Structure and breakdown of invariant tori in a 4-D mapping model of accelerator dynamics

M. N. Vrahatis, H. Isliker, T. C. Bountis

Research output: Contribution to journalArticle

40 Citations (Scopus)

Abstract

We study sequences of periodic orbits and the associated phase space dynamics in a 4-D symplectic map of interest to the problem of beam stability in circular particle accelerators. The increasing period of these orbits is taken from a sequence of rational approximants to an incommensurate pair of irrational rotation numbers of an invariant torus. We find stable (elliptic-elliptic) periodic orbits of very high period and show that smooth rotational tori exist in their neighborhood, on which the motion is regular and bounded at large distances away from the origin. Perturbing these tori in parameter and/or initial condition space, we find either chains of smaller rotational tori or certain twisted tube-like tori of remarkable morphology. These tube-tori and tori chains have small scale chaotic motions in their surrounding vicinity and are formed about invariant curves of the 4-D map, which are either single loops or are composed of several disconnected loops, respectively. These smaller chaotic regions as well as the non-smoothness properties of large rotational tori under small perturbations, leading to eventual escape of orbits to infinity, are studied here by the computation of correlation dimension and Lyapunov exponents.

Original languageEnglish
Pages (from-to)2707-2722
Number of pages16
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume7
Issue number12
Publication statusPublished - Dec 1997
Externally publishedYes

Fingerprint

Invariant Tori
Accelerator
Particle accelerators
Breakdown
Torus
Orbits
Periodic Orbits
Model
Tube
Orbit
Invariant Curves
Correlation Dimension
Irrational number
Rotation number
Chaotic Motion
Small Perturbations
Lyapunov Exponent
Phase Space
Initial conditions
Infinity

ASJC Scopus subject areas

  • General
  • Applied Mathematics

Cite this

Structure and breakdown of invariant tori in a 4-D mapping model of accelerator dynamics. / Vrahatis, M. N.; Isliker, H.; Bountis, T. C.

In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, Vol. 7, No. 12, 12.1997, p. 2707-2722.

Research output: Contribution to journalArticle

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