Contents

Introduction vii

1. Notation and main results 1

1.1. Notation and hypotheses 1

1.2. Main results 3

1.3. Example 5

2. Analytic properties of the homoclinic orbit of the unperturbed system 7

2.1. Introduction and main results 7

2.2. Proof of Proposition 2.1 10

3. Parameterization of local invariant manifolds 15

3.1. Introduction 15

3.2. Definitions and main result 15

3.3. Averaging of the equation 17

3.4. Estimates for the Poincare map 23

3.5. The operators B and B 30

3.6. Proof of Theorem 3.1 32

4. Flow box coordinates 37

4.1. Introduction 37

4.2. Definitions and main result 38

4.3. A preliminary change of variables 40

4.4. The unperturbed case 41

4.5. Flow box coordinates in a complex domain 42

4.6. Proof of Theorem 4.2 59

5. The Extension Theorem 63

6. Splitting of separatrices 65

6.1. Introduction 65

6.2. The splitting function 66

6.3. Proof of Theorem 1.1 and its corollary 73

6.4. Proof of Lemma 6.4 75

6.5. Proof of Corollary 1.1 80

References 82