Kaloshin Vadim and Zhang Ke
- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.003.0010
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter proves the geometric picture of double resonance described in Chapter 4. There are two cases. In the simple critical homology case, the chapter shows the homoclinic orbit can be extended ...
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This chapter proves the geometric picture of double resonance described in Chapter 4. There are two cases. In the simple critical homology case, the chapter shows the homoclinic orbit can be extended to periodic orbits both in positive and negative energy. The union of these periodic orbits forms a normally hyperbolic invariant manifold (which is homotopic to a cylinder with a puncture). In the non-simple homology case, the chapter demonstrates that for positive energy, there exist periodic orbits. The strategy is to prove the existence of these periodic orbits as hyperbolic fixed points of composition of local and global maps. A main technical tool to prove the existence and uniqueness of these fixed points is the Conley-McGehee isolation block.Less
This chapter proves the geometric picture of double resonance described in Chapter 4. There are two cases. In the simple critical homology case, the chapter shows the homoclinic orbit can be extended to periodic orbits both in positive and negative energy. The union of these periodic orbits forms a normally hyperbolic invariant manifold (which is homotopic to a cylinder with a puncture). In the non-simple homology case, the chapter demonstrates that for positive energy, there exist periodic orbits. The strategy is to prove the existence of these periodic orbits as hyperbolic fixed points of composition of local and global maps. A main technical tool to prove the existence and uniqueness of these fixed points is the Conley-McGehee isolation block.
