### 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 Poincare 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. (C) 2004 Elsevier Ltd. All rights reserved.

Original language | English |
---|---|

Pages (from-to) | 947-956 |

Number of pages | 9 |

Journal | Chaos, Solitons & Fractals |

Volume | 24 |

DOIs | |

Publication status | Published - 2005 |

### Keywords

- OPEN HYDRODYNAMICAL FLOWS
- LINEAR-ARRAY
- VORTICES
- SYSTEMS
- INSTABILITY
- BOUNDARIES
- ADVECTION
- DYNAMICS
- SADDLES

### Cite this

*Chaos, Solitons & Fractals*,

*24*, 947-956. https://doi.org/10.1016/j.chaos.2004.09.059

**Intersections of Stable and Unstable Manifolds: the Skeleton of Lagrangian Chaos.** / Grebogi, Celso; Kurths, J.; Gellert, M.; Witt, A.; Feudel, F.

Research output: Contribution to journal › Article

*Chaos, Solitons & Fractals*, vol. 24, pp. 947-956. https://doi.org/10.1016/j.chaos.2004.09.059

}

TY - JOUR

T1 - Intersections of Stable and Unstable Manifolds: the Skeleton of Lagrangian Chaos

AU - Grebogi, Celso

AU - Kurths, J.

AU - Gellert, M.

AU - Witt, A.

AU - Feudel, F.

PY - 2005

Y1 - 2005

N2 - 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 Poincare 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. (C) 2004 Elsevier Ltd. All rights reserved.

AB - 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 Poincare 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. (C) 2004 Elsevier Ltd. All rights reserved.

KW - OPEN HYDRODYNAMICAL FLOWS

KW - LINEAR-ARRAY

KW - VORTICES

KW - SYSTEMS

KW - INSTABILITY

KW - BOUNDARIES

KW - ADVECTION

KW - DYNAMICS

KW - SADDLES

U2 - 10.1016/j.chaos.2004.09.059

DO - 10.1016/j.chaos.2004.09.059

M3 - Article

VL - 24

SP - 947

EP - 956

JO - Chaos, Solitons & Fractals

JF - Chaos, Solitons & Fractals

SN - 0960-0779

ER -