UID:
almafu_9960117279902883
Umfang:
1 online resource (xii, 285 pages) :
,
digital, PDF file(s).
Ausgabe:
First edition.
ISBN:
1-316-46695-7
,
1-316-46870-4
,
1-139-34347-5
Inhalt:
Lyapunov exponents lie at the heart of chaos theory, and are widely used in studies of complex dynamics. Utilising a pragmatic, physical approach, this self-contained book provides a comprehensive description of the concept. Beginning with the basic properties and numerical methods, it then guides readers through to the most recent advances in applications to complex systems. Practical algorithms are thoroughly reviewed and their performance is discussed, while a broad set of examples illustrate the wide range of potential applications. The description of various numerical and analytical techniques for the computation of Lyapunov exponents offers an extensive array of tools for the characterization of phenomena such as synchronization, weak and global chaos in low and high-dimensional set-ups, and localization. This text equips readers with all the investigative expertise needed to fully explore the dynamical properties of complex systems, making it ideal for both graduate students and experienced researchers.
Anmerkung:
Title from publisher's bibliographic system (viewed on 05 Feb 2016).
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Cover -- Half-title -- Title page -- Copyright information -- Table of contents -- Preface -- 1 Introduction -- 1.1 Historical considerations -- 1.1.1 Early results -- 1.1.2 Biography of Aleksandr Lyapunov -- 1.1.3 Lyapunov's contribution -- 1.1.4 The recent past -- 1.2 Outline of the book -- 1.3 Notations -- 2 The basics -- 2.1 The mathematical setup -- 2.2 One-dimensional maps -- 2.3 Oseledets theorem -- 2.3.1 Remarks -- 2.3.2 Oseledets splitting -- 2.3.3 "Typical perturbations" and time inversion -- 2.4 Simple examples -- 2.4.1 Stability of fixed points and periodic orbits -- 2.4.2 Stability of independent and driven systems -- 2.5 General properties -- 2.5.1 Deterministic vs. stochastic systems -- 2.5.2 Relationship with instabilities and chaos -- 2.5.3 Invariance -- 2.5.4 Volume contraction -- 2.5.5 Time parametrisation -- 2.5.6 Symmetries and zero Lyapunov exponents -- 2.5.7 Symplectic systems -- 3 Numerical methods -- 3.1 The largest Lyapunov exponent -- 3.2 Full spectrum: QR decomposition -- 3.2.1 Gram-Schmidt orthogonalisation -- 3.2.2 Householder reflections -- 3.3 Continuous methods -- 3.4 Ensemble averages -- 3.5 Numerical errors -- 3.5.1 Orthogonalisation -- 3.5.2 Statistical error -- 3.5.3 Near degeneracies -- 3.6 Systems with discontinuities -- 3.6.1 Pulse-coupled oscillators -- 3.6.2 Colliding pendula -- 3.7 Lyapunov exponents from time series -- 4 Lyapunov vectors -- 4.1 Forward and backward Oseledets vectors -- 4.2 Covariant Lyapunov vectors and the dynamical algorithm -- 4.3 Dynamical algorithm: numerical implementation -- 4.4 Static algorithms -- 4.4.1 Wolfe-Samelson algorithm -- 4.4.2 Kuptsov-Parlitz algorithm -- 4.5 Vector orientation -- 4.6 Numerical examples -- 4.7 Further vectors -- 4.7.1 Bred vectors -- 4.7.2 Dual Lyapunov vectors -- 5 Fluctuations, finite-time and generalised exponents -- 5.1 Finite-time analysis.
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5.2 Generalised exponents -- 5.3 Gaussian approximation -- 5.4 Numerical methods -- 5.4.1 Quick tools -- 5.4.2 Weighted dynamics -- 5.5 Eigenvalues of evolution operators -- 5.6 Lyapunov exponents in terms of periodic orbits -- 5.7 Examples -- 5.7.1 Deviation from hyperbolicity -- 5.7.2 Weak chaos -- 5.7.3 Hénon map -- 5.7.4 Mixed dynamics -- 6 Dimensions and dynamical entropies -- 6.1 Lyapunov exponents and fractal dimensions -- 6.2 Lyapunov exponents and escape rate -- 6.3 Dynamical entropies -- 6.4 Generalised dimensions and entropies -- 6.4.1 Generalised Kaplan-Yorke formula -- 6.4.2 Generalised Pesin formula -- 7 Finite-amplitude exponents -- 7.1 Finite vs. infinitesimal perturbations -- 7.2 Computational issues -- 7.2.1 One-dimensional maps -- 7.3 Applications -- 8 Random systems -- 8.1 Products of random matrices -- 8.1.1 Weak disorder -- 8.1.2 Highly symmetric matrices -- 8.1.3 Sparse matrices -- 8.1.4 Polytomic noise -- 8.2 Linear stochastic systems and stochastic stability -- 8.2.1 First-order stochastic model -- 8.2.2 Noise-driven oscillator -- 8.2.3 Khasminskii theory -- 8.2.4 High-dimensional systems -- 8.3 Noisy nonlinear systems -- 8.3.1 LEs as eigenvalues and supersymmetry -- 8.3.2 Weak-noise limit -- 8.3.3 Synchronisation by common noise and random attractors -- 9 Coupled systems -- 9.1 Coupling sensitivity -- 9.1.1 Statistical theory and qualitative arguments -- 9.1.2 Avoided crossing of LEs and spacing statistics -- 9.1.3 A statistical-mechanics example -- 9.1.4 The zero exponent -- 9.2 Synchronisation -- 9.2.1 Complete synchronisation and transverse Lyapunov exponents -- 9.2.2 Clusters, the evaporation and the conditional Lyapunov exponent -- 9.2.3 Synchronisation on networks and master stability function -- 10 High-dimensional systems: general -- 10.1 Lyapunov density spectrum -- 10.1.1 Infinite systems.
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10.2 Chronotopic approach and entropy potential -- 10.3 Convective exponents and propagation phenomena -- 10.3.1 Mean-field approach -- 10.3.2 Relationship between convective exponents and chronotopic analysis -- 10.3.3 Damage spreading -- 10.4 Examples of high-dimensional systems -- 10.4.1 Hamiltonian systems -- 10.4.2 Differential-delay models -- 10.4.3 Long-range coupling -- 11 High-dimensional systems: Lyapunov vectors and finite-size effects -- 11.1 Lyapunov dynamics as a roughening process -- 11.1.1 Relationship with the KPZ equation -- 11.1.2 The bulk of the spectrum -- 11.2 Localisation of the Lyapunov vectors and coupling sensitivity -- 11.3 Macroscopic dynamics -- 11.3.1 From micro to macro -- 11.3.2 Hydrodynamic Lyapunov modes -- 11.4 Fluctuations of the Lyapunov exponents in space-time chaos -- 11.5 Open system approach -- 11.5.1 Lyapunov spectra of open systems -- 11.5.2 Scaling behaviour of the invariant measure -- 12 Applications -- 12.1 Anderson localisation -- 12.2 Billiards -- 12.3 Lyapunov exponents and transport coefficients -- 12.3.1 Escape rate -- 12.3.2 Molecular dynamics -- 12.4 Lagrangian coherent structures -- 12.5 Celestial mechanics -- 12.6 Quantum chaos -- Appendix A Reference models -- A.1 Lumped systems: discrete time -- A.2 Lumped systems: continuous time -- A.3 Lattice systems: discrete time -- A.4 Lattice systems: continuous time -- A.5 Spatially continuous systems -- A.6 Differential-delay systems -- A.7 Global coupling: discrete time -- A.8 Global coupling: continuous time -- Appendix B Pseudocodes -- Appendix C Random matrices: some general formulas -- C.1 Gaussian matrices: discrete time -- C.2 Gaussian matrices: continuous time -- Appendix D Symbolic encoding -- Bibliography -- Index.
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English
Weitere Ausg.:
ISBN 1-107-03042-0
Sprache:
Englisch
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