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Index

absolute covariant derivative, 330, 337

abstract ergodic theory, 661

abstraction of Euclidean geometry, 368

achieved micro–states, 654

action, 316, 466, 511

action of a new theory, 425

action potential, 155

action principle, 316

action–amplitude formalism, 425

action–amplitude picture, 425, 637

action–angle coordinates, 344

action–at–a–distance electrodynamics,652

active polarized state, 226

adaptive, 705

adaptive Lie–derivative controller, 703

adaptive motor control, 709

adaptive path integral, 437, 637, 639,640

adaptive path measure, 653, 656

adiabatic elimination, 203

affine connection, 312

affine Hamiltonian function , 700

affine transformation, 39

agreement, 225

algebraic topology, 380

Aliev–Panfilov model, 159

almost Lorentzian metric, 404

alpha skeletomotor neurons, 701

Ambrose–Singer theorem, 117

American option, 593

amplitude, 432

amplitude of the driving force, 3

Andronov, 33Andronov–Hopf bifurcation, 33Anosov diffeomorphism, 33Anosov flow, 34, 661Anosov map, 33anti–control of chaos, 7approach to equilibrium, 77arc–element, 331arc–length, 382, 659area–preserving map, 53Arnold cat map, 33arrow of time, 625asymmetric dilution, 614atlas, 306, 318atmospheric convection, 45attaching a handle, 396attachment of handles, 672attractor, 4, 44, 79, 102attractor neural networks, 245autocatalator, 64autogenetic motor servo, 701autonomous system, 3average, 348, 352average degree of chaoticity, 348average energy, 193Axelrod model, 224Axiom A, 79Axiom–A systems, 35

Bar–Eiswirth model, 159background–independent, 487Baker map, 55Banach manifold, 320

832 Index

Banach space, 320, 471Barkley model, 159Basic consequence: a different way for

calculating probabilities., 431basic sets, 79basin of attraction, 44, 80, 145basins of attraction, 4, 53Bekenstein–Hawking entropy, 574Belouzov–Zhabotinski reaction, 127Belusov–Zhabotinsky reaction, 155Berlin–Kac spherical model, 673Bernoulli map, 33Bernoulli shift dynamics, 109, 113Bernoulli systems, 32Bernstein’s motor coordination and

dexterity, 709Bessel functions, 268, 275Betti number, 417Betti numbers, 309, 375, 377, 667, 703Bianchi identity, 314Bianchi symmetry condition, 336bifurcation, 6bifurcation diagram, 51bifurcation point, 17Big–Bang, 397, 398bijection, 306billiard, 32binary systems, 101bio–diversity, 50biomechanical force–velocity relation,

233biomorph, 66biomorphic systems, 66biphasic pacing, 165Birkhoff, 26, 30, 32Birkhoff curve shortening flow, 32black hole dynamics, 572Black–Scholes–Merton formula, 584block entropy, 21body motion manifold, 709Boltzmann, 25Boltzmann constant, 192Boltzmann entropy, 72Bolzmann constant, 71Borde–Sorkin conjecture, 693Bose–Einstein condensate, 433bosonic string theory, 535Bott’s extended Morse theory, 677boundaries, 377

boundary condition, 468boundary conditions, 463boundary of a boundary, 377boundary operator, 377bra–covectors, 433bra–ket, 433brain, 708brain dynamics, 7brain motion manifold, 709brane, 532Brouwer degree, 341Brownian dynamics, 428Brownian motion, 243, 585BRST–operator, 517brute–force, 143building blocks, 394Burgers dynamical system, 116, 579butterfly effect, 46, 110

calculus of variations, 316Campbell–Baker–Hausdorff formula,

335Cantor set, 37, 103Cantori, 98capacity dimension, 65, 130carrying capacity, 49Cartesian product, 307Cartwright, 33Catalan numbers, 501catastrophes, 6category, 43Cauchy Theorem, 558causal continuity, 404causal discontinuity, 408, 682, 687Central–limit–theorem, 352chain, 376chain complex, 377chain coupling, 234chaos, 27chaos control, 128, 158chaos theory, 29, 108, 200chaotic, 3, 102chaotic attractor, 4, 102, 145chaotic behavior, 109chaoticity, 343Chapman–Kolmogorov equation, 429,

588Chapman–Kolmogorov integro–

differential equation, 430

Index 833

Chapman–Kolmogorov law, 328characteristic equation, 195characteristic Lyapunov exponents, 129chart, 306chemical kinetics, 64Chern class, 576Chern–Simons gauge theory, 512, 571Chern–Simons theory, 515Christoffel symbols, 86, 313, 331Chua’s circuit, 62Chua–Matsumoto circuit, 127circle, 306circle map, 54Clifford, 309closed string theories, 529closed timelike curves, 395co–area formula, 665coarse graining, 72coarse system, 33cobordism, 393code, 625cognitive intention Lagrangian, 655collapse of wave packets, 77colored multiplicative noise, 202combinatorial route, 434compact manifold, 307compact manifolds, 375complex number, 433complexity conjecture, 694computational complexity, 694conatural projection, 322condensed, 624conditioned training, 694configuration manifold, 317, 321configuration–space path integral, 442configurational partition function, 362conformal, 350conformal z−map, 65conformal field theory, 534congruence of geodesics, 348connection homotopy, 334consensus, 224conservation law, 48constant relative degree, 702continuity, 368continuous deformation, 368continuous phase transition, 231continuous phase transitions, 176continuous–time regularization, 452

contravariant velocity equation, 699control law, 702control parameter, 180, 231control parameters, 229, 230control parameters are iteratively

adjusted, 709convective Benard fluid flow, 45conventional chaos theory, 344convolution, 463coordinate ball, 319coordinate chart, 319coordinate domain, 319coordinate map, 319correlation function, 243, 464correspondence principle, 431cortical motion control, 709cotangent bundle, 322cotangent space, 322countably additive measure, 451coupling, 706covariant derivative, 659covariant differentiation, 330covariant force equation, 699covariant force functor, 86, 651covariant force law, 233, 651covariant formalism on smooth

manifolds, 84cover, 306critical phenomena, 178critical point, 176, 381, 382, 404, 671critical points, 370critical slowing down, 231, 232, 472critical submanifolds, 675, 679critical value, 381cross–section, 321cube, 703cumulative distribution function, 427Curie–Weiss law, 260curvature, 309, 348curvature singularities, 395curved space, 490CW complexes, 375cycle, 22cycles, 377cyclic forms of human motion, 710

damped pendulum, 10, 56damping parameter, 3damping rate, 145

834 Index

De Rham theorem, 703Defuzzification, 705degeneracy density, 677degenerate critical point, 371degree of order, 230desired micro–states, 654determinism, 343deterministic, 658deterministic chaos, 1, 45deterministic chaotic system, 26diffeomorphicity, 672diffeomorphism, 37, 79, 320, 665, 709diffeomorphism invariant, 666difference equation, 50different, 666diffusion equation, 450, 463diffusion fluctuations, 429dilaton field, 535dimensionality, 428direct strategy, 225directed attachment, 225Dirichlet branes, 537discontinuous phase transition, 232discontinuous phase transitions, 176discrete–time models, 50discrete–time steps, 30discretized, 704disjoint union, 104, 369disorder ⇒ order, 232disorder–averaged partition function,

601dissipation, 428, 630dissipative structures, 70, 425distribution function, 426, 450Donaldson theory, 516double-scaling limit, 503drift, 429driven nonlinear pendulum, 3driven pendulum, 57Duffing, 32Duffing map, 54Duffing oscillator, 60Duffing–Van der Pol equation, 60dynamical, 661dynamical edge of chaos, 173dynamical intuition, 317dynamical invariant, 32dynamical phase transition, 206dynamical similarity, 109

dynamical system, 16dynamical systems, 26dynamics, 8dynamics of conservative systems, 657Dyson–Wyld diagrammatic analysis,

119

eddy, 113edges, 309, 368effective gaussian process, 367effective instability equation, 661Ehrenfest classification scheme, 175eigenvalue relation, 221eigenvalues, 195eigenvectors, 195Einstein equation, 475Einstein tensor, 314Einstein–Hilbert action, 475Eisenhart metric, 659elastic pendulum, 63electron–phonon interaction, 181elementary cobordism, 396elliptic geometry, 309energy functionals, 382energy surface, 98ensemble averages, 657entangled state, 629entropic function, 259entropy, 32, 70, 174, 229environment, 708equilibrium point, 4equilibrium statistical mechanics, 245,

257equipartition, 428ergodic hypothesis, 25, 32, 71, 78, 657,

661ergodic hypothesis of Boltzmann, 25ergodic system, 248ergodicity breaking, 245error function, 702escape rate, 25Euclidean, 495Euclidean 3D space, 308Euclidean action, 462Euclidean chart, 318Euclidean geometry, 305Euclidean image, 318Euclidean metric, 320Euclidean nD space, 309

Index 835

Euclidean path integral, 462Euclidean path–integral, 489Euclidean quantum gravity, 496, 503Euclidean spaces, 305Euclidean triangulations, 484Euler, 25, 309Euler character, 528Euler characteristic, 27, 309, 328, 341,

368, 371, 651, 666Euler class, 527Euler’s formula, 369Euler–Lagrange equations, 315Euler–Lagrangian equations, 334Euler–Poincare characteristic, 662, 703European option, 586excitable media, 166excited state, 155existence & uniqueness theorems for

ODEs, 16expanding Jacobian, 80expansion principle, 434exponential law, 227extended Hamilton oscillator, 237external configuration manifold, 84extract order from chaos, 229extrinsic view, 308

faces, 309, 368Faddeev–Popov procedure, 519Farey construction, 99feedback control, 704Feigenbaum cascade, 47Feigenbaum constant, 47Feigenbaum number, 51ferromagnet, 231Feynman diagram, 531, 578Feynman kernel, 436Feynman path integral, 194, 425, 430,

434, 637Feynman–Vernon formalism, 627fibre, 321field, 8field action functional, 194finite–time probability distribution, 591finitely additive, 451Finsler manifold, 654Finsler metric function, 655Finsler tensor, 655first–order phase transitions, 176

Fitzhugh–Nagumo model, 156fixed–point, 36, 41, 44flame front, 116Floquet multiplicator, 136Floquet stability analysis, 136flow, 16, 38, 79, 114, 327flow line, 324flow pattern, 11flow property, 327fluctuating force, 243fluctuation theorem, 77fluctuation–dissipation theorem, 244,

295fluctuation–dissipation theorems, 296foamlike structure, 394Fock space, 537Fock state, 432Fokker–Planck equation, 188, 200, 429,

430, 582folding, 4, 35force equation, 233force HBE servo–controller, 700force–field psychodynamics, 637forced nonlinear oscillators, 11forced Van der Pol oscillator, 58formal exponential, 327formalism of jet bundles, 11forward–Euler integration scheme, 167Fourier decomposition, 469Fourier transform, 108fractal attractor, 4, 46fractal dimension, 44fractal microstructure, 511fractal pattern, 51fractal set, 37fractals, 65fractional dimension, 65fragmentation, 224Fredholm integral equation, 464Fredholm operator, 525Fredholm theory, 464free Abelian group, 376free energy, 193, 257free energy potential, 174, 231free particle, 442free string, 529frequency, 3friction, 428Frobenius–Perron theorem, 675

836 Index

from a classical point of view, 506full coupling, 234functional integral, 455functional manifold, 116functor, 380Fuzzification, 700, 704fuzzy inference system, 704

gamma fusimotor neurons, 701gauge condition, 458Gauss, 308Gauss map, 341Gauss’ Theorema Egregium, 349Gauss–Bonnet formula, 328, 341Gauss–Bonnet theorem, 369, 662Gauss–Bonnet–Hopf theorem, 667Gauss–Kronecker curvature, 667Gaussian approximation, 597Gaussian curvature, 328, 480, 663Gaussian integrals, 441Gaussian measure, 597Gaussian multiplicative noise, 199Gaussian saddlepoint approximation,

578Gaussian–distributed Markov process,

661general sense, 127generalized Henon map, 53generalized solution, 127generalized SRB measure, 81generating functional, 194, 466genus, 369geodesic, 310, 325, 334geodesic equation, 313, 334geodesic equations, 659geodesic spray, 325geodetic separation field, 348geometrical intuition, 318geometrodynamical functor, 639ghost number, 537Gibbs entropy, 71Ginzburg–Landau equation, 114, 180,

