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| Gumowski I., Mira Ch. — Recurrences and Discrete Dynamic Systems |
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| Предметный указатель |
Abel, functional equation of 4 97
Accumulation of bifurcations 86
Admissible perturbations 18
Almost identity transformation 92 95 98 124
Analytical invariant curve (in the sense of Poincare) 67 70
Antecedent 12 29
Antecedent, half-trajectory 12
Antecedent, sequence 6
Asymptotic properties of cycles 112 113
Attractor, composite 216—219 223 233
Attractor, preferential (priviledged) 196 198 200—202 215
Attractor, strange 10 216 229 249
Automorphic function 4 46 67 210 245 246
Autonomous differential equation 17 18 101 102 104
Autonomous recurrence 2 10 11 15
Baker transformation 26
Bifurcation 6 21 36 33 40 94 115 129 136 146 187 196
Bifurcation, chain 37 40 178 208 216 224
Bifurcation, function 129
Bifurcation, leading to a regular closed invariant curve 26 91 95 216 225—
Bifurcation, leading to an invariant segment 38 43
Bifurcation, leading to the disappearance of a closed invariant curve 219 220
Bifurcation, scheme 36 37 115 119 130 133 144 168 177—179 213—216
Bifurcation, structure 41—43
Bifurcation, value 39 40 129 147 177 191 193 199
Biological model in the form of a recurrence 23 223
Birkhoff's, classes of functions 10
Birkhoff's, instability rings 17 105 110 111 115
Boettcher, functional equation of 4
Boundary, of a no-escape region 33 185 226
Boundary, of an influence domain 32—34 83 91 92 185
Box-within-a-box structure, in parameter space 41
Box-within-a-box structure, in phase space 177—179 196 201
Branching process 6
Canonical form 71—74 78 92 98 102 122 126 128 133 140 148 149 181
Centre 17 72—74
Centre, unstable 167 169
Chain of cycles 41
Chaos, generated by a first order recurrence 44
Chaos, generated by a second order recurrence 122 136 137 148 159 169 173 185 222—233 249
Chaos, in biology 23 223
Chaos, in economics 25
Chaos, in the sense of Poincare 19
Characteristic, equation 63 66
Characteristic, number (in the sense of Liapunov) 94
Characteristic, qualifiers 21 149
Characterization of a function by means of singularities 6
Cigala constants 92
Closed phase plane trajectory 3 91—94 177 216—221 225
Collective variables 12 20 104
Combinatorial explosion 29
Complex (or chaotic) dynamics 12 51 52 185
Composite, fixed point or cycle 47 50 149
Composite, stability region 34 36
Conjugate transformation 13 55 241
Consequent 8 29
Consequent, sequence 6
Conservation of symmetry 55
Continuous, dependence on data 19
Continuous, dynamic system 5 17 30 179
Control problem with hysterisis 63
Correctly set problem 19
Critical, critical case (in the sense of Liapunov stability) 17 30 36 47 122 132
Critical, curve 83 88 89 185 203 204 207 217 226 227
Critical, effective order 119 121
Critical, point (in the sense of Julia — Fatou) 12 32
Cusp 133 148 178 197
CYCLE 9 15 29
Cycle with unequal number of antecedents 204 205
Cycle, external 150—152 160—164 207
Cycle, internal 149—152 160—164
Cycle, mixed 154—159 164—169 173 197 198
Cycle, secondary 161
Cyclic curve segments 220
Damping threshold 215
Deterministic parity violation 54—57
Difference equation 4 65 66 69
Differentiation operator, non-integer 99
Diffusion region 122 152
Disappearance of homoclinic points 213
Discontinuity of antecedent invariant curves 204 207—210
Discrete (half-) trajectory 6
Discrete (half-) trajectory, located on an invariant curve 70 216 232
Discreteness of measurements 5
Disintegration of an island structure 117—119 143—146
Disorderiliness 14
Display of discrete trajectories 6 156 157 186 222 226
Dominant (or dominating) terms 76 123 124 126—129 181 191 194 203
Dynamic system, continuous 5
Dynamic system, discrete 5
Dynamics, complex or chaotic 20 122 136 152 153
Dynamics, orderly 41 51 52 58 81 212 216 249
Economic model 23—25 62
Effective order (or k/r ratio) 116 187 197 199 201 212
Eigendirections 71 77
Eigenvalue 30 66 94
Eigenvalue, problem for a functional equation 98 238—240
Eigenvalue, relation to stability 67 72
Equivalence, between recurrences and difference equations 4
