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308 BIBLIOGRAPHY Beyn, W.-J., Champneys, A., Doedel, E., Govaerts, W., Kuznetsov, Yu.A. & Sandstede, B. [2002], Numerical continuation, and computation of normal forms, in B. Fiedler, ed., Handbook of Dynamical Systems, Vol. 2, Elsevier Science, Amsterdam, pp. 149 219. Brin, M. & Stuck, G. [2002], Introduction to Dynamical Systems, Cambridge University Press, Cambridge. Carr, J. [1981], Applications of Center Manifold Theory, Springer-Verlag, New York. Chow, S. N. & Hale, J. K. [1982], Methods of Bifurcation Theory, Vol. 251 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], Springer-Verlag, New York. Coullet, P. & Eckmann, J.-P. [1980], Iterated Maps on the Interval as a Dynamical System, Birkhauser, Boston, MA. Devaney, R. L. [1989], An Introduction to Chaotic Dynamical Systems, Addison- Wesley Studies in Nonlinearity, second edn, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA. Dumortier, F., Llibre, J. & Artés, J.C.. [2006], Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin. Guckenheimer, J. & Holmes, P. [1983], Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York. Hale, J. & Koçak, H. [1991], Dynamics and Bifurcations, Springer-Verlag, New York. Hartman, P. [2002], Ordinary Differential Equations, Vol. 38 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Hasselblatt, B. & Katok, A. [2003], A First Course in Dynamics: With a Panorama of Recent Developments, Cambridge University Press, New York. Hirsch, M. & Smale, S. [1974], Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York. Ilyashenko, Yu. & Li, Weigu. [1999], Nonlocal Bifurcations, American Mathematical Society, Providence, RI. Iooss, G. [1979], Bifurcations of Maps and Applications, North-Holland, Amsterdam. Iooss, G. & Adelmeyer, M. [1992], Topics in Bifurcation Theory and Applications, World Scientific, Singapore. Iooss, G. & Joseph, D. D. [1990], Elementary Stability and Bifurcation Theory, Undergraduate Texts in Mathematics, second edn, Springer-Verlag, New York.
BIBLIOGRAPHY 309 Irwin, M. [1980], Smooth Dynamical Systems, Academic Press, New York. Kato, T. [1980], Perturbation Theory for Linear Operators, Springer-Verlag, New York. Katok, A. & Hasselblatt, B. [1995], Introduction to the Modern Theory of Dynamical Systems, Vol. 54 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge. Kielhöfer, H. [2004], Bifurcation Theory: An Introduction with Applications to PDEs, Springer-Verlag, New York. Kirchgraber, U. & Palmer, K. J. [1990], Geometry in the neighborhood of invariant manifolds of maps and flows and linearization, Vol. 233 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow. Kuznetsov, Yu. A. [2004], Elements of Applied Bifurcation Theory, Vol. 112 of Applied Mathematical Sciences, third edn, Springer-Verlag, New York. Lyubich, M. [2000], The quadratic family as a qualitatively solvable model of chaos, Notices Amer. Math. Soc. 47, 1042 1052. Marsden, J. E. & Ratiu, T. S. [1999], Introduction to Mechanics and Symmetry, Vol. 17 of Texts in Applied Mathematics, second edn, Springer-Verlag, New York. Moser, J. [1973], Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, NJ. Nitecki, Z. [1971], Differentiable Dynamics, MIT Press, Cambridge, MA. Perko, L. [2001], Differential Equations and Dynamical systems, Vol. 7 of Texts in Applied Mathematics, third edn, Springer-Verlag, New York. Shilnikov, L.P., Shilnikov, A.L., Turaev, D.V. & Chua, L. [1998], Methods of Qualitative Theory in Nonlinear Dynamics. Part I, World Scientific, Singapore. Shilnikov, L.P., Shilnikov, A.L., Turaev, D.V. & Chua, L. [2001], Methods of Qualitative Theory in Nonlinear Dynamics. Part II, World Scientific, Singapore. Shub, M. [2005], What is...a horseshoe?, Notices Amer. Math. Soc. 52(5), 516 517. van Strien, S. [1991], Interval dynamics, in E. van Groesen & E. de Jager, eds, Structures in Dynamics, Vol. 2 of Studies in Mathematical Physics, North- Holland, Amsterdam, pp. 111 160. Vanderbauwhede, A. [1989], Centre manifolds, normal forms and elementary bifurcations, Dynamics Reported 2, 89 169.
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Index, 9, 9, 9, 9, 261 C n, 9 R n, 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 Hénon 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 Hénon 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 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 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 INDEX 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 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
INDEX 313 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 Hénon map, 232, 295 Hadamard Graph Transform, 219, 235 Hamilton-Cayley Theorem, 38 Hamiltonian, 144, 165 system, 144 Heisenberg equation, 70 heteroclinic structure, 170 homeomorphism, 26, 279, 288 homoclinic explosion, 303, 305 homoclinic structure, 103, 284 of cycle, 170 Implicit Function Theorem, 88, 119, 153, 176, 179, 181, 183, 186, 190, 192, 209, 210, 223 Intermediate Value Theorem, 262, 283 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 limit cycle, 121 stable, 285 linearization, 287 Liouville formula, 86 Theorem, 166 Liouville-Arnold Theorem, 168 Lorenz attractor, 304 system, 112, 258, 300, 305 Lorenz system, 294 Lotka-Volterra system, 126 Lyapunov function, 109, 132 Theorem, 84
314 INDEX 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 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
INDEX 315 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 Poincaré map, 88 for Hopf bifurcation, 189 for Lorenz system, 301 near homoclinic orbit, 290 Recurrence Theorem, 167 Poincaré map, 34 Poincaré-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 Rössler 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 INDEX Shilnikov saddle-focus, 285 Shoshitaishvilly Theorem, 252 singularity, 177, 201 fold, 178 Smale Horseshoe, 279, 285, 291, 294, 295 Smale horseshoe map and a homoclinic structure, 284 space Banach, 16 metric, 16 phase, 15 state, 15 complex, 16 infinite-dimensional, 16 spectral bound, 81 projector, 47 resolvent formula, 51 radius, 43, 58 spectrum, 38 stable manifold local, 102 state, 15 steady, 24 strip horizontal, 278 vertical, 278 subshift dynamics, 20 symbolic dynamics, 20 one-sided, 266 system autonomous, 17 conservative, 163 continuous, 17 deterministic, 16 discrete-time, 19 dynamical, 18 first-order homogeneous, 158 Hamiltonian, 144 2m-dimensional, 165 integrable, 165, 168 invertible, 17 linear planar, 54 locally defined, 17 nonautonomous, 29 noninvertible, 17 parameters, 17 planar potential, 147 slow-fast, 158 potential 2m-dimensional, 166 examples of, 148 reversible, 164 slow-fast, 298 smooth, 17 systems conjugate, 27 diffeomorphic, 28 equivalent, 26 orbitally, 28 time, 15 continuous, 15 discrete, 15 reparametrization, 190, 287 time-series, 22 trace, 9 transition matrix, 20 translation, 18 transverse segment, 117 trapping region, 89 travelling wave, 296 periodic, 298 unit sphere, 9 unstable manifold local, 102 variation of constants, 81 vector field, 21 divergence of, 146, 166 Hamiltonian, 167