTEGRAL HOMOCLINIC CENTER MANIFOLDS GENERIC HOMOCLINIC BIFURCATIONS IN RN APPENDIX 1: FOCUS-FOCUS HOMOCLINIC BIFURCATION IN FOUR-DIMENSIONAL SYSTEMS OTHER ONE-PARAMETER BIFURCATIONS IN CONTINUOUS-TIME DYNAMICAL SYSTEMS CODIM 1 BIFURCATIONS OF HOMOCLINIC ORBITS TO NONHYPERBOLIC EQUILIBRIA SADDLE-NODE HOMOCLINIC BIFURCATION ON THE PLANE “EXOTIC” BIFURCATIONS NONTRANSVERSAL HOMOCLINIC ORBIT TO A HYPERBOLIC CYCLE HOMOCLINIC ORBITS TO A NONHYPERBOLIC LIMIT CYCLE BIFURCATIONS ON INVARIANT TORI REDUCTION TO A POINCAR´E MAP ROTATION NUMBER AND ORBIT STRUCTURE STRUCTURAL STABILITY AND BIFURCATIONS PHASE LOCKING NEAR A NEIMARK-SACKER BIFURCATION: ARNOLD TONGUES BIFURCATIONS IN SYMMETRIC SYSTEMS GENERAL PROPERTIES OF SYMMETRIC SYSTEMS Z2-EQUIVARIANT SYSTEMS CODIM 1 BIFURCATIONS OF EQUILIBRIA IN Z2-EQUIVARIANT SYSTEMS CODIM 1 BIFURCATIONS OF CYCLES IN Z2-EQUIVARIANT SYSTEMS APPENDIX 1: BIBLIOGRAPHICAL NOTES TWO-PARAMETER BIFURCATIONS OF EQUILIBRIA IN CONTINUOUS-TIME DYNAMICAL SYSTEMS LIST OF CODIM 2 BIFURCATIONS OF EQUILIBRIA CODIM 1 BIFURCATION CURVES CODIM 2 BIFURCATION POINTS CUSP BIFURCATION NORMAL FORM DERIVATION BIFURCATION DIAGRAM OF THE NORMAL FORM E?ECT OF HIGHER-ORDER TERMS BAUTIN (GENERALIZED HOPF) BIFURCATION BOGDANOV-TAKENS (DOUBLE-ZERO) BIFURCATION FOLD-HOPF (ZERO-PAIR) BIFURCATION DERIVATION OF THE NORMAL FORM BIFURCATION DIAGRAM OF THE TRUNCATED NORMAL FORM HOPF-HOPF BIFURCATION APPENDIX 1: LIMIT CYCLES AND HOMOCLINIC ORBITS OF BOGDANOV NORMAL FORM TWO-PARAMETER BIFURCATIONS OF FIXED POINTS IN DISCRETE-TIME DYNAMICAL SYSTEMS LIST OF CODIM 2 BIFURCATIONS OF ?XED POINTS GENERALIZED ?IP BIFURCATION CHENCINER (GENERALIZED NEIMARK-SACKER) BIFURCATION STRONG RESONANCES APPROXIMATION BY A ?OW 1:1 RESONANCE 1:2 RESONANCE 1:3 RESONANCE 1:4 RESONANCE CODIM 2 BIFURCATIONS OF LIMIT CYCLES 10 NUMERICAL ANALYSIS OF BIFURCATIONS 10 1 NUMERICAL ANALYSIS AT ?XED PARAMETER VALUES 10 1 1 EQUILIBRIUM LOCATION 10 1 2 MODI?ED NEWTON’S METHODS 10 1 3 EQUILIBRIUM ANALYSIS 10 1 4 LOCATION OF LIMIT CYCLES 10 2 ONE-PARAMETER BIFURCATION ANALYSIS 10 2 1 CONTINUATION OF EQUILIBRIA AND CYCLES 10 2 2 DETECTION AND LOCATION OF CODIM 1 BIFURCATIONS 10 2 3 ANALYSIS OF CODIM 1 BIFURCATIONS 10 2 4 BRANCHING POINTS 10 3 TWO-PARAMETER BIFURCATION ANALYSIS 10 3 1 CONTINUATION OF CODIM 1 BIFURCATIONS OF EQUILIBRIA AND ?XED POINTS 10 3 2 CONTINUATION OF CODIM 1 LIMIT CYCLE BIFURCATIONS 10 3 3 CONTINUATION OF CODIM 1 HOMOCLINIC ORBITS 10 3 4 DETECTION AND LOCATION OF CODIM 2 BIFURCATIONS 10 4 CONTINUATION STRATEGY 10 5 EXERCISES 10 6 APPENDIX 1: CONVERGENCE THEOREMS FOR NEWTON METHODS 10 7 APPENDIX 2: DETECTION OF CODIM 2 HOMOCLINIC BIFURCATIONS 10 7 1 SINGULARITIES DETECTABLE VIA EIGENVALUES 10 7 2 ORBIT AND INCLINATION ?IPS 10 7 3 SINGULARITIES ALONG SADDLE-NODE HOMOCLINIC CURVES 10 8 APPENDIX 3: BIBLIOGRAPHICAL NOTES