579Ginzburg–Landau model, 180global chaos control, 158globular cluster, 52goal–driven interaction, 654golden numbe, 98Goldstone theorem, 624

Golgi tendon organs, 701gradient-like, 684gradient-like vector–field, 419graphs, 369Green’s function, 241, 463grows exponentially, 5grows linearly, 5growth of the entropy, 493growth rate, 49

H–cobordism theorem, 34Henon map, 30, 52, 142Henon strange attractor, 52Hadamard, 32Haken’s synergetics, 111halo orbit, 31Hamilton, 25Hamilton oscillator, 236Hamilton’s principle, 316Hamiltonian, 317Hamiltonian action, 658Hamiltonian chaos, 95, 144, 658Hamiltonian dynamics, 323Hamiltonian system, 53Hamming distance, 101, 102, 106handle, 369, 405handle decomposition, 403handle-body decomposition, 415harmonic oscillator, 443Hausdorff dimension, 504Hausdorff space, 319Hayashi, 32heat bath, 193, 428Hebbian synapses, 601Heisenberg commutation relations, 452Heisenberg picture, 453Hermitian inner product, 433Hessian, 333, 404high–dimensional chaos theory, 44Hilbert manifold, 320Hilbert space, 26, 320hippocampus, 187history, 431, 681Hodgkin–Huxley, 160holographic hypothesis, 624homeomorphic, 368homoclinic point, 29, 38, 42homoclinic tangle, 27, 41homology group, 376, 377

Index 837

homology theory, 368homomorphism, 377homotopy, 379homotopy equivalent, 371, 676Hopf, 32Hopf bifurcation, 145Hopf–like bifurcation, 62Hopf–Rinow Theorem, 310Hopfield discrete–time network, 239Hopfield overlap, 280Hopfield overlaps, 262human–like locomotor system, 233hurricane, 112hybrid dynamical system of variable

structure, 126hybrid dynamics, 428hybrid systems, 127hyperbolic fixed–point, 43hyperbolic geometry, 309hyperbolic system, 17hysteresis effect, 231, 232

ideal thermostat, 75imaginary time, 465Implantable Cardioverter–Defibrillator,

167impossible to integrate, 706index, 370, 381, 396, 404, 677, 679index of the critical point, 671infinite–dimensional neural network,

637infinity, 709information, 129information theory, 33inner product, 309instability, 11, 343instability sequence, 231instanton vacua, 578integral curve, 324integrate–and–fire neuron, 188integration, 16interaction, 225internal configuration manifold, 84intrinsic curvature, 490intrinsic definition for differentiable

manifolds, 309intrinsic dynamical instability, 658intrinsic view, 308intuition, 26

invariant distribution, 427invariant set, 37, 39invariant torus, 97inverted driven pendulum, 63irregular and unpredictable, 1irreversibility, 70Ising–spin Hamiltonian, 257isolated closed trajectory, 5isometric, 310isotropic, 349Ito lemma, 585iterated map, 30iteration of conditioned reflexes, 709iterative maps, 17Ito stochastic differential equation, 429Ito stochastic integral, 429Ito stochastic integral equation, 429

Jacobi equation, 659Jacobi equation of geodesic deviation,

337Jacobi fields, 337, 383Jacobi metric, 659Jacobi variation field, 659Jacobi–Levi-Civita equation, 347jerk, 702jerk function, 144jet space, 332joint action, 654

Kahler gauge, 572Kahler potential, 574Kaluza–Klein monopole, 690Kaluza–Klein theories, 395KAM–torus, 99Kaplan–Yorke dimension, 131, 157Kaplan–Yorke map, 55Karder–Parisi–Zhang equation, 119Karma model, 159, 171Kepler, 27ket–vectors, 433kick equation, 241Killing vector–field, 423kinetic energy, 194kinetic–energy estimator, 473Klein–Gordon Lagrangian, 458Kolmogorov, 32, 33Kolmogorov–Arnold–Moser (KAM)

