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Regular and Chaotic Dynamics (Applied Mathematical Sciences) Softcover reprint of edition by Lichtenberg, Allan, Lieberman, Michael (2010) Paperback
This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.
- GenresPhysics
Paperback
First published June 24, 1992
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February 22, 2024
The authors’ careers coincided with the flourishing of modern dynamical systems theory and the present work summarizes their lifetime’s work. A.J. Lichtenberg and M.A. Lieberman’s classic Regular and Chaotic Dynamics (Springer-Verlag Applied Mathematical Sciences, 1991, now in its second edition) is not a textbook, however, but more like an extended review article, useful as a reference for researchers in the field (running to nearly 700 pages). Almost everything one will have heard of in physics-department colloquia or in popular expositions is hit upon one place or another in this sprawling monster of a text. The main omission would be an explicit account of celestial mechanics, for which see C.L. Siegel and J.K. Moser’s Lectures in Celestial Mechanics or, for a more recent treatment, Kenneth R. Meyer, Glenn R. Hall and Dan Offin’s Introduction to Hamiltonian Dynamical Systems and the N-Body Problem (reviewed by us here resp. here).
Rapid overview of contents: chapter one considers the theory of canonical transformations in integrable and near integrable systems (illustrated with the Hénon-Heiles problem). Chapter two: the introductory section stresses that the averaging procedure of classical perturbation theory does not account for the change in topology of phase space. NB, the averaging procedure always yields an integrable system that is supposed to approximate the true non-integrable one. If far from the separatrix, the discrepancy will be small but with resonance overlaps around the separatrix the method breaks down. A good example of how Lichtenberg-Lieberman show everything but don’t explain the theory that well may be found in section 2.3d on Kruskal’s method. Still, the chapter is rich in material, with a detailed discussion of secular perturbation theory, higher-order resonances, resonant wave-particle interaction, global removal of resonances, the Lie transform, adiabatic invariants and superconvergent methods.
Chapter three turns to the Hamiltonian case, phase-space structure and KAM theory. Then chapter four takes a closer look at the transition to global stochasticity (illustrated in detail with the standard map), resonance overlap criteria, growth of second-order islands near elliptic fixed points resp. near the separatrix, and lastly the stability of high-order fixed points.
Chapters five and six offer no rigorous theory (though Arnold has done such) but clever stochastic models of diffusion in action space capable of accounting for the effects of the separatrix layer, thick layers, trapping at island chains (cantorus), resonance streaming, extrinsic noise etc. in a Fokker-Planck-type formalism: heuristic, supported by numerical experiments.
Chapter seven on bifurcation and transition to chaos in dissipative systems nicely exhibits the phenomena (strange attractors, period doubling, onset of chaos, critical behavior and universality, intermittency, boundary crises, onset of homoclinic intersection) with examples, but 1) there is precious little physics in it and 2) does not teach the material.
Similarly for chapter eight on chaotic motion in dissipative systems (transient chaos, invariant distributions on strange attractors, multifractals). Section 8.4 on reconstructing the dynamics from time series may not be intrinsically interesting, except for section 8.5b on self-organized criticality (little theory, mostly inferences from numerical experiments) and 8.6b transition to turbulence. The Landau picture is not generally consistent with observations, hence Ruelle-Takens’ view and some other models are advanced (the authors observe that there cannot be a single mechanism that explains all cases).
The appendix contains a very compressed discussion of several physical problems (celestial mechanics, accelerators and beams, plasma confinement, chemical dynamics, quantum systems): good for all the references to the literature they have compiled, but little else.
Three stars. As soon as one immerses oneself in the early chapters of this text, one encounters what is typically a generous plenty of material, it is true, but described maybe without sufficient development of the mathematical techniques themselves (more or less assumed known to the reader). Thus, Lichtenberg-Lieberman’s treatment will be found good indeed 1) for its many informative diagrams and figures, 2) for the numerous explicit formulae occurring in popular model systems that crop up constantly in the literature, and 3) for refining one’s understanding of certain points not necessarily covered in elementary textbooks.
Therefore, skim through Lichtenberg-Lieberman for a richly documented overview of the entire field of chaos theory and non-linear dynamical systems, without seeking to pick up any rigorous mathematics, though. Here, one seems to have the opposite problem to Siegel-Moser’s Lectures in Celestial Mechanics: the latter work is crammed with far too much detail for one to take in readily but, all the same, offers little in the way of high-level explanation, whereas here with Lichtenberg and Lieberman one gets numerous explanations at too high a level without enough mathematical justification. Perhaps, then, one should afterwards go on to a text like Ferdinand Verhulst’s Nonlinear Differential Equations and Dynamical Systems really to master the theory of non-linear dynamics of ordinary differential equations, as this latter work does include sufficient illustrative homework exercises (just reviewed by us here).
Rapid overview of contents: chapter one considers the theory of canonical transformations in integrable and near integrable systems (illustrated with the Hénon-Heiles problem). Chapter two: the introductory section stresses that the averaging procedure of classical perturbation theory does not account for the change in topology of phase space. NB, the averaging procedure always yields an integrable system that is supposed to approximate the true non-integrable one. If far from the separatrix, the discrepancy will be small but with resonance overlaps around the separatrix the method breaks down. A good example of how Lichtenberg-Lieberman show everything but don’t explain the theory that well may be found in section 2.3d on Kruskal’s method. Still, the chapter is rich in material, with a detailed discussion of secular perturbation theory, higher-order resonances, resonant wave-particle interaction, global removal of resonances, the Lie transform, adiabatic invariants and superconvergent methods.
Chapter three turns to the Hamiltonian case, phase-space structure and KAM theory. Then chapter four takes a closer look at the transition to global stochasticity (illustrated in detail with the standard map), resonance overlap criteria, growth of second-order islands near elliptic fixed points resp. near the separatrix, and lastly the stability of high-order fixed points.
Chapters five and six offer no rigorous theory (though Arnold has done such) but clever stochastic models of diffusion in action space capable of accounting for the effects of the separatrix layer, thick layers, trapping at island chains (cantorus), resonance streaming, extrinsic noise etc. in a Fokker-Planck-type formalism: heuristic, supported by numerical experiments.
Chapter seven on bifurcation and transition to chaos in dissipative systems nicely exhibits the phenomena (strange attractors, period doubling, onset of chaos, critical behavior and universality, intermittency, boundary crises, onset of homoclinic intersection) with examples, but 1) there is precious little physics in it and 2) does not teach the material.
Similarly for chapter eight on chaotic motion in dissipative systems (transient chaos, invariant distributions on strange attractors, multifractals). Section 8.4 on reconstructing the dynamics from time series may not be intrinsically interesting, except for section 8.5b on self-organized criticality (little theory, mostly inferences from numerical experiments) and 8.6b transition to turbulence. The Landau picture is not generally consistent with observations, hence Ruelle-Takens’ view and some other models are advanced (the authors observe that there cannot be a single mechanism that explains all cases).
The appendix contains a very compressed discussion of several physical problems (celestial mechanics, accelerators and beams, plasma confinement, chemical dynamics, quantum systems): good for all the references to the literature they have compiled, but little else.
Three stars. As soon as one immerses oneself in the early chapters of this text, one encounters what is typically a generous plenty of material, it is true, but described maybe without sufficient development of the mathematical techniques themselves (more or less assumed known to the reader). Thus, Lichtenberg-Lieberman’s treatment will be found good indeed 1) for its many informative diagrams and figures, 2) for the numerous explicit formulae occurring in popular model systems that crop up constantly in the literature, and 3) for refining one’s understanding of certain points not necessarily covered in elementary textbooks.
Therefore, skim through Lichtenberg-Lieberman for a richly documented overview of the entire field of chaos theory and non-linear dynamical systems, without seeking to pick up any rigorous mathematics, though. Here, one seems to have the opposite problem to Siegel-Moser’s Lectures in Celestial Mechanics: the latter work is crammed with far too much detail for one to take in readily but, all the same, offers little in the way of high-level explanation, whereas here with Lichtenberg and Lieberman one gets numerous explanations at too high a level without enough mathematical justification. Perhaps, then, one should afterwards go on to a text like Ferdinand Verhulst’s Nonlinear Differential Equations and Dynamical Systems really to master the theory of non-linear dynamics of ordinary differential equations, as this latter work does include sufficient illustrative homework exercises (just reviewed by us here).
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