Equivalence, between recurrences and differential equations 3 101—103
Ergodic theory 12 51 104
Exceptional case (in the sense of Cigala) 72 122 124 125 127 133—135 138 143 181 187 193 201
Excess, antecedent 46 81 203
Excess, antecedent curve 83
Excess, cycle 136
Feedback system (non linear) 26
Fixed point 8 15 29
Focus 17 73 74
Fractional iterate 17 70 71 96 100 101
Functional equation of automorphic functions 4 16 46 245
Functional equation of invariant curves 16 67—69 77
Functional equation of iterative square roots 255
Functional equation of translation 74 94
Functional equation, Abel 4 97
Functional equation, Boettcher 4
Functional equation, dependence on an arbitrary function 68 242
Functional equation, dynamically relevant 4 52
Functional equation, involving involutions 252—254
Functional equation, Perron — Frobenius 4 12 51 228 238 249 254
Functional equation, Picard 53 238
Functional equation, Schroeder 4 46 49 79 80 83 97 98 210
Functional equation, Schroeder, for an invariant segment 238
Functional iterate 4 9 13 97
General solution of a recurrence 52 238—239
Generating bifurcation 40 91 95 233
Germ of an invariant curve 76 77 229 232
Grossier (system or equation) 75 76
Hadamard, correctly set problem 19
Hat function 26 53 56
Heteroclinic point, accumulation point of cycles 115
Heteroclinic point, conservative 116 121 159
Heteroclinic point, N-type 195 196 199 201
Heteroclinic point, non conservative 194 197 218
Homoclinic point 110 112 113 116 133 144 155 159 193 232
Homoclinic point, accumulation point of cycles 113 115 152 177 197 199 200
Image cycle 169
Implicit function theorem 181
Inert (system or equation) 75 76
Influence domain 30 56 81—91 193
Influence domain, analytical 83—85 87
Influence domain, composite 34 40 50 90 184 185 227
Influence domain, immediate 13 31 47 49 87 98
Influence domain, multiply connected (or disjoined) 30—31 49 87—89
Influence domain, singly connected 30
Influence domain, total 31 33 81 87 93
Instability ring (of Birkhoff) 17 105 110 111 115
Invariant angle (or direction) 125—128 140 201
Invariant curve 16 67 70
Invariant curve, analytical (in the sense of Poincare) 67 70
Invariant curve, antecedent branch 83
| Invariant curve, closed 91 94 177 216—221 225
Invariant curve, consequent branch 83
Invariant curve, coordinate dependence 68
Invariant curve, dependence (apparent) on an arbitrary function 70
Invariant curve, excess antecedent branch 203 207—210
Invariant curve, explicitly known 101 252—256
Invariant curve, functional equation of 16 67—69 77
Invariant curve, main 112 122 137 148 174 203
Invariant curve, parametric description 70
Invariant curve, regular 79 81 135 177
Invariant curve, relation to the Schroeder equation 79
Invariant curve, S-loops of 192—194 198 216
Invariant curve, self-intersecting 203 208—210 217 218 222
Invariant curve, separation into branches 69
Invariant curve, singular 67 79 82 131 177
Invariant curve, stability of 94
Invariant density 12 14 50 56 227 228
Invariant manifold 7
Invariant segment 9—11 17 32 34 38 40 238
Invariant segment, cyclic 33 43—46 56 57 240 241
Inverse recurrence 11
Involution 239 251—255
Island structure 110 115 136 137 144 153
Iterate, continuous 17 70 71 96—98 100 101
Iterate, fractional 17 70—7 1 96—98 100 101
Iterated recurrence 8 15 17 29
Iteration of quadratic polynomials 23 96
Iterative square root 96 97 255—257
Jacobian determinant, equal to unity 61 102 104 182 252
Jacobian determinant, non vanishing, not implying unique antecedents 181 182
Jacobian determinant, vanishing 12
Julia — Fatou theorem (on critical points) 32 40
Koenigs — Lemeray staircase construction 30
Liapunov V-function 81 196
Limit cycle 3 26 64
Linear differential equation with periodic coefficients 63 65
Linear recurrence 2 17 30 47 65—75 100
Liouville's theorem (area conservation) 104 184 254
Logarithmic convexity 99
Longitudinal motion of a particle 61
Lorenz problem (meteorology) 25
m-th order form 123 152
Macroscopic variables 12
Main fixed point 112 122 137 148 174 187 191 194 195 197 198 201 216
Main invariant curve 112 140 159 160 169 174 187 189 190
Mean mass density 50 56 227 228
