### Abstract

Language | English |
---|---|

Journal | Chaos, Solitons and Fractals |

Volume | 19 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 2009 |

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### Keywords

- chaos
- nonlinear dynamical systems
- partial differential equations

### Cite this

*Chaos, Solitons and Fractals*,

*19*(3). https://doi.org/10.1063/1.3222860

}

*Chaos, Solitons and Fractals*, vol. 19, no. 3. https://doi.org/10.1063/1.3222860

**On the use of stabilizing transformations for detecting unstable periodic orbits in high-dimensional flows.** / Crofts, J.J.; Davidchack, R.L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the use of stabilizing transformations for detecting unstable periodic orbits in high-dimensional flows

AU - Crofts, J.J.

AU - Davidchack, R.L.

PY - 2009/9

Y1 - 2009/9

N2 - We explore the possibility of extending the stabilizing transformations approach [ J. J. Crofts and R. L. Davidchack, SIAM J. Sci. Comput. (USA) 28, 1275 (2006) ]. to the problem of locating large numbers of unstable periodic orbits in high-dimensional flows, in particular those that result from spatial discretization of partial differential equations. The approach has been shown to be highly efficient when detecting large sets of periodic orbits in low-dimensional maps. Extension to low-dimensional flows has been achieved by the use of an appropriate Poincaré surface of section [ D. Pingel, P. Schmelcher, and F. K. Diakonos, Phys. Rep. 400, 67 (2004) ]. For the case of high-dimensional flows, we show that it is more efficient to apply stabilizing transformations directly to the flows without the use of the Poincaré surface of section. We use the proposed approach to find many unstable periodic orbits in the model example of a chaotic spatially extended system-the Kuramoto-Sivashinsky equation. The performance of the proposed method is compared against other methods such as Newton-Armijo and Levenberg-Marquardt algorithms. In the latter case, we also argue that the Levenberg-Marquardt algorithm, or any other optimization-based approach, is more efficient and simpler in implementation when applied directly to the detection of periodic orbits in high-dimensional flows without the use of the Poincaré surface of section or other additional constraints.

AB - We explore the possibility of extending the stabilizing transformations approach [ J. J. Crofts and R. L. Davidchack, SIAM J. Sci. Comput. (USA) 28, 1275 (2006) ]. to the problem of locating large numbers of unstable periodic orbits in high-dimensional flows, in particular those that result from spatial discretization of partial differential equations. The approach has been shown to be highly efficient when detecting large sets of periodic orbits in low-dimensional maps. Extension to low-dimensional flows has been achieved by the use of an appropriate Poincaré surface of section [ D. Pingel, P. Schmelcher, and F. K. Diakonos, Phