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Arrowsmith D.K., Place C.M. — Dynamical systems. Differential equations, maps and chaotic behaviour
Arrowsmith D.K., Place C.M. — Dynamical systems. Differential equations, maps and chaotic behaviour

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Название: Dynamical systems. Differential equations, maps and chaotic behaviour

Авторы: Arrowsmith D.K., Place C.M.

Аннотация:

This text discusses the qualitative properties of dynamical systems including both differential equations and maps, The approach taken relies heavily on examples (supported by extensive exercises, hints to solutions and diagrams to develop the material including a treatment of chaotic behaviour. The unprecedented popular interest shown in recent years in the chaotic behaviour of discrete dynamic systems including such topics as chaos and fractals has had its impact on the undergraduate and graduate curriculum. The book is aimed at courses in dynamics, dynamical systems and differential equations and dynamical systems for advanced undergraduates and graduate students. Applications in physics, engineering and biology are considered and introduction to fractal imaging and cellular automata are given.


Язык: en

Рубрика: Физика/Динамические системы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1992

Количество страниц: 330

Добавлена в каталог: 07.12.2004

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Action-angle variables      149
Afline system      55 175
Algebraic type      52
Animal conflict model      218
Area-preserving map, Henon      263
Area-preserving map, Poincare      155 259
Area-preserving map, twist      261
Asymptotic stability      84 89 229
Attracting set      103 140 141
Attractor      9 141
Aubry — Mather theorem      261
Autonomous equations      6 12
Basin of attraction      91
Beats      175
Bifurcation      212
Bifurcation point      224
Bifurcation, cusp      237
Bifurcation, flip      242 245 258
Bifurcation, fold      240 242
Bifurcation, Hopf      225 228
Bifurcation, period-doubling      246 259
Bifurcation, pitchfork      293
Bifurcation, saddle-node, flows      224 226
Bifurcation, saddle-node, maps      258
Bifurcation, saddle-node, symmetric      291
Braided structure      258
Canonical local family, flip      245 293
Canonical local family, fold      242 293
Canonical system      42
Cantor set      276 280
Cantor set, fractal dimension      281
Cantor set, horseshoe map      277 278
Cantor set, iterated function scheme      281
Cantor set, tent map      276
Cellular automata      284
Centre      46
Chaotic (strange) attractor, Duffing oscillator      271
Chaotic (strange) attractor, folded band (Rossler)      251 257
Chaotic (strange) attractor, Henon      295
Chaotic (strange) attractor, iterated function scheme      282
Chaotic orbit, area preserving map      263
Chaotic orbit, homoclinic tangles      264
Chaotic orbit, horseshoe map      278
Chaotic orbit, logistic map      250
Chaotic region      266
Characteristic surface      216
Characteristic, cubic      189
Characteristic, folded      194
Characteristic, neon      194
Characteristic, resistor      189
Characteristic, triode      201
Chemical oscillator      231
Closed orbit      46 102 129
Closed orbit, hyperbolic      132
Collage      284
Competing species      180
Competitive exclusion      182
Complementary function      56
Conservative system      97
Coupled pendula      172
Critical damping      165
Cusp bifurcation      237
Decoupled system      17 58
Derivative along a curve      88
Derivative, directional      96
Diffeomorphism      132
Differentiable manifold      128
Dissipative map      268
Domain of stability      91
Doubling map      300
Duffing oscillator, twin well      254
Dynamical equations      162 223
Dynamical equations, attracting/repelling      104
Dynamical equations, capacitor      169
Dynamical equations, elliptic      253 260 261 266
Dynamical equations, hyperbolic      132
Dynamical equations, inductor      168
Dynamical equations, mutual-inductance      201
Dynamical equations, Newton’s law of cooling      11
Dynamical equations, Newton’s second law of motion      163
Dynamical equations, Poincare map/diffeomorphism      104 131
Dynamical equations, stable/unstable manifold      132
Economic model      170
Eigenspace, centre      156
Eigenspace, stable/unstable      121
Electrical circuit theory      167
Elliptic fixed point      253 260 261 266
Evolution operator      23 52
Exact differential equation      27 28
Exponential matrix      52
Family of differential equations      212 223
Feigenbaum number      246 250 280
First integral      96 147
Fixed point, phase portrait/flow      9 14
Fixed point, phase portrait/flow, hyperbolic      80 120 122
Fixed point, phase portrait/flow, isolated      11 14 16
Fixed point, phase portrait/flow, neutrally stable      86
Fixed point, phase portrait/flow, non-hyperbolic      125
Fixed point, phase portrait/flow, non-simple      46 81
Fixed point, phase portrait/flow, simple      43 77
Fixed point, phase portrait/flow, stable      84 89
Fixed point, phase portrait/flow, stable/unstable manifold      123 298
Fixed point, phase portrait/flow, unstable      87 92
Flip bifurcation      242 245 258
Flow      23
Flow box theorem      94
Focus, linear      46
Focus, non-linear      80
Fold bifurcation      240 242
Folded band (Rossler), attractor      251 257
Forcing terms      176
Fractal basin boundary      270
Fractal dimension      280
Fractal dimension, Cantor set      281
Fractal dimension, Sierpinski carpet      301
Fractal dimension, Sierpinski gasket      283 301
Generic property      226
Global phase portrait      71 237
Green’s Theorem      109
Hamiltonian      98 148
Hamiltonian flow      154
Hamiltonian system      148
Hamiltonian, generalized coordinates/momenta      148
Hamiltonian, integrable      149
Hamiltonian, normal form      149
Hamilton’s equations      98 148
Harmonic oscillator, forced      176
Harmonic oscillator, free      165
Harmonic oscillator, overdamped      166
Harmonic oscillator, second order form      163 170 171
Harmonic oscillator, underdamped      165
Heartbeat model      212
Henon, attractor      295
Henon, map, area-preserving      263 297 298
Henon, map, quadratic      258 294 296
Heteroclinic tangle      265
Holling — Tanner model      185
Homoclinic point      264
Homoclinic tangle, periodic orbit      265
Homoclinic tangle, saddle point      264
Homogeneous differential, equation      28
Hopf bifurcation      225 228 239
Hopf bifurcation, theorem      229
Improper node, linear      45
Improper node, non-linear      80
Integrating factor      27
Invariant circle      142 260 261
Invariant set      107
Invariant torus      152
Island chain      261
Isocline      5 20
Iterated function scheme      281
Iteration      104
Jordan form, $2\times2$      38
Jordan form, $3\times3$      57
Jordan form, $4\times4$      61
Jordan form, $n\times n$      62
Jump assumption      193
Kirchhoff Laws      168
Kolmogorov — Arnold — Moser (KAM) theorem      261
Kolmogorov — Arnold — Moser (KAM) theorem, KAM-circle      261
Left shift      272
Level curve      88 96
Level surface      220
Liapunov, function      88
Liapunov, stability theorem      89
Liapunov, strong      90
Liapunov, weak      90
Lienard equation      191
Lienard plane      191
Limit cycle      103
Limit cycle in modelling      185
Limit cycle, criterion for non-existence      109
Limit cycle, semi-stable      104
Limit cycle, stable      104
Limit cycle, unstable      104
Limit point      101
Limit set      101
Linear change of variable      35
Linear diffeomorphism, classification      134
Linear diffeomorphism, hyperbolic      133
Linear diffeomorphism, orientation-preserving      134
Linear diffeomorphism, orientation-reversing      134
Linear mapping      35 48
Linear part      74
Linear system      11 35
Linear system, algebraic type      52
Linear system, classification of      52
Linear system, coefficient matrix of      35
Linear system, homogeneous      55
Linear system, non-homogeneous      55
Linear system, non-simple      46
Linear system, qualitative (topological) type      52
Linear system, simple      43
Linearization theorem, differential equations      77 123
Linearization theorem, Poincare maps/diffeomorphism      133 135
Linearization, Poincare map/diffeomorphism      132
Linearization, vector field      74
Linearized system      74
Liouville’s theorem      154
Local bifurcation      225
Local coordinates      75
Local family      242
Local phase portrait      71
Logistic law      12
Logistic map      245
Logistic map, chaotic orbits      250
Logistic map, eventually periodic orbits      294
Maximal solution      1
Momentum      163
Momentum, generalized      148
Natural frequency      165
Negative definite      88
Negative semi-definite      88
Neighbourhood      71
Node, linear      43
Node, non-linear      80
Non-autonomous equation      32 34
Normal coordinates      174
Normal modes      174
Ohm’s Law      168
Orbit in phase portrait      13
Orbit in terms of flow      26
Orbit of Poincare map/diffeomorphism      131
Ordinary point      93
Parametrically forced, pendulum      251 254
Partially decoupled system      18
Particular integral      56
Partitioned matrices      40 57 61 62
Partitioned matrices, application of      173
Period-doubling bifurcation, one-dimension      246
Period-doubling bifurcation, two-dimensions      257 259 296
Periodic orbit      136
Periodic orbit, Poincare — Birkhoff      261
Periodic orbit, stable/unstable manifold      137
Periodic point, phase portrait/flow      46 127
Phase line      11
Phase plane      13
Phase point      11 15 27
Phase portrait, construction of      17
Phase portrait, one-dimension      8
Phase portrait, qualitative type of      52
Phase portrait, restriction of      71
Phase portrait, two-dimensions      13
Piecewise modelling      195
Pitchfork bifurcation      293
Plant-pollinator model      226
Poincare (first return) map      104 129
Poincare (first return) map for Hamiltonian flow      153 155
Poincare (first return) map, area-preserving      155 259
Poincare map/diffeomorphism      136
Poincare — Bendixson theorem      106
Poincare — Birkhoff theorem      261
Polar coordinates      17
Positive definite      88
Positive semi-definite      88
Positively invariant set      107
Prey-predator problem      183 185
Principal directions      48 81
q-cycle      136
Qualitative behaviour      4 8 14 16 120 125 128 132
Qualitative equivalence      4 9 10 16 51
Qualitative equivalence of families of differential equations      239
Quasi-periodic motion      128 138
Rationally independent numbers      127
Rayleigh equation      209 292
Regularization      193
Relaxation oscillations      190
Renormalization      280 300 301
Repellor      9
Resonance      177
resonant frequency      179
Robust system      185
Rossler attractor      251 257
Rotation interval      261 297
Rotation number      261 297
Saddle connection      95 99 105 107
Saddle connection, bifurcation      239
Saddle connection, heteroclinic      265
Saddle connection, homoclinic      264
Saddle point, linear      44 121
Saddle point, non-linear      80 123
Saddle point, spiral      122 156
Saddle-node bifurcation, flows      224
Saddle-node bifurcation, maps      258
Saddle-node bifurcation, symmetric      291
Sawtooth oscillations      197
Sensitive dependence on, initial conditions      274
Separable differential equation      27
Separalrix      44 80
Separation of variables      6 27
Shunt      9
Sierpinski carpet      301
Sierpinski gasket      283 285
Similar matrices      36
Similarity classes      37
Similarity types      38
simple pendulum      208
Simply connected region      109
Smale horseshoe map      276
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