George was a very creative scientist and a very good teacher whose former students and collaborators are working now in America, Europe and Asia. He authored and co-authored 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 watercolors was issued in 2009 in Moscow.

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.

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