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Intersections of stable and unstable manifolds: the skeleton of Lagrangian chaos

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Listed:
  • Feudel, F.
  • Witt, A.
  • Gellert, M.
  • Kurths, J.
  • Grebogi, C.
  • Sanjuán, M.A.F.

Abstract

We study Hamiltonian chaos generated by the dynamics of passive tracers moving in a two-dimensional fluid flow and describe the complex structure formed in a chaotic layer that separates a vortex region from the shear flow. The stable and unstable manifolds of unstable periodic orbits are computed. It is shown that their intersections in the Poincaré map as an invariant set of homoclinic points constitute the backbone of the chaotic layer. Special attention is paid to the finite time properties of the chaotic layer. In particular, finite time Lyapunov exponents are computed and a scaling law of the variance of their distribution is derived. Additionally, the box counting dimension as an effective dimension to characterize the fractal properties of the layer is estimated for different duration times of simulation. Its behavior in the asymptotic time limit is discussed. By computing the Lyapunov exponents and by applying methods of symbolic dynamics, the formation of the layer as a function of the external forcing strength, which in turn represents the perturbation of the originally integrable system, is characterized. In particular, it is shown that the capture of KAM tori by the layer has a remarkable influence on the averaged Lyapunov exponents.

Suggested Citation

  • Feudel, F. & Witt, A. & Gellert, M. & Kurths, J. & Grebogi, C. & Sanjuán, M.A.F., 2005. "Intersections of stable and unstable manifolds: the skeleton of Lagrangian chaos," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 947-956.
    • Handle: RePEc:eee:chsofr:v:24:y:2005:i:4:p:947-956
    DOI: 10.1016/j.chaos.2004.09.059
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    References listed on IDEAS

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    1. Hassan Aref, 1999. "Order in chaos," Nature, Nature, vol. 401(6755), pages 756-758, October.
    2. D. Rothstein & E. Henry & J. P. Gollub, 1999. "Persistent patterns in transient chaotic fluid mixing," Nature, Nature, vol. 401(6755), pages 770-772, October.
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    Cited by:

    1. Harle, M. & Feudel, U., 2007. "Hierarchy of islands in conservative systems yields multimodal distributions of FTLEs," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 130-137.
    2. Siewe, M. Siewe & Cao, Hongjun & Sanjuán, Miguel A.F., 2009. "Effect of nonlinear dissipation on the basin boundaries of a driven two-well Rayleigh–Duffing oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1092-1099.
    3. Chen, Yen-Sheng & Chang, Chien-Cheng, 2009. "Impulsive synchronization of Lipschitz chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1221-1228.

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