Theorem, 97

838 Index

Kolmogorov–Sinai, 131Kolmogorov–Sinai entropy, 20, 129, 131Kosterlitz–Thouless transition, 662Kramers–Moyal expansion, 277Krylov, 32Kunneth formula, 417Kuramoto–Sivashinsky equation, 579

lack of memory, 427Lagrange, 25Lagrange’s equations, 315Lagrange’s points, 31Lagrangian chaos, 87Lagrangian density, 458, 642Lagrangian mechanics, 315laminar flow, 111Landau, 229Landau free energy, 181Landau’s theory of phase transitions,

179Landau–Ginzburg equation, 158Langevin rate equation, 243Langevin rate equations, 428Laplace–Beltrami operator, 333large system, 74largest Lyapunov exponent, 129, 343lattice regularization, 449laws of probability, 433learning, 225learning rate,, 654Legendre transformation, 317Levi–Civita connection, 86, 312, 330Lewinian force–field theory, 638Lickorish–Wallace theorem, 689, 692Lie, 309Lie bracket, 312, 330, 334, 335, 702Lie derivative, 125, 330, 348, 667, 702Lie derivative formalism, 702Lie group, 307Lie structure equations, 117Lie super–algebra, 517limit cycle, 12, 44, 57limit–cycle attractor, 5linear homotopy ODE, 126linear homotopy segment, 127linear operator equation, 463linearization, 3Liouville equation, 429Liouville measure, 230

Liouville theorem, 98, 230Liouville tori, 98Lipschitz condition, 326Lissajous curves, 64Littlewood, 33local Bernstein’s adaptation process,

653local chaos control, 158locally stable, 80locally unstable, 80locally–optimal solution, 633logistic equation, 49, 200logistic growth, 49logistic map, 50, 52, 104, 133long–range correlation, 624Lorentz–invariant theories, 530Lorentzian cobordism, 395, 412, 681Lorentzian dynamical triangulations,

477Lorentzian space-time, 492, 495Lorentzian structure, 659Lorenz attractor, 5, 125Lorenz equations, 64, 143Lorenz flow, 110Lorenz mask, 46, 110Lorenz system, 18, 46, 52lower limit of complexity, 694Luo–Rudy I model, 160Lyapunov, 33Lyapunov dimension, 130Lyapunov exponent, 2, 19, 68, 660, 661,

671Lyapunov exponents, 44, 81Lyapunov function, 279Lyapunov spectrum, 157Lyapunov stability, 17Lyapunov time, 2

Mobius strip, 378Mobius band, 397macroscopic change, 673macroscopic entanglement, 694macroscopic neuro–mechanical

entanglement, 709magnitude, 677major topology change, 664Malthus model, 49Malthusian parameter, 49, 51Mamdani inference, 705

Index 839

Mandelbrot and Julia sets, 65manifold, 305, 317manifold structure, 319manifoldness, 309map, 50map sink, 44Markov assumption, 430Markov chain, 21, 247, 427, 470Markov partitions, 34, 79Markov process, 21Markov stochastic process, 427, 430,

641Markovian networks, 227Master equation, 430material metric tensor, 699Mathai–Quillen formula, 525mathematical induction, 309Matrix Product Formalism, 219matrix–symplectic explicit integrator,

705Maupertius action principle, 333Maupertuis’ action, 658Maupertuis’ principle, 346Maurer–Cartan connection, 527maximal geodesic, 325maximal integral curve, 324Maxwell, 25Maxwell–Haken laser equations, 64, 111mean first–passage time, 207mean kinetic energy, 244mean square limit, 429mean–field, 236mean–field approximation, 199mean–field theory, 101, 176measure theory, 666Mel’nikov integrals, 344membrane potential, 156memory, 225memory term, 146mental force law, 651Mermin–Wagner theorem, 668metal–insulator transition, 181metastable state, 77metric space, 310metric tensor, 313Metropolis algorithm, 470Mexican–hat coupling, 190micro–canonical ensemble, 71, 78Minkowski action, 462

Mittag–Leffler, 28mixing, 81mixing Anosov diffeomorphism, 79model space, 320momentum phase–space, 703Monte Carlo method, 52, 468Monte Carlo simulation, 206Monte–Carlo simulations, 492Morse, 32Morse cobordisms, 682Morse function, 381, 382, 404, 413, 672,

682Morse index, 413, 683Morse inequalities, 404Morse Lemma, 671Morse lemma, 381, 663Morse metric, 413Morse metrics, 407, 683Morse number, 381, 684Morse singularities, 683Morse structure, 414Morse theory, 32, 35, 380, 381, 404, 406,

412, 413, 511, 672Morse trousers, 407motor conditioned reflexes, 705multi–kick equation, 241multi–spiral strange attractor, 62multicanonical sampling, 473multigrid method, 472multiplicative zero–mean Gaussian

white noise, 200

Nambu–Goto action, 534natural projection, 322Navier–Stokes equations, 45, 91, 108,

115nearest neighbor coupling, 234Necker cube, 238Neumann boundary condition, 538neural information processing, 187neural path integral, 619neutral strategy, 225Newton, 25Newton’s Second Law, 315Newtonian mechanics, 16, 315no–flux Neumann boundary conditions,

167noble numbers, 98Noether Theorem, 317

840 Index

noise, 654noise–free limit, 253noise–induced phase transitions, 210non–autonomous 2D continuous

systems, 11non–autonomuous system, 9non–conserving dynamics, 218non–equilibrium phase transition, 218non–equilibrium statistical mechanics,

245non–equilibrium steady state, 77Non–Euclidean geometry, 309non–wandering set, 37, 79non-degenerate, 381nonholonomic coordinates, 332nonlinear control theory, 531nonlinear non–equilibrium systems, 199nonlinear oscillators, 25nonlinear Schrodinger equation, 180nonlinear sigma model, 534Nonlinear velocity, 702normal vector–field, 323number of mechanical degrees of

freedom, 694, 697number of mechanical DOF, 694

observational resolution, 694, 697, 709ODEs, 17one–parameter group of diffeomor-

phisms, 327open string theories, 529opinion dynamics, 225opinion–dynamics, 227optimal policy, 654orbit, 16, 22, 38order, 677, 679order parameter, 177, 203, 205, 229, 231order parameter equation, 205, 232, 264order parameters, 181, 245ordered symmetry–breaking state, 199ordering chaos, 7oriented cobordism, 402oriented strings, 530Ornstein–Uhlenbeck noise, 203Ornstein–Uhlenbeck process, 289oscillatory cortical–control, 710oscillatory dynamics, 710oscillatory synchronization, 694Oseledec theorem, 81

Ott–Grebogi–Yorke map, 55output–space dimension, 709overdrive pacing, 165

pacing response diagrams, 165Panfilov model, 159, 170parabolic Einstein equation, 341parallel transport, 312parametric instability, 350, 661, 662partition function, 34, 192, 220, 257,

460path integral, 425, 580path–integral approach, 405path–integral expression, 456path–integral formalism, 454, 599path–integral formulation, 317, 453path–integral quantization, 452pendulum angle, 3period doubling bifurcations, 51period–doubling bifurcation, 47, 133periodic orbit, 29, 41periodic orbit theory, 24, 116periodic solutions, 11perturbative path integral, 474perturbative string theory, 535Pesin formula, 82phase, 4, 432phase point, 11phase portrait, 4, 11phase space, 308phase transition, 6, 173, 174, 231phase transition of first order, 232phase transition of second order, 231phase transitions, 472phase–flow, 10, 16phase–locking, 710phase–space, 4phase–space flow, 30phase–space path integral, 441, 449phase–transition effects, 231phase–transition theory, 229physical Hamiltonian function, 699physically desirable histories, 682Pickover’s biomorphs, 67pinball game, 1pinned Wiener measure, 451Planck length, 529playground swing, 3Poincare, 26, 309

Index 841

Poincare conjecture, 34, 309, 378Poincare duality, 369Poincare section, 5, 27, 30, 60, 135Poincare map, 53Poincare section, 24, 52Poincare–Bendixson theorem, 6, 11, 27Poincare–Birkhoff Theorem, 96Poincare–Birkhoff theorem, 344Poincare–Fermi theorem, 353Poincare–Hopf index theorem, 27Poincare section, 147Poisson detection statistics, 432Poisson process, 243polarization, 224Polyakov action, 534Pontryagin, 33Pontryagin Maximum Principle, 580population models, 48positional stiffness, 701positive leading Lyapunov exponent,

130potential energy, 194, 198Prandtl number, 46predictability, 343, 658predictability time, 130predictability/controllability, 694Principle of stationary action, 317probabilistic description, 426probability amplitude, 453probability density, 200probability density function, 430probability of acknowledged influence,

225product topology, 114product–topology theorem, 703propagator, 436, 438, 449, 460, 488protozoan morphology, 66pruning, 22pull–back, 702pulse–coupled oscillators, 188Pyragas control, 136

quantum behavior, 509quantum brain, 637quantum coherent state, 431quantum entanglement, 708quantum field theory, 194quantum gravity, 474quantum modifications, 510

quantum probability, 430quantum space-time, 487quantum statistical mechanics, 192quantum theory, 507quantum universe, 503quantum–mechanics propagator, 454quasi–isotropic manifold, 351quasi-isotropy, 662quotient space, 377

Rossler, 143random thermostat, 75random variable, 426random walk, 427rate of error growth, 129Rayleigh–Benard convection, 18, 127reaction–diffusion systems, 218recovery period, 155rectified, 704recurrent neural networks, 244recursive homotopy dynamics, 649reduced curvature 1–form, 118reentrant excitations, 156Regge calculus, 479, 490Regge geometries, 476Regge geometry, 490Regge simplicial action, 481regular level, 686reinforcement learning, 654reinforcement training, 709relative degree, 125relaxation oscillator, 57reliable predictor, 49repeller, 4, 24representative point, 318resting state, 155return map, 24reverse strategy, 225Reynolds number, 109, 111ribbon, 397Ricci curvature, 660Ricci flow, 341Ricci scalar curvature, 314Ricci tensor, 314, 329, 336, 660Riemann, 309Riemann curvature tensor, 313, 328,

334, 650, 660Riemann sphere, 311Riemann surface, 310

842 Index

Riemannian geometry, 311Riemannian manifold, 309Riemannian manifolds, 308Riemannian metric tensor, 328Rossler, 61Rossler system, 61route to chaos, 6, 34, 145route to turbulence, 113Rudolphine Tables, 27Ruelle, 34

saddle point, 4saddle–point integration, 259Sard’s Theorem, 383scalar curvature, 329, 336scalar Gaussian curvature, 337scale factor, 509scatterers, 663Schrodinger equation, 431, 448Schrodinger picture, 453scroll waves, 156Second Law of thermodynamics, 70second variation formula, 338second–countable space, 319second–order phase transition, 199second–order phase transitions, 176sectional curvature, 329self–assembly, 694, 709self–consistency relation, 204self–limiting process, 49self–organized, 653, 656semantic integration, 656sensitive dependence on initial

conditions, 102sensitivity to initial conditions, 5sensitivity to parameters, 5sequence of period doublings, 145servo–controllers, 702servoregulatory loops, 701set, 43Shannon, 33shape operator, 667Sherrington–Kirkpatrick model, 617short–term predictability, 5short–time evolution, 205signal, 654signature, 492simplicial complexes, 375simplicial homology, 376

simplicity, 694Sinai, 33sine–Gordon equation, 113singular homology, 376singularities, 395Smale, 32Smale horseshoe, 99Smale horseshoe map, 35Smale–Zehnder Theorem, 99small system, 74smooth manifold triad, 412smooth manifolds, 308solution, 127source, 466sources and sinks, 425space entropy, 80space–time, 308spaces, 492spatio-temporal chaos, 156specific heat capacity, 174spectral decomposition, 79spectral dimension, 510sphere, 307spherical modification, 402spin glass, 77spin networks, 475spindle receptors, 701spiral turbulence, 157spiral waves, 156spontaneous rotational symmetry

breaking, 120squeezing, 4, 35stability, 11, 343, 658stable and unstable manifold, 38stable eigen–direction, 99stable manifold, 53, 99staging algorithm, 472standard Hamiltonian systems, 671standard Hamiltonians, 658standard map, 53state, 8state vector, 453stationary probability density, 200statistical error, 470Steifel–Whitney and Pontrjagin

numbers, 412step size, 654Stiefel–Whitney numbers, 681stochastic force, 428

Index 843

Stochastic forces, 699stochastic influence, 428stochastic integral, 428stochastic oscillator equation, 660stochastic oscillator problem, 354stochastic system, 10strange attractor, 4, 7, 27, 34, 44–46,

60, 110Stratonovitch interpretation, 200stream function, 88stretch–and–fold, 60stretching, 4, 35string corrections, 529string tension, 534string–field, 571string–field–theory action, 537stroboscopic section, 41structural instability, 230structural stability, 17structurally stable, 33Sturm–Liouville problem, 463subsumption architectures, 653sum–over–histories, 431sum–over–topologies, 493super–field, 563super–space, 564supercell thunderstorms, 112superposition, 488, 694superstring theory, 535supersymmetry, 535supervised, 709supervised gradient descent learning,

654, 656support of a vector–field, 326surface of Earth, 305surgery, 402, 684survival probability, 25symbolic dynamics, 22, 34, 38, 40, 79,

101symmetric affine connection, 330symmetric dilution, 614symmetry, 463symmetry breaking instability, 231symmetry–breaking, 177symmetry–breaking oscillation, 206symmetry–breaking transition, 668symmetry–restoring oscillation, 206symplectic manifolds, 308synchronization, 710

synergetics, 229, 238system parameters, 233systematic error, 469

tachyon field, 536Takens, 34tangent bundle, 310, 322tangent dynamics equation, 660tangent dynamics equation , 671tangent map, 322tangent space, 309, 321tangent vector–field, 323tensor–field, 16theoretical ecology, 50theory of quantum gravity, 486theory of turbulence, 34thermal equilibrium, 229thermodynamics, 70three–body problem, 25, 27threshold, 155time averages, 657time entropy, 80time–dependent Schrodinger equation,

430time–dependent vector–field, 324, 327time–flow, 395time–ordered product, 463time–phase plot, 11time–reversal invariance, 72time–reversal symmetry, 177topological cobordism, 395, 412, 681topological entropy, 2topological hypothesis, 665, 669, 672topological invariant, 309, 368, 669topological manifold, 308topological property, 309topological quantum field theory, 511topological theorem, 671topological transition, 676topologically transitive, 102topology, 26, 368, 493topology change, 394, 412, 496, 671topology changing, 681topology changing processes, 416topology–changing transition, 681tornado, 112torsion, 312torus, 307, 703total system complexity, 694

844 Index

trace, 686trajectory, 11, 16, 22transient chaos, 113transition amplitude, 405, 425, 431, 433,

447transition energy, 671transition functions, 318transition map, 306transition probability, 431, 433transition probability amplitude, 438transition probability distribution, 590transition temperature, 671transitive Anosov flow, 79trapping region, 102Trotter decomposition formula, 471trousers, 394, 397, 398, 400, 407turbulence, 17, 108, 159turbulent flow, 111twist map, 96two–point function, 464

umbrella sampling, 473uncorrelated configuration model, 227undamped pendulum, 55universality class, 179unpredictability, 27, 658unstable eigen–direction, 99unstable manifold, 53, 99unstable periodic orbits, 25unsupervised, 653, 656

vacuum state, 433, 629Van der Pol, 32Van der Pol oscillator, 36, 125vector–field, 16velocity equation, 233velocity phase–space manifold, 321velocity vector–field, 321ventricular fibrillation, 166

Verhulst model, 200vertices, 309, 368virial estimator, 473virtual particles, 428virtual paths, 425, 488visual cortex, 187volatility, 585Volume(T), 493von Neumann, 26vortex, 111vortices, 156vorticity dynamics, 112

W–cycle, 473wandering point, 79water vapor, 230wave–particle duality, 432Weierstrass, 28weights, 653Weyl, 309Whitney, 309Wick rotation, 483, 540Wiener, 33Wiener measure, 450Wiener process, 429Wigner function, 626winding number, 95Witten’s TQFT, 511world–sheet, 529wormhole creation, 400

Yang–Baxter equation, 515Yang–Mills gauge theory, 512Yang–Mills relation, 578yarmulke, 397, 398, 407, 683yarmulke cobordism, 682

Zamolodchikov metric, 568Zorn’s lemma, 311