Noise in Nonlinear Dynamical Systems

doi:10.1017/CBO9780511897818

Nature is inherently noisy and nonlinear. It is noisy in the sense that all macroscopic systems are subject to the fluctuations of their environments and also to internal fluctuations. It is nonlinear in the sense that the restoring force on a system displaced from equilibrium does not usually vary linearly with the size of the displacement. To calculate the properties of stochastic (noisy) nonlinear systems is in general extremely difficult, although considerable progress has been made in the past. The three volumes that make up Noise in Nonlinear Dynamical Systems comprise a collection of specially written authoritative reviews on all aspects of the subject, representative of all the major practitioners in the field. The first volume deals with the basic theory of stochastic nonlinear systems. It includes an historical overview of the origins of the field, chapters covering some developed theoretical techniques for the study of coloured noise, and the first English-language translation of the landmark 1933 paper by Pontriagin, Andronov and Vitt.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-viii
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List of contributors
pp ix-xii
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Preface
pp xiii-xiv
- By Frank Moss, University of Missouri, P. V. E. McClintock, University of Lancaster
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Introduction to Volume 1
pp xv-xvi
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1 - Noise-activated escape from metastable states: an historical view
pp 1-15
- By Rolf Landauer, IBM Thomas J. Watson Research Center
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2 - Some Markov methods in the theory of stochastic processes in nonlinear dynamical systems
pp 16-71
- By R. L. Stratonovich, Moscow State University
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3 - Langevin equations with colored noise
pp 72-109
- By J. M. Sancho, Universidad de Barcelona, M. San Miguel, Universität de les Illes Balears
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4 - First passage time problems for non-Markovian processes
pp 110-160
- By Katja Lindenberg, University of California, Bruce J. West, University of California, Jaume Masoliver, University of California
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5 - The projection approach to the Fokker–Planck equation: applications to phenomenological stochastic equations with colored noises
pp 161-190
- By Paolo Grigolini, Consiglio Nazionale delle Ricerche
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6 - Methods for solving Fokker–Planck equations with applications to bistable and periodic potentials
pp 191-224
- By H. Risken, Universität Ulm, H. D. Vollmer, Universität Ulm
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7 - Macroscopic potentials, bifurcations and noise in dissipative systems
pp 225-278
- By Robert Graham, Universität-Gesamthochschule-Essen
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8 - Transition phenomena in multidimensional systems – models of evolution
pp 279-306
- By W. Ebeling, Humboldt-Universität zu Berlin, L. Schimansky-Geier, Humboldt-Universität zu Berlin
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9 - Colored noise in continuous dynamical systems: a functional calculus approach
pp 307-328
- By Peter Hänggi, Universität Augsburg
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Appendix: On the statistical treatment of dynamical systems
pp 329-348
- By L. Pontryagin, Stecklov Institute, A. Andronov, Scientific Research Institute Moscow, A. Vitt, Scientific Research Institute Moscow
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Index
pp 349-353
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Noise in Nonlinear Dynamical Systems

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Contents

Frontmatter
pp i-iv

Contents
pp v-viii

List of contributors
pp ix-xii

Preface
pp xiii-xiv

Introduction to Volume 1
pp xv-xvi

1 - Noise-activated escape from metastable states: an historical view
pp 1-15

2 - Some Markov methods in the theory of stochastic processes in nonlinear dynamical systems
pp 16-71

3 - Langevin equations with colored noise
pp 72-109

4 - First passage time problems for non-Markovian processes
pp 110-160

5 - The projection approach to the Fokker–Planck equation: applications to phenomenological stochastic equations with colored noises
pp 161-190

6 - Methods for solving Fokker–Planck equations with applications to bistable and periodic potentials
pp 191-224

7 - Macroscopic potentials, bifurcations and noise in dissipative systems
pp 225-278

8 - Transition phenomena in multidimensional systems – models of evolution
pp 279-306

9 - Colored noise in continuous dynamical systems: a functional calculus approach
pp 307-328

Appendix: On the statistical treatment of dynamical systems
pp 329-348

Index
pp 349-353

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