Random dynamical systems.

*(English)*Zbl 0906.34001
Springer Monographs in Mathematics. Berlin: Springer. xi, 586 p. (1998).

The monograph presents the first systematic presentation of the theory of measure-preserving random dynamical systems in form of a (relatively) selfcontained book. As the author says in the preface, it can be understood as a continuation of that by J. Guckenheimer and P. Holmes [Nonlinear oscillations, dynamical systems, and bifurcations of vector fields; Springer, New York (1983; Zbl 0515.34001)].

The presented theory involves qualitative aspects of measure-preserving random mappings, random and stochastic differential equations based mainly on the author’s central view on Oseledets’ multiplicative ergodic theorem for linear random dynamical systems. Therefore, the local theory of nonlinear random systems plays a significant role in this exposition.

The exposition is organized in four parts as follows. Part I. Basic definitions. Invariant measures (definition of random dynamical systems (RDS), perfection of cocycles, invariant measures for measurable and continuous RDS, generation of RDS). Part II. Multiplicative ergodic theory. (The multiplicative ergodic theorem, Lyapunov exponents, tempered random variables, RDS on manifolds, random Lyapunov metrics, random norms, cocycles on Lie groups, Furstenberg-Khasminskii formulas, RDS on Grassmannians, rotation numbers). Part III. Smooth random dynamical systems. (Invariant manifolds, random Hartman-Grobman theorem, local and global invariant manifolds, normal forms and center manifolds, bifurcation theory, noisy Duffing-van der Pol oscillator, Hopf and pitchfork bifurcation). Part IV. Appendices. (Measurable dynamical systems, ergodic theory, stochastic processes, stationary and Markov processes, smooth dynamical systems, autonomous dynamical systems, flows on manifolds).

The author provides numerous instructive examples which are treated both analytically and numerically. However, the author follows the obvious intention to provide a very comprehensive and foundational source for further research and applications at the interface of probability theory and dynamical systems. Recent publications concerning topological dynamics, Pesin’s theory of random dynamical systems, the beautiful geometry of stochastic flows and progess in infinite-dimensional random systems which the author has to omit within one volume would even give a need and enough subjects for a continuation of the presented theory in a further volume.

All in all, the book presents the state-of-the-art of the 90’ies at the end of 20th century and is very well-written. Therefore, and because of the high mathematical standard presented, it is a definite must to have it as a standard reference manual in the bookshelve of any scientist who is dealing with nonlinear dynamics.

The presented theory involves qualitative aspects of measure-preserving random mappings, random and stochastic differential equations based mainly on the author’s central view on Oseledets’ multiplicative ergodic theorem for linear random dynamical systems. Therefore, the local theory of nonlinear random systems plays a significant role in this exposition.

The exposition is organized in four parts as follows. Part I. Basic definitions. Invariant measures (definition of random dynamical systems (RDS), perfection of cocycles, invariant measures for measurable and continuous RDS, generation of RDS). Part II. Multiplicative ergodic theory. (The multiplicative ergodic theorem, Lyapunov exponents, tempered random variables, RDS on manifolds, random Lyapunov metrics, random norms, cocycles on Lie groups, Furstenberg-Khasminskii formulas, RDS on Grassmannians, rotation numbers). Part III. Smooth random dynamical systems. (Invariant manifolds, random Hartman-Grobman theorem, local and global invariant manifolds, normal forms and center manifolds, bifurcation theory, noisy Duffing-van der Pol oscillator, Hopf and pitchfork bifurcation). Part IV. Appendices. (Measurable dynamical systems, ergodic theory, stochastic processes, stationary and Markov processes, smooth dynamical systems, autonomous dynamical systems, flows on manifolds).

The author provides numerous instructive examples which are treated both analytically and numerically. However, the author follows the obvious intention to provide a very comprehensive and foundational source for further research and applications at the interface of probability theory and dynamical systems. Recent publications concerning topological dynamics, Pesin’s theory of random dynamical systems, the beautiful geometry of stochastic flows and progess in infinite-dimensional random systems which the author has to omit within one volume would even give a need and enough subjects for a continuation of the presented theory in a further volume.

All in all, the book presents the state-of-the-art of the 90’ies at the end of 20th century and is very well-written. Therefore, and because of the high mathematical standard presented, it is a definite must to have it as a standard reference manual in the bookshelve of any scientist who is dealing with nonlinear dynamics.

Reviewer: Henri Schurz (BogotĂˇ)

##### MSC:

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

34F05 | Ordinary differential equations and systems with randomness |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

37Hxx | Random dynamical systems |

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

37H05 | General theory of random and stochastic dynamical systems |

37H10 | Generation, random and stochastic difference and differential equations |

37H15 | Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents |

37H20 | Bifurcation theory for random and stochastic dynamical systems |

93E15 | Stochastic stability in control theory |