Table of contents

CONTENTS Introduction 1. Chaotic dynamical systems 1.1. A stochastic attractor. Hyperbolic systems. The method of symbolic dynamics 1.2. Dynamical systems with singularities. Piecewise stretching maps. The operator approach 1.3. The Li-Yorke chaos 2. Random perturbations of stochastic attractors 2.1. Random perturbations of hyperbolic systems 2.2. Random perturbations of systems with singularities 2.3. Stabilization of unstable invariant measures 2.4. The "typical" ("generic") property and properties of ε- and ε-a-trajectories 2.5. The most probable trajectories 3. Space discretization in chaotic systems 3.1. Definitions and basic examples 3.2. Properties of the periodic trajectories under discretization 3.3. Statistical probability under discretizations 3.4. Stability of stochastic attractors under space discretizations 3.5. Chaos under a partial space discretization 4. Time discretization in dynamical systems 4.1. Chaos under time discretizations References

CONTENTS Introduction Chapter I. Hamiltonian theory of systems of hydrodynamic type § 1. General properties of Poisson brackets § 2. Hamiltonian formalism of systems of hydrodynamic type and Riemannian geometry § 3. Generalizations: differential-geometric Poisson brackets of higher orders, differential-geometric Poisson brackets on a lattice, and the Yang-Baxter equation § 4. Riemann invariants and the Hamiltonian formalism of diagonal systems of hydrodynamic type. Novikov's conjecture. Tsarev's theorem. The generalized hodograph method Chapter II. Equations of hydrodynamics of soliton lattices § 5. The Bogolyubov-Whitham averaging method for field-theoretic systems and soliton lattices. The results of Whitham and Hayes for Lagrangian systems § 6. The Whitham equations of hydrodynamics of weakly deformed soliton lattices for Hamiltonian field-theoretic systems. The principle of conservation of the Hamiltonian structure under averaging § 7. Modulations of soliton lattices of completely integrable evolutionary systems. Krichever's method. The analytic solution of the Gurevich-Pitaevskii problem on the dispersive analogue of a shock wave § 8. Evolution of the oscillatory zone in the KdV theory. Multi-valued functions in the hydrodynamics of soliton lattices. Numerical studies § 9. Influence of small viscosity on the evolution of the oscillatory zone References

CONTENTS Introduction § 0. Preliminaries: fuzzy sets § 1. Fuzzy topological spaces: the basic categories of fuzzy topology § 2. Fundamental interrelations between the category Top of topological spaces and the categories of fuzzy topology § 3. Local structure of fuzzy topological spaces § 4. Convergence structures in fuzzy spaces § 5. Separation in fuzzy spaces § 6. Normality and complete regularity type properties in fuzzy topology § 7. Compactness in fuzzy topology § 8. Connectedness in fuzzy spaces § 9. Fuzzy metric spaces and metrization of fuzzy spaces § 10. The fuzzy real line and its subspaces § 11. Fuzzy modification of a linearly ordered space § 12. Fuzzy probabilistic modification of a topological space § 13. The interval fuzzy real line § 14. On hyperspaces of fuzzy sets § 15. Another view of the subject of fuzzy topology and certain categorical aspects of it Conclusion: some reflections on the role and significance of fuzzy topology References