Mean mass density, equation of 54 228
Microtron 61 174
Myrberg recurrence 38 96 237 240 257
No-escape region 33 36 50 156 157 184 223 226 228—233
No-escape segment 12 36
Node, (simple) 17 71 73 77
Node, critical 75 77 80
Node, di-critical (or star-node) 74 75 80
Non homogeneous linear functional equation 78 79
Non linear feedback system 26
Open optical resonators 61
Orderly dynamics (or motion) 41 51 52 58 81 212 216 249
Perron — Frobenius functional equation 4 12 51 211 228 238 249 254
Perturbation, admissible 18 21
Perturbation, non linear 17 76 79 102
Perturbation, structural 20
Phase (space) cell 14 79 109 123 159
Phase advance (or rotation angle) 67
Piecewise linear recurrence 25 52
Poincare's surface of section 2 64 179
Point of alternance 83 85 88 207—210 217
Point singularity 7 15 16
Point singularity, stochastic 250 251
Point singularity, types of 17
Points of a stability boundary 31 83
Polynomials of Myrberg 38
Population dynamics 23 62 223
Preferential attractor 196 198 200—202
Pseudo-eigenvalue 59
Quadratic polynomials, iteration of 23
Quadratic recurrence, first order 38 96 237 240 257
Quadratic recurrence, second order 106—137 187—201 216—223 229—233
random number generation 10 15 27
Randomness, relation to Birkhoff's function classes 14
Randomness, relation to topological entropy 14 59 60
Randomness, tests 14 59 60
Recurrence 2 15
Recurrence, biological model 23 62 223
Recurrence, bounded quadratic 106 148—159 197—201 212
Recurrence, cubic 106 137—148 201
Recurrence, cycle of 9 29
Recurrence, economic model 23—25
Recurrence, explicitly known invariant curves of 101 252—256
Recurrence, explicitly known solutions 44 237—239 242 243
Recurrence, exponential 106 159—169
Recurrence, fixed point of 8 29
Recurrence, general solution 3 15
Recurrence, generic examples 5
Recurrence, inverse 11
Recurrence, iterated 8 15
Recurrence, linear 2 65—75
Recurrence, piecewise linear 25
Recurrence, quadratic 38 96 106—137 187—201 216—223 229—233 237 240 257
Recurrence, sinusoidal 106 169—173
Recurrence, sinusoidal unsymmetric 106 173—176
Recurrence, solution of 3
Recurrence, solution of, analytic definition 5
Recurrence, solution of, Cauchy problem 4 6
Recurrence, solution of, explicit known 44 66 237—239 242 243
Recurrence, solution of, indirect definition 6
Recurrence, solution of, operational definition 6
Recurrence, solution of, relation to fractional and continuous iterates 244 256 257
Recurrence, solution of, relation to solutions of some functional equations 52 97 238 239
Recursive system of equations 50 54 77 80 100 149
Redundancy tests 82 105
Regularization of an incorrectly set problem 20
Resonance, harmonic 9 63
Resonance, main 9 72
Resonance, subharmonic 9 29 63
Resonance, weak 74 187
Rotation angle (or phase advance) 67 72 146 169
Rotation number 108 109 112 159 187 196 199 229
Rotation sequence 45 229
Saddle (simple) 17 72 73
Saddle (simple), multibranch 127 128 136 181 201
Schwarzian derivative 33
Secular terms in functional equations 55
Segment without contact 28
Sensitivity analysis 20
Sensitivity coefficients 20 45 59 220 237 247 248
Separatrix 79 105
Sequence of antecedents and consequents 6
Servo with feedback 23
Singularity of a recurrence 6 7 69 71 97
Singularity structure 6 14 18 20 36 38 44 45 200 220
Singularity structure, composite 7 219 221
Singularity structure, elementary 7
Singularity structure, generic 21
Small parameter expansion 49 54 55 73 92—94 191 193 199
Solution of a recurrence see "Recurrence solution
Stability (in the sense of Liapunov) 29 94
Stability domain see "Influence domain"
Stagnation of discrete trajectories 216
Star-node 27 28 74 75 80 83 84
Stationary state 7—10 234
Stationary state, Birkhoff's classification 10
Stationary state, orderly see "Orderly dynamics"
Stationary state, qualitative change 26 27
Stationary state, stability of 8 9
Stationary state, stochastic (or chaotic) 12—14 17 27 28 see
Statistical physics 12 14 104
Statistical tests see "Randomness tests"
Stochastic instability (and region of) 122 136 152 153
Stochastic point singularity 250 251
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