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Title: Efficient manifolds tracing for planar maps

Abstract

Here, we present an exact calculation method and an approximation technique for tracing the invariant manifolds of unstable periodic orbits of planar maps. The exact approach relies on an adaptive refinement procedure that prevents redundant calculations occurring in other approaches, and the approximated method relies on a novel interpolation approach based on normal displacement functions. The resulting approximated manifold is precise when compared to the exact one, and its relative computational cost falls like the inverse of the manifold length. To present the tracing method, we obtain the invariant manifolds of the Chirikov-Taylor map, and as an application we illustrate the transition from homoclinic to heteroclinic chaos in the Duffing oscillator that leads from localized chaos to global chaotic motion.

Authors:
ORCiD logo [1]; ORCiD logo [2];  [3]; ORCiD logo [4]
  1. Federal Univ. of Paraná (Brazi); Univ. of São Paulo (Brazil)
  2. Univ. of São Paulo (Brazil)
  3. Federal Univ. of Paraná (Brazi)
  4. General Atomics, San Diego, CA (United States)
Publication Date:
Research Org.:
General Atomics, San Diego, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1540200
Alternate Identifier(s):
OSTI ID: 1469428
Grant/Contract Number:  
[FC02-04ER54698; SC0012706; FC02-04ER54698, DE- SC0012706]
Resource Type:
Accepted Manuscript
Journal Name:
Chaos: An Interdisciplinary Journal of Nonlinear Science
Additional Journal Information:
[ Journal Volume: 28; Journal Issue: 9]; Journal ID: ISSN 1054-1500
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS

Citation Formats

Ciro, David, Caldas, Iberê L., Viana, Ricardo L., and Evans, Todd E. Efficient manifolds tracing for planar maps. United States: N. p., 2018. Web. doi:10.1063/1.5027698.
Ciro, David, Caldas, Iberê L., Viana, Ricardo L., & Evans, Todd E. Efficient manifolds tracing for planar maps. United States. doi:10.1063/1.5027698.
Ciro, David, Caldas, Iberê L., Viana, Ricardo L., and Evans, Todd E. Tue . "Efficient manifolds tracing for planar maps". United States. doi:10.1063/1.5027698. https://www.osti.gov/servlets/purl/1540200.
@article{osti_1540200,
title = {Efficient manifolds tracing for planar maps},
author = {Ciro, David and Caldas, Iberê L. and Viana, Ricardo L. and Evans, Todd E.},
abstractNote = {Here, we present an exact calculation method and an approximation technique for tracing the invariant manifolds of unstable periodic orbits of planar maps. The exact approach relies on an adaptive refinement procedure that prevents redundant calculations occurring in other approaches, and the approximated method relies on a novel interpolation approach based on normal displacement functions. The resulting approximated manifold is precise when compared to the exact one, and its relative computational cost falls like the inverse of the manifold length. To present the tracing method, we obtain the invariant manifolds of the Chirikov-Taylor map, and as an application we illustrate the transition from homoclinic to heteroclinic chaos in the Duffing oscillator that leads from localized chaos to global chaotic motion.},
doi = {10.1063/1.5027698},
journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
number = [9],
volume = [28],
place = {United States},
year = {2018},
month = {9}
}

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