Index

activation 327 desynchronization 288, 297 anti–integrable limit 219 diffusion 155, 247, 269 22, 71, 183, 301, 329 discommensuration 218 Milnor 183 distribution 268 DNA 213, 310 baker’s maps 74 155, 182 basin of attraction 22, 25, 182, 200 lattice 15, 154 bifurcation 12, 38, 60, 66, 194, 198, multicomponent 14 252, 291, 316, 331, 355 reversible 75 diagram 59, 291, 294 translational 155 saddle-node 293, 300, 303 with noise 189 bistable 266, 316 boundary condition 26 energy 81, 103, 210, 246 bounded variation 119, 128, 134 entropy 139, 160 escape factor 44 Cantor set 42, 48, 106, 107, 218 expanding map 82, 96 cellular automaton 17, 69 Stavskaya’s 73 flow 242 Cherry flow 294 Frenkel–Kontorova model 215, 246 cluster 191, 288, 289 front 267 configuration 17, 153, 217, 266 metastable 219, 234 genetic regulatory network 309 saddle 217 global orbits 329 convolution 128, 268 globally coupled 36, 191, 286 correlation 46, 96, 147 gradient dynamics 220, 246 Coupled Map Lattice 15, 35, 36, 67, ground state 217 84, 95, 116, 166, 266, 267 Coupled Map Network 326 H¨older condition 85 coupling 35, 67, 84, 97, 116, 126, 166, Hamiltonian 46, 51, 53, 99 191, 225, 267, 286, 348 Hausdorff dimension 102, 187, 295 diffusive 16, 35, 116 heteroclinic point 163 finite range 116, 136 Hill function 314 homoclonic point 161 decay of interaction 97, 173 hyperchaos 304 360 Index hysteresis 316 sample 22 SRB 21, 45, 80, 97, 108 indecomposable CML 72 Miller–Huse model 38 inhibition 313, 327 mixing 71, 125 integrable limit 218 spatio–temporal 39, 98, 117 interaction 14, 99, 246, 313, 327 mollifier 128 short range 47 monotonicity 70, 243, 256, 269 interface 273 multistability 289, 301 intermingled sets 188 mutual–inhibition network 316, 332 invariant manifold 293, 297, 303 Ising model model 341 kinetic 59, 76 one-dimensional 47 order preserving dynamics 243 two-dimensional 54 oscillator 344 isotropy overdamped system 220, 253 lattice 191 subgroup 190 Perron–Frobenius 44, 98, 118 iterated function scheme 40 Peyrard-Bishop-Dauxois model 214 phase 65, 80, 266, 286 Josephson junction 211 Gibbs 79 oscillators 291 Kuramoto model 287 pure 67 space 14, 35, 80, 96, 153, 303 La Salle’s Invariance Principle 249 phase transition 25, 37, 110 Lasota–Yorke inequality 123, 130, 138 Picard iterations 244 law of large numbers 117 piecewise affine map 19, 41, 66, 183, limit set 327 α- 182 piecewise contracting 327 ω- 182 piecewise expanding 118 Lyapunov exponent 166, 192, 299 piecewise linear differential tangential 192 321 transverse 192 Lyapunov function 249 ratchet effect 223 repeller 42, 172 majority voter 17, 70 resonant tongue 295, 303 marginals 134 riddled set 187 Markov rotation number 278 chain 18, 48, 81, 89, 172 rotationally ordered 216, 248, 258 partition 18, 43 master equation 56 saddle node 252 mean spacing 216, 247, 277 space-time chaos 11, 36 measure spatial chaos 160 conditional invariant 44 spatially–homogeneous solutions 164 image 97 spectral gap 125, 140, 145 invariant 43, 97 stability natural 22, 80, 109, 192 asymptotic 183 observable 109, 117, 148 stochastic 21, 117 physical 21, 80, 117, 148 transverse 192, 299 regular 97 standard map 215 Index 361 state 80, 242, 254 matrix 44 qualitative 325 operator 22, 79, 118, 126, 137 state transition graph 325 transient 37, 72 symbolic dynamics 40, 80, 329 transversal mapping 105 synapse 343 synchronization 167, 285, 345 velocity telescoping 101 irrational 172, 267 Theorem rational 172, 267 Birkhoff Recurrence 251 Helly Selection 275 wave Implicit Function 173 lurching 351 Ionescu-Tulcea Marinescu 124 thalamic 350 thermodynamic limit 36, 55 travelling 167, 252, 267, 277, 348 theta model 349 winding ratio 294 transfer Winfree model 287