Hamiltonian maps in the complex plane
Following Arnol'd's proof of the KAM theorem, an analogy with the vertical pendulum, and some general arguments concerning maps in the complex plane, detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency. The functional dependence of the coordinate q on the canonical angle variable theta is analytically continued into the complex theta-plane, where natural boundaries are found at constant absolute values of Im theta. The boundaries represent the appearance of chaotic motion in the complex plane. Two independent numerical methods based on Fourier analysis in the angle variable were used, one based on a variation-annihilation method and the other on a double expansion. The results were further checked by direct solution of the complex equations of motion. The numerically simpler, but intrinsically complex, semipendulum and semistandard map are also studied. We conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or KAM surfaces of general nonintegrable systems.
- Research Organization:
- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States); Queen Mary Coll., London (UK)
- DOE Contract Number:
- AM02-76-CH03073
- OSTI ID:
- 6647266
- Report Number(s):
- PPPL-1744; TRN: 81-005350
- Country of Publication:
- United States
- Language:
- English
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