##
**Regular and chaotic dynamics.
2. ed.**
*(English)*
Zbl 0748.70001

Applied Mathematical Sciences. 38. New York: Springer-Verlag. xxii, 692 p. (1992).

[See the review of the first edition (1983; Zbl 0506.70016.)]

The original title of our book, “Regular and stochastic motion”, was chosen to emphasize Hamiltonian dynamics and the physical motion of bodies. The new edition is more evenhanded, with considerably more discussion of dissipative systems and dynamics not involving physical motion. To reflect this partial change of emphasis, we have substituted the more general terms in our title. The common usage of the new terms clarifies the emphasis of the book.

The main change in the book has been to expand the sections on dissipative dynamics, including discussion of renormalization, circle maps, intermittancy, crises, transient chaos, multifractals, reconstruction, and coupled mapping systems. These topics were either mainly in the mathematical literature or essentially unstudied when our first edition was written. The volume of work in these areas has surpassed that in Hamitonian dynamics within the past few years.

We have also made changes in the Hamiltonian sections, adding many new topics such as more general transformation and stability theory, connected stochasticity in two-dimensional maps, converse KAM theory, new topics in diffusion theory, and an approach to equilibrium in many dimensions. Other sections such as mapping models have been revised to take into account new perspectives. We have also corrected a number of misprints and clarified various arguments with the help of colleagues and students. We have again chosen not to treat quantum chaos, partly due to our own lack of acquaintance with the subject. Since writing the first edition, a number of monographs and texts complementary to our book have appeared that are included in the new bibliography.

The original title of our book, “Regular and stochastic motion”, was chosen to emphasize Hamiltonian dynamics and the physical motion of bodies. The new edition is more evenhanded, with considerably more discussion of dissipative systems and dynamics not involving physical motion. To reflect this partial change of emphasis, we have substituted the more general terms in our title. The common usage of the new terms clarifies the emphasis of the book.

The main change in the book has been to expand the sections on dissipative dynamics, including discussion of renormalization, circle maps, intermittancy, crises, transient chaos, multifractals, reconstruction, and coupled mapping systems. These topics were either mainly in the mathematical literature or essentially unstudied when our first edition was written. The volume of work in these areas has surpassed that in Hamitonian dynamics within the past few years.

We have also made changes in the Hamiltonian sections, adding many new topics such as more general transformation and stability theory, connected stochasticity in two-dimensional maps, converse KAM theory, new topics in diffusion theory, and an approach to equilibrium in many dimensions. Other sections such as mapping models have been revised to take into account new perspectives. We have also corrected a number of misprints and clarified various arguments with the help of colleagues and students. We have again chosen not to treat quantum chaos, partly due to our own lack of acquaintance with the subject. Since writing the first edition, a number of monographs and texts complementary to our book have appeared that are included in the new bibliography.

### MSC:

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

70Gxx | General models, approaches, and methods in mechanics of particles and systems |

70Hxx | Hamiltonian and Lagrangian mechanics |

70K55 | Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics |

34Cxx | Qualitative theory for ordinary differential equations |

37Dxx | Dynamical systems with hyperbolic behavior |

37Exx | Low-dimensional dynamical systems |

37Hxx | Random dynamical systems |