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FROM SINGLE WELL CHAOS TO CROSS WELL CHAOS: A DETAILED EXPLANATION IN TERMS OF MANIFOLD INTERSECTIONS

Publication ,  Journal Article
KATZ, AL; DOWELL, EH
Published in: International Journal of Bifurcation and Chaos
August 1994

The study of stable and unstable manifolds, and their intersections with each other, is a powerful technique for interpreting complex bifurcations of nonlinear systems. The escape phenomenon in the twin-well Duffing oscillator is one such bifurcation that is elucidated through the analysis of manifold intersections. In this paper, two escape scenarios in the twin-well Duffing oscillator are presented. In each scenario, the relevant manifold structures are examined for parameter values on either side of the escape bifurcation. Included is a description of the role of the hilltop saddle stable manifolds, which are known to separate the single well basins (should single well attractors exist). In each of the two bifurcation scenarios, it is shown through a detailed analysis of Poincaré maps that a homoclinic intersection of the manifolds of a specific period-3 saddle implies the destruction of the single well chaotic attractor. Although the Duffing oscillator is used to illustrate the ideas advanced here, it is thought that the approach will be useful for a variety of dynamical systems.

Duke Scholars

Published In

International Journal of Bifurcation and Chaos

DOI

EISSN

1793-6551

ISSN

0218-1274

Publication Date

August 1994

Volume

04

Issue

04

Start / End Page

933 / 941

Publisher

World Scientific Pub Co Pte Lt

Related Subject Headings

  • Fluids & Plasmas
  • 4903 Numerical and computational mathematics
  • 4901 Applied mathematics
  • 0913 Mechanical Engineering
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics
 

Citation

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KATZ, A. L., & DOWELL, E. H. (1994). FROM SINGLE WELL CHAOS TO CROSS WELL CHAOS: A DETAILED EXPLANATION IN TERMS OF MANIFOLD INTERSECTIONS. International Journal of Bifurcation and Chaos, 04(04), 933–941. https://doi.org/10.1142/s0218127494000666
KATZ, ANDREW L., and EARL H. DOWELL. “FROM SINGLE WELL CHAOS TO CROSS WELL CHAOS: A DETAILED EXPLANATION IN TERMS OF MANIFOLD INTERSECTIONS.” International Journal of Bifurcation and Chaos 04, no. 04 (August 1994): 933–41. https://doi.org/10.1142/s0218127494000666.
KATZ AL, DOWELL EH. FROM SINGLE WELL CHAOS TO CROSS WELL CHAOS: A DETAILED EXPLANATION IN TERMS OF MANIFOLD INTERSECTIONS. International Journal of Bifurcation and Chaos. 1994 Aug;04(04):933–41.
KATZ, ANDREW L., and EARL H. DOWELL. “FROM SINGLE WELL CHAOS TO CROSS WELL CHAOS: A DETAILED EXPLANATION IN TERMS OF MANIFOLD INTERSECTIONS.” International Journal of Bifurcation and Chaos, vol. 04, no. 04, World Scientific Pub Co Pte Lt, Aug. 1994, pp. 933–41. Crossref, doi:10.1142/s0218127494000666.
KATZ AL, DOWELL EH. FROM SINGLE WELL CHAOS TO CROSS WELL CHAOS: A DETAILED EXPLANATION IN TERMS OF MANIFOLD INTERSECTIONS. International Journal of Bifurcation and Chaos. World Scientific Pub Co Pte Lt; 1994 Aug;04(04):933–941.
Journal cover image

Published In

International Journal of Bifurcation and Chaos

DOI

EISSN

1793-6551

ISSN

0218-1274

Publication Date

August 1994

Volume

04

Issue

04

Start / End Page

933 / 941

Publisher

World Scientific Pub Co Pte Lt

Related Subject Headings

  • Fluids & Plasmas
  • 4903 Numerical and computational mathematics
  • 4901 Applied mathematics
  • 0913 Mechanical Engineering
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics