1 Fractional Zaslavsky and Henon Discrete Maps
Vasily E. Tarasov
1.1 Introduction
1.2 Fractional derivatives
1.2.1 Fractional Riemann-Liouville derivatives
1.2.2 Fractional Caputo derivatives
1.2.3 Fractional Liouville derivatives
1.2.4 Interpretation of equations with fractional derivatives.
1.2.5 Discrete maps with memory
1.3 Fractional Zaslavsky maps
1.3.1 Discrete Chirikov and Zaslavsky maps
1.3.2 Fractional universal and Zaslavsky map
1.3.3 Kicked damped rotator map
1.3.4 Fractional Zaslavsky map from fractional differential equations
1.4 Fractional H6non map
1.4.1 Henon map
1.4.2 Fractional Henon map
1.5 Fractional derivative in the kicked term and Zaslavsky map
1.5.1 Fractional equation and discrete map
1.5.2 Examples
1.6 Fractional derivative in the kicked damped term and generalizations of Zaslavsky and Henon maps
1.6.1 Fractional equation and discrete map
1.6.2 Fractional Zaslavsky and Henon maps
1.7 Conclusion
References
2 Self-similarity, Stochasticity and Fractionality
Vladimir V Uchaikin
2.1 Introduction
2.1.1 Ten years ago
2.1.2 Two kinds of motion
2.1.3 Dynamic self-similarity
2.1.4 Stochastic self-similarity
2.1.5 Self-similarity and stationarity
2.2 From Brownian motion to Levy motion
2.2.1 Brownian motion
2.2.2 Self-similar Brownian motion in nonstationary nonhomogeneous environment
2.2.3 Stable laws
2.2.4 Discrete time Levy motion
2.2.5 Continuous time Levy motion
2.2.6 Fractional equations for continuous time Levy motion
2.3 Fractional Brownian motion
2.3.1 Differential Brownian motion process
2.3.2 Integral Brownian motion process
2.3.3 Fractional Brownian motion
2.3.4 Fractional Gaussian noises
2.3.5 Barnes and Allan model
2.3.6 Fractional Levy motion
2.4 Fractional Poisson motion
2.4.1 Renewal processes
2.4.2 Self-similar renewal processes
2.4.3 Three forms of fractal dust generator
2.4.4 nth arrival time distribution
2.4.5 Fractional Poisson distribution
2.5 Fractional compound Poisson process
2.5.1 Compound Poisson process
2.5.2 Levy-Poisson motion
2.5.3 Fractional compound Poisson motion
2.5.4 A link between solutions
2.5.5 Fractional generalization of the Levy motion
Acknowledgments
Appendix. Fractional operators
References
3 Long-range Interactions and Diluted Networks
Antonia Ciani, Duccio Fanelli and Stefano Ruffo
3.1 Long-range interactions
3.1.1 Lack of additivity
3.1.2 Equilibrium anomalies: Ensemble inequivalence, negative specific heat and temperature jumps
3.1.3 Non-equilibrium dynamical properties
3.1.4 Quasi Stationary States
3.1.5 Physical examples
3.1.6 General remarks and outlook
3.2 Hamiltonian Mean Field model: equilibrium and out-of- equilibrium features
3.2.1 The model
3.2.2 Equilibrium statistical mechanics
3.2.3 On the emergence of Quasi Stationary States: Non-
equilibrium dynamics
3.3 Introducing dilution in the Hamiltonian Mean Field model
3.3.1 Hamiltonian Mean Field model on a diluted network
3.3.2 On equilibrium solution of diluted Hamiltonian Mean Field
3.3.3 On Quasi Stationary States in presence of dilution
3.3.4 Phase transition
3.4 Conclusions
Acknowledgments
References
4 Metastability and Transients in Brain Dynamics: Problems and Rigorous Results
Valentin S. Afraimovich, Mehmet K. Muezzinoglu and
Mikhail I. Rabinovich
4.1 Introduction: what we discuss and why now
4.1.1 Dynamical modeling of cognition
4.1.2 Brain imaging
4.1.3 Dynamics ...

1 Fractional Zaslavsky and Henon Discrete Maps Vasily E. Tarasov 1.1 Introduction 1.2 Fractional derivatives 1.2.1 Fractional Riemann-Liouville derivatives 1.2.2 Fractional Caputo derivatives 1.2.3 Fractional Liouville derivatives 1.2.4 Interpretation of equations with fractional derivatives. 1.2.5 Discrete maps with memory 1.3 Fractional Zaslavsky maps 1.3.1 Discrete Chirikov and Zaslavsky maps 1.3.2 Fractional universal and Zaslavsky map 1.3.3 Kicked dam...