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Index in control, 258 in Lorenz system, 258, 300 subcritical, 187, 304 supercritical, 187 limit point, 177, 201 local, 173 Neimark-Sacker, 202 in adaptive control, 259 in delayed logistic map, 228 of limit cycle, 257 supercritical, 215 period-doubling, 202, 296 of limit cycles, 257 pitchfork, 182 in Lorenz system, 258 saddle-node, 253 torus, 257 transcritical, 182 in CSTR, 186

2, 9 3, 9 ◦, 9 ⊕, 9 ⇀, 261 Cn , 9 Rn , 9 adjacency matrix, 20 Arnold diffusion, 170 attractor, 26 Lorenz, 304 geometric, 304 strange, 26 Bautin example, 231 bifurcation, 174 Andronov-Hopf, 186 diagram, 175 flip, 202 Feigenbaum cascade, 213, 276 in adaptive control, 259 in H´enon map, 232 in Ricker map, 212 of limit cycles, 257 subctitical, 208 supercritical, 208 fold, 34, 177, 201 in a population model, 207, 230 in H´enon map, 232 in the logistic map, 264 of limit cycles, 255 homoclinic, 285, 295 to saddle-node, 175 Hopf, 186 examples of, 230 in a predator-prey model, 231 in Brusselator, 201

Cantor set, 271 capacity carrying, 129, 230 digestion/handling, 132 center, 57 characteristic polynomial, 38 Classical Mechanics, 143 codimension, 174 condition bifurcation, 174 cone, 281 genericity, 173, 176 conjugacy smooth, 27 topological, 27, 283, 305 constant Feigenbaum, 213, 273 Lipschitz, 94 311

312 of motion, 128 continuous-flow stirred tank reactor, 183 contraction linear, 46, 51 Contraction Mapping Principle, 95 Criterion Bendixson, 124 Dulac, 125 Routh-Hurwitz, 111 cross-section, 87, 301 curve horizontal, 276 vertical, 277 cycle, 24 exponentially orbitally stable, 90 hyperbolic, 109 saddle, 109 simple, 86 slow-fast, 160 decomposition Jordan, 71 orthogonal, 185 partial fraction, 48 determinant, 9 diameter, 278 diffeomorphism, 26 direct sum, 39 distance, 16 domain connected, 124 doubly-connected, 125 fundamental, 63, 64, 203–205, 209 simply connected, 124 dynamical system, 18 chaotic, 273 ecological modelling, 126 eigenspace generalized for complex eigenvalues, 41 stable, 47 unstable, 47 eigenvalue, 37 critical, 173, 233 determining, 286

INDEX multiple, 40 simple, 39 eigenvector, 38 generalized, 40 energy, 144, 165 kinetic, 147, 166 potential, 147, 166 Kepler, 163 equation branching, 183 Brusselator, 199 characteristic, 38 logistic, 129 van der Pol, 153 equilibrium, 24 hyperbolic, 108 internal, 127 nontrivial, 127 positive, 127 saddle, 108 saddle-focus, 285 saddle-node, 253 saddle-saddle, 33 simple, 146 equivalence local, 28 orbital, 139, 287 smooth, 26 topological, 26, 305 of complex ODEs, 188 of families, 175 evolution, 16 Feichtinger model, 231 Feigenbaum cascade, 213 constant, 273 universality, 294 Fibonacci law, 30 numbers, 70 first Lyapunov coefficient, 197 FitzHugh-Nagumo model, 296 fixed point, 24 hyperbolic, 100

313

INDEX saddle, 100 infinite-dimensional, 275 Floquet multiplier, 86 flow, 18 gradient, 146 Hamiltonian, 166 focus, 56 stable, 57 unstable, 57 formula Cauchy integral, 44 Fredholm decomposition, 47 solvability condition, 246 function Hamiltonian, 144 multilinear, 245 generator, 19 infinitesimal, 21 of translation, 30 Green Theorem, 124 Grobman-Hartman Theorem for maps, 100 global, 99 for ODEs, 108 global, 103 Gronwall Lemma, 81 group property, 18

invariant curve closed, 215 stable, 222, 230 invariant set, 25 asymptotically stable, 26 globally, 26 stable, 26 strange, 26 invariant torus, 169, 257 Inverse Function Theorem, 77, 181, 198, 206 Lipschitz, 96 involution, 168 isoclines, 126 Jacobi identity, 168 Jordan block, 40 chain, 40 length of, 40 curve, 117 equivalent, 125 decomposition, 71 Jury criteria, 81

Law of Mass Action, 126 Leibnitz rule, 168 Lemeray diagram, 23 Li–Yorke Theorem, 262 Lie algebra, 168 Lienard system, 160 H´enon map, 232, 295 limit cycle, 121 Hadamard Graph Transform, 219, 235 stable, 285 Hamilton-Cayley Theorem, 38 linearization, 287 Hamiltonian, 144, 165 Liouville system, 144 formula, 86 Heisenberg equation, 70 Theorem, 166 heteroclinic structure, 170 Liouville-Arnold Theorem, 168 homeomorphism, 26, 279, 288 Lorenz homoclinic explosion, 303, 305 attractor, 304 homoclinic structure, 103, 284 system, 112, 258, 300, 305 of cycle, 170 Lorenz system, 294 Implicit Function Theorem, 88, 119, 153, Lotka-Volterra system, 126 176, 179, 181, 183, 186, 190, 192, Lyapunov function, 109, 132 209, 210, 223 Theorem, 84 Intermediate Value Theorem, 262, 283

314 Lyapunov-Schmidt reduction, 182 manifold center, 240 parameter-dependent, 251 center–unstable, 234 local, 239 equilibrium, 178 invariant, 286, 296, 300, 304 slow, 298 map t-shift, 18 adaptive control, 259 correspondence, 88 delayed logistic, 228 deleyed logistic, 110 horseshoe, 278, 282 linear planar, 42 linear hyperbolic, 47 in Banach space, 52 Lipschitz, 94 logistic, 265, 266 saw-tooth, 303, 305 surjective, 26 tent, 266 Markov graph, 262 matrix adjoint, 246 monodromy, 86 nilpotent, 71 semisimple, 71 transpose, 9 matrix product bialternate, 81 method of unknown coefficients, 194 metric, 16 equivalent, 30 minimal period, 86 model chemostat, 156 feedback control, 258 Lotka-Volterra, 126, 132, 144, 164 generalized, 132 perturbed, 149

INDEX of infectious diseases, 157 population, 207, 212, 228, 230 prey-predator, 155 rock-scissor-paper, 162 Rosenzweig-MacArthur, 135 monodromy matrix, 86 multiplier, 86 nontrivial, 86 trivial, 86 nerve impulse, 296 Newton Law, 147 node, 56 stable, 56 unstable, 56 norm, 16 equivalent, 45 Euclidean, 9 Lyapunov, 60 operator, 43 supremum, 18 normal coordinates, 93 normal form, 173 for flip bifurcation, 207 on the critical center manifold, 249 for fold bifurcation, 178, 202 on the critical center manifold, 246, 249 for Hopf bifurcation, 186 on the critical center manifold, 247 for Neimark-Sacker bifurcation, 214 for Neimerk-Sacker bifurcation on the critical center manifold, 250 for pitchfork bifurcation, 182 for transcritical bifurcation, 182 on a center manifold, 245 topological, 176 null-space, 245 operator doubling, 274 evolution, 18 orbit, 22 connecting, 25 heteroclinic, 25 homoclinic, 25, 285, 289

315

INDEX in Lorenz system, 301 periodic, 24, 281, 302, 304 in planar ODE, 119 isolated, 125 on closed invariant curve, 222 quasi-periodic, 168 ordering lexicographic, 305 Sharkovsky, 265 ordinary differential equation, 22 linear, 52 hyperbolic, 60 orthogonal complement, 246 oscillator Duffing’s, 148 harmonic, 148, 154 nonlinear, 110 van der Pol, 158, 230 part imaginary, 9 real, 9 pendulum elastic, 169 ideal, 148 phase portrait, 24 of a complex system, 188 of planar linear ODEs, 54 Poincar´e map, 88 for Hopf bifurcation, 189 for Lorenz system, 301 near homoclinic orbit, 290 Recurrence Theorem, 167 Poincar´e map, 34 Poincar´e-Bendixson Theorem, 120 point ω-limit, 113 bifurcation, 174 equilibrium, 24 fixed, 24 turning, 177 Poisson bracket, 165, 168 predator, 126 prey, 126

product matrix, 37 scalar, 9, 16 Quantum Mechanics, 70 R¨ossler system, 29 rate growth, 230 harvest, 230 Rayleigh equation, 230 recurrence, 207, 212, 228 Reduction Principle, 240, 244 repellor, 26 resolvent, 44 resonant term, 196, 226 response functional, 132 Holling Type II, 132 numerical, 132 Ricker map, 29, 212, 273 saddle, 55 standard, 177, 245 saddle quantity, 285, 296 saddle-focus, 285, 288, 294, 296, 298 Schwarzian derivative, 295 segment cone, 78 line, 283 semiflow, 18 sequence, 279 kneading, 304 symbolic, 302 set ω-limit, 113 compact, 280 connected, 74 invariant, 25, 222, 283, 298 closed, 281 stable, 102 unstable, 102 level, 145 Sharkovsky Theorem, 262, 265 shift dynamics, 20 shift map, 267

316 nonautonomous, 29 Shilnikov saddle-focus, 285 noninvertible, 17 Shoshitaishvilly Theorem, 252 parameters, 17 singularity, 177, 201 planar fold, 178 potential, 147 Smale Horseshoe, 279, 285, 291, 294, 295 slow-fast, 158 Smale horseshoe map potential and a homoclinic structure, 284 2m-dimensional, 166 space examples of, 148 Banach, 16 reversible, 164 metric, 16 slow-fast, 298 phase, 15 smooth, 17 state, 15 systems complex, 16 conjugate, 27 infinite-dimensional, 16 diffeomorphic, 28 spectral equivalent, 26 bound, 81 orbitally, 28 projector, 47 resolvent formula, 51 time, 15 radius, 43, 58 continuous, 15 spectrum, 38 discrete, 15 stable manifold reparametrization, 190, 287 local, 102 time-series, 22 state, 15 trace, 9 steady, 24 transition matrix, 20 strip translation, 18 horizontal, 278 transverse segment, 117 vertical, 278 trapping region, 89 subshift dynamics, 20 travelling wave, 296 symbolic dynamics, 20 periodic, 298 one-sided, 266 system unit sphere, 9 autonomous, 17 unstable manifold conservative, 163 local, 102 continuous, 17 variation of constants, 81 deterministic, 16 vector field, 21 discrete-time, 19 divergence of, 146, 166 dynamical, 18 Hamiltonian, 167 first-order homogeneous, 158 Hamiltonian, 144 2m-dimensional, 165 integrable, 165, 168 invertible, 17 linear planar, 54 locally defined, 17

INDEX