Transversal Homoclinic Orbits near Elliptic Fixed Points of Area-preserving Diffeomorphisms of the Plane -- 1. Introduction -- 2. Elements of the Theory of Minimal States -- 3. A Priori Lipschitz Estimates for Minimal Orbits -- 4. First Perturbation: Isolation and Hyperbolicity of Minimal Periodic Orbits -- 5. Second Perturbation: Nondegeneracy of Homoclinic Orbits -- 6. Application to Mather Sets -- 7. Special Classes of Diffeomorphisms -- References -- Asymptotic Periodicity of Markov and Related Operators -- 1. Basic Notions and Results -- 2. The Reduction Procedure -- 3. Asymptotic Periodicity of Constrictive Marcov Operators -- 4. Weakly Almost Periodic Operators -- 5. Asymptotic Periodicity of Power Bounded Operators -- 6. Asymptotic Periodicity of Operators on Signed Measures -- References -- A Nekhoroshev-Like Theory of Classical Particle Channeling in Perfect Crystals -- I. Introduction -- 2. Background and Outline of Main Results -- 3. Formulation of the Channeling Problem -- 4. Construction of the Normal Forms -- 5. The Generalized Continuum Models -- 6. Concluding Remarks -- References -- The Adiabatic Invariant in Classical Mechanics -- I The Classical Adiabatic Invariant Theory -- 1. Introduction -- 2. Action-Angle Variables -- 3. Perturbation Theory -- 4. The Adiabatic Invariant -- 5. Explicit Approach to Action-Angle Variables -- 6. Extension of Perturbation Theory to the Case of Unbounded Period -- II Transition Through a Critical Curce -- 1. Introduction -- 2. Neighborhood of an Homoclinic Orbit -- 3. The Autonomous Problem Close to the Equilibrium -- 4. The Autonomous Problem Close to the Homoclinic Orbit -- 5. Traverse from Apex to Apex -- 6. Probability of Capture -- 7. Time of Transit -- 8. Change in the Invariant -- III The Paradigms -- 1. Introduction -- 2. The Pendulum -- 3. The Second Fundamental Model -- 4. The Colombo's Top -- 5. Dissipative Forces -- IV Applications -- 1. Introduction -- 2. Passage Through Resonance of a Forced Anharmonic Oscillator -- 3. Particle Motion in a Slowly Modulated Wave -- 4. The Magnetic Bottle -- 5. Orbit-Orbit Resonances in the Solar System -- 6. Spin-Orbit Resonance in the Solar System -- Appendix 1: Variational Equations -- Appendix 2: Fixing the Unstable Equilibrium and the Time Scale. -- Appendix 3: Mean Value of Ri(?i, Ji, ?) 1?i?2 -- Appendix 4: Mean Value of R3 (?3, J3, ?) -- Appendix 5: Estimation of the Trajectory Close to the Equilibrium. -- Appendix 6: Computation of the True Time of Transit -- Appendix 7: The Diffusion Parameter in Non-Symmetric Cases -- Appendix 8: Remarks on the Paper "On the Generalization of a Theorem of A. Liapounoff", by J. Moser (Comm. P. Appl. Math. 9, 257-271, 1958) -- References -- List of Contributors.