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The iteration-approximation decoupling in the reversible KAM theory
1Department of Chemistry, H. C. Orsted Institute, University of Copenhagen, DK-2100 O, Denmark.
Chaos (Woodbury, N.Y.)
|September 1, 1995
Summary
New theorems prove the persistence of quasiperiodic motions in reversible systems using a novel embedding method. This approach applies Diophantine approximations to construct invariant tori under weak nondegeneracy conditions.
Area of Science:
- Dynamical Systems
- Mathematical Physics
- Differential Geometry
Background:
- Quasiperiodic motions are fundamental in understanding complex dynamical systems.
- Reversible systems, common in physics, exhibit unique stability properties.
- Persistence of motion under perturbation is a key theoretical challenge.
Purpose of the Study:
- To establish general theorems on the persistence of quasiperiodic motions in reversible systems.
- To introduce a novel method for analyzing these motions under weak nondegeneracy conditions.
- To extend these results to both continuous flows and discrete diffeomorphisms.
Main Methods:
- Embedding the reversible system into a multiparameter family of reversible systems.
- Applying Diophantine approximation results to Whitney-smooth Cantor foliations.
- Constructing invariant tori for permissible parameter values.
Main Results:
- General theorems on the persistence of quasiperiodic motions are established.
- Invariant tori are successfully constructed for vector fields and diffeomorphisms.
- The method is effective under very weak nondegeneracy conditions.
Conclusions:
- The novel embedding method provides a powerful tool for analyzing reversible dynamical systems.
- The findings contribute to the theoretical understanding of stability in quasiperiodic motions.
- The study offers a unified approach for flows and diffeomorphisms.