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TitleDynamics Reported [electronic resource] : Expositions in Dynamical Systems / edited by C. K. R. T. Jones, U. Kirchgraber, H. O. Walther
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1995
Connect tohttp://dx.doi.org/10.1007/978-3-642-61215-2
Descript IX, 269 p. online resource

SUMMARY

DYNAMICS REPORTED reports on recent developments in dynamical systems. Dynamical systems of course originated from ordinary differential equations. Today, dynamical systems cover a much larger area, including dynamical processes described by functional and integral equations, by partial and stochastic differential equations, etc. Dynamical systems have involved remarkably in recent years. A wealth of new phenomena, new ideas and new techniques are proving to be of considerable interest to scientists in rather different fields. It is not surprising that thousands of publications on the theory itself and on its various applications are appearing DYNAMICS REPORTED presents carefully written articles on major subjects in dyยญ namical systems and their applications, addressed not only to specialists but also to a broader range of readers including graduate students. Topics are advanced, while detailed exposition of ideas, restriction to typical results - rather than the most general one- and, last but not least, lucid proofs help to gain the utmost degree of clarity. It is hoped, that DYNAMICS REPORTED will be useful for those entering the field and will stimulate an exchange of ideas among those working in dynamical systems Summer 1991 Christopher K. R. T Jones Drs Kirchgraber Hans-Otto Walther Managing Editors Table of Contents The "Spectral" Decomposition for One-Dimensional Maps Alexander M. Blokh Introduction and Main Results 1. 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 0. 1. 1. Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. 2. A Short Description of the Approach Presented . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. 3. Solenoidal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Basic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 4


CONTENT

The "Spectral" Decomposition for Oneโ{128}{147}Dimensional Maps -- 1. Introduction and Main Results -- 2. Technical Lemmas -- 3. Solenoidal Sets -- 4. Basic Sets -- 5. The Decomposition -- 6. Limit Behavior for Maps Without Wandering Intervals -- 7. Topological Properties of the Sets Per f, ?(f) and ?(f) -- 8. Transitive and Mixing Maps -- 9. Corollaries Concerning Periods of Cycles -- 10. Invariant Measures -- 11. Discussion of Some Recent Results of Block and Coven and Xiong Jincheng -- References -- A Constructive Theory of Lagrangian Tori and Computer-assisted Applications -- 1. Introduction -- 2. Quasi-Periodic Solutions and Invariant Tori for Lagrangian Systems: Algebraic Structure -- 3. Quasi-Periodic Solutions and Invariant Tori for Lagrangian Systems: Quantitative Analysis -- 4. KAM Algorithm -- 5. A KAM Theorem -- 6. Application of the KAM Algorithm to Problems with Parameters -- 7. Power Series Expansions and Estimate of the Error Term -- 8. Computer Assisted Methods -- 9. Applications: Three-Dimensional Phase Space Systems -- 10. Applications: Symplectic Maps -- Appendices -- References -- Ergodicity in Hamiltonian Systems -- 0. Introduction -- 1. A Model Problem -- 2. The Sinai Method -- 3. Proof of the Sinai Theorem -- 4. Sectors in a Linear Symplectic Space -- 5. The Space of Lagrangian Subspaces Contained in a Sector -- 6. Unbounded Sequences of Linear Monotone Maps -- 7. Properties of the System and the Formulation of the Results -- 8. Construction of the Neighborhood and the Coordinate System -- 9. Unstable Manifolds in the Neghborhood U -- 10. Local Ergodicity in the Smooth Case -- 11. Local Ergodicity in the Discontinous Case -- 12. Proof of Sinai Theorem -- 13. โ{128}{152}Tail Boundโ{128}{153} -- 14. Applications -- References -- Linearization of Random Dynamical Systems -- 1. Introduction -- 2. Random Difference Equations -- 3. Random Dynamical Systems -- 4. Local Results -- 5. Appendix -- References


Mathematics Mathematical analysis Analysis (Mathematics) Probabilities Physics Mathematics Analysis Probability Theory and Stochastic Processes Theoretical Mathematical and Computational Physics



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