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Foreward ix I. INTRODUCTION 3 1. The stability problem 3 2.Historical comments 3 3. Other problems 8 4. Unstable and statisticalbehavior 14 5. Plan 18 II. STABILITY PROBLEM 21 1. A model problem in thecomplex 21 2. Normal forms for Hamiltonian and reversible systems 30 3.Invariant manifolds 38 4. Twist theorem 50 III. STATISTICAL BEHAVIOR 61 1.Bernoulli shift. Example 61 2. Shift as a topological mapping 66 3. Shiftas a subsystem 68 4. Alternate conditions for C'-mappings 76 5. Therestricted three-body problem 83 6. Homoclinic points 99 IV. FINAL REMARKS113 V. EXISTENCE PROOF IN THE PRESENCE OF SMALL DIVISORS 113 1.Reformulation of Theorem 2.9 113 2. Construction of the root of a function120 3. Proof of Theorem 5.1 127 4. Generalities 138 A. Appendix to ChapterV 149 a. Rate of convergence for scheme of s.2b) 149 b. The improved schemeby Hald 151 VI. PROOFS AND DETAILS FOR CHAPTER III 153 1. Outline 153 2.Behavior near infinity 154 3. Proof of Lemmas 1 and 2 of Chapter III 160 4.Proof of Lemma 3 of Chapter III 163 5. Proof of Lemma 4 of Chapter III 1676. Proof of Lemma 5 of Chapter III 171 7. Proof of Theorem 3.7, concerninghomoclinic points 181 8. Nonexistence of intergals 188 BOOKS AND SURVEYARTICLES 191