内容简介:
Hamiltonian Chaos Beyond the KAM Theory Dedicated to George M. Zaslavsky (1935-2008) covers the recent developments and advances in the theory and application of Hamiltonian chaos in nonlinear Hamiltonian systems. The book is dedicated to Dr. George Zaslavsky, who was one of three founders of the theory of Hamiltonian chaos. Each chapter in this book was written by well-established scientists in the field of nonlinear Hamiltonian systems. The development presented in this book goes beyond the KAM theory, and the onset and disappearance of chaos in the stochastic and resonant layers of nonlinear Hamiltonian systems are predicted analytically, instead of qualitatively.

The book is intended for researchers in the field of nonlinear dynamics in mathematics, physics and engineering.

目录:
1 Stochastic and Resonant Layers in Nonlinear Hamiltonian Systems Albert C.J. Luo

1.1 Introduction

1.2 Stochastic layers

1.2.1 Geometrical description

1.2.2 Approximate criterions

1.3 Resonant layers

1.3.1 Layer dynamics

1.3.2 Approximate criterions

1.4 A periodically forced Duffing oscillator

1.4.1 Approximate predictions

1.4.2 Numerical illustrations

1.5 Discussions

References

2 A New Approach to the Treatment of Separatrix Chaos and Its Applications

S.M. Soskin, R. Mannella, O.M. Yevtushenko, LA. Khovanov, P V.E. McClintock

2.1 Introduction

2.1.1 Heuristic results

2.1.2 Mathematical and accurate physical results

2.1.3 Numerical evidence for high peaks in ΔE(ωf) and their rough estimations

2.1.4 Accurate description of the peaks and of the related phenomena

2.2 Basic ideas of the approach

2.3 Single-separatrix chaotic layer

2.3.1 Rough estimates. Classification of systems

2.3.2 Asymptotic theory for systems of type I

2.3.3 Asymptotic theory for systems of type II

2.3.4 Estimate of the next-order corrections

2.3.5 Discussion

2.4 Double-separatrix chaos

2.4.1 Asymptotic theory for the minima of the spikes

2.4.2 Theory of the spikes'' wings

2.4.3 Generalizations and applications

2.5 Enlargement of a low-dimensional stochastic web

2.5.1 Slow modulation of the wave angle

2.5.2 Application to semiconductor superlattices

2.5.3 Discussion

2.6 Conclusions

2.7 Appendix

2.7.1 Lower chaotic layer

2.7.2 Upper chaotic layer

References

3 Hamiltonian Chaos and Anomalous Transport in Two Dimensional Flows Xavier Leoncini

3.1 Introduction

3.2 Point vortices and passive tracers advection

3.2.1 Definitions

3.2.2 Chaotic advection

3.3 A system of point vortices

3.3.1 Definitions

3.4 Dynamics of systems with two or three point vortices

3.4.1 Dynamics of two vortices

3.4.2 Dynamics of three vortices

3.5 Vortex collapse and near collapse dynamics of point vortices

3.5.1 Vortex collapse

3.5.2 Vortex dynamics in the vicinity of the singularity

3.6 Chaotic advection and anomalous transport

3.6.1 A brief history

3.6.2 Definitions

3.6.3 Anomalous transport in incompressible flows

3.6.4 Tracers (passive particles) dynamics

3.6.5 Transport properties

3.6.6 Origin of anomalous transport

3.6.7 General remarks

3.7 Beyond characterizing transport

3.7.1 Chaos of field lines

3.7.2 Local Hamiltonian dynamics

3.7.3 An ABC type flow

3.8 Targeted mixing in an array of alternating vortices

3.9 Conclusion

References

4 Hamiltonian Chaos with a Cold Atom in an Optical Lattice S. V. Prants

4.1 Short historical background

4.2 Introduction

4.3 Semiclassical dynamics

4.3.1 Hamilton-Schrrdinger equations of motion

4.3.2 Regimes of motion

4.3.3 Stochastic map for chaotic atomic transport

4.3.4 Statistical properties of chaotic transport

4.3.5 Dynamical fractals

4.4 Quantum dynamics

4.5 Dressed states picture and nonadiabatic transitions

4.5.1 Wave packet motion in the momentum space

4.6 Quantum-classical correspondence and manifestations of dynamical chaos in wave-packet atomic motion

References

5 Using Stochastic Webs to Control the Quantum Transport of Electrons in Semiconductor Superlattices

T.M. Fromhold, A.A. Krokhin, S. Bujkiewicz, P.B. Wilkinson,

D. Fowler, A. Patane, L. Eaves, D.PA. Hardwick, A.G. Balanov,

M.T. Greenaway, A. Henning

5.1 Introduction

5.2 Superlattice structures

5.3 Semiclassical electron dynamics

5.4 Electron drift velocity

5.5 Current-voltage characteristics: theory and experiment...

5.6 Electrostatics and charge domain structure

5.7 Tailoring the SL structure to increase the number of conductance resonances

5.8 Energy eigenstates and Wigner functions

5.9 Summary and outlook

References

6 Chaos in Ocean Acoustic Waveguide

A.L. Virovlyansky

6.1 Introduction

6.2 Basic equations

6.2.1 Parabolic equation approximation

6.2.2 Geometrical optics. Hamiltonian formalism

6.2.3 Modal representation of the wave field

6.2.4 Ray-based description of normal modes

6.3 Ray chaos

6.3.1 Statistical description of chaotic rays

6.3.2 Environmental model

6.3.3 Wiener process approximation

6.3.4 Distribution of ray parameters

6.3.5 Smoothed intensity of the wave field

6.4 Ray travel times

6.4.1 Timefront

6.4.2 Statistics of ray travel times

6.5 Modal structure of the wave field under conditions of ray chaos..

6.5.1 Coarse-grained energy distribution between normal modes

6.5.2 Transient wave field

6.6 Conclusion

References

前言:
George M. Zaslavsky was bom in Odessa, Ukraine in 1935 in a family of an artillery officer. He received education at the University of Odessa and moved in 1957 to Novosibirsk, Russia. In 1965, George joined the Institute of Nuclear Physics where he became interested in nonlinear problems of accelerator and plasma physics.Roald Sagdeev and Boris Chirikov were those persons who formed his interest in the theory of dynamical chaos. In 1968 George introduced a separatrix map that became one of the major tools in theoretical study of Hamiltonian chaos. The work "Stochastical instability of nonlinear oscillations" by G. Zaslavsky and B. Chirikov,published in Physics Uspekhi in 1971, was the first review paper "opened the eyes"of many physicists to power of the theory of dynamical systems and modem ergodic theory. It was realized that very complicated behavior is possible in dynamical sys-tems with only a few degrees of freedom. This complexity cannot be adequately described in terms of individual trajectories and requires statistical methods. Typi-cal Hamiltonian systems are not integrable but chaotic, and this chaos is not homo-geneous. At the same values of the control parameters, there coexist regions in the phase space with regular and chaotic motion. The results obtained in the 1960s were summarized in the book "Statistical Irreversibility in Nonlinear Systems" (Nauka,Moscow, 1970).

The end of the 1960s was a hard time for George. He was forced to leave the Institute of Nuclear Physics in Novosibirsk for signing letters in defense of some Soviet dissidents. George got a position at the Institute of Physics in Krasnoyarsk,not far away from Novosibirsk. There he founded a laboratory of the theory of non-linear processes which exists up to now. In Krasnoyarsk George became interested in the theory of quantum chaos. The first rigorous theory of quantum resonance was developed in 1977 in collaboration with his co-workers. They introduced the impor-tant notion of quantum break time (the Ehrenfest time) after which quantum evolu-tion begins to deviate from a semiclassical one. The results obtained in Krasnoyarsk were summarized in the book "Chaos in Dynamical Systems" (Nauka, Moscow and Harwood, Amsterdam, 1985).

In 1984, R. Sagdeev invited George to the Institute of Space Research in Moscow.There he has worked on the theory of degenerate and almost degenerate Hamilto-nian systems, anomalous chaotic transport, plasma physics, and theory of chaos in waveguides. The book "Nonlinear Physics: from the Pendulum to Turbulence and Chaos" (Nauka, Moscow and Harwood, New York, 1988), written with R. Sagdeev,is now a classical textbook for everybody who studies chaos theory. When studying interaction of a charged particle with a wave packet, George with colleagues from the Institute discovered that stochastic layers of different separatrices in degenerated Hamiltonian systems may merge producing a stochastic web. Unlike the famous Arnold diffusion in non-degenerated Hamiltonian systems, that appears only if the number of degrees of freedom exceeds 2, diffusion in the Zaslavsky webs is possible at one and half degrees of freedom. This diffusion is rather universal phenomenon and its speed is much greater than that of Arnold diffusion. Beautiful symmetries of the Zaslavsky webs and their properties in different branches of physics have been described in the book "Weak chaos and Quasi-Regular Structures" (Nauka,Moscow, 1991 and Cambridge University Press, Cambridge, 1991) coauthored with R. Sagdeev, D. Usikov, and A. Chernikov.

In 1991, George emigrated to the USA and became a Professor of Physics and Mathematics at Physical Department of the New York University and at the Courant Institute of Mathematical Sciences. The last 17 years of his life he de-voted to principal problems of Hamiltonian chaos connected with anomalous kinet-ics and fractional dynamics, foundations of statistical mechanics, chaotic advection,quantum chaos, and long-range propagation of acoustic waves in the ocean. In his New York period George published two important books on the Hamiltonian chaos:"Physics of Chaos in Hamiltonian Systems" (Imperial College Press, London, 1998)and "Hamiltonian chaos and Fractional Dynamics" (Oxford University Press, NY,2005). His last book "Ray and wave chaos in ocean acoustics: chaos in waveguides"(World Scientific Press, Singapore, 2010), written with D. Makarov, S. Prants, and A. Virovlynsky, reviews original results on chaos with acoustic waves in the under-water sound channel.

George was a very creative scientist and a very good teacher whose former stu-dents and collaborators are working now in America, Europe and Asia. He authored and coauthored 9 books and more than 300 papers in journals. Many of his works are widely cited. George worked hard all his life. He loved music, theater, literature and was an expert in good vines and food. Only a few people knew that he loved to paint. In the last years he has spent every summer in Provence, France, working,writing books and papers and painting in water colors. The album with his water colors was issued in 2009 in Moscow.

George Zaslavsky was one of the key persons in the theory of dynamical chaos and made many important contributions to a variety of other subjects. His books and papers influenced very much in advancing modern nonlinear science.

Sergey Prants

Albert C.J. Luo

Valentin Afralmovich

March, 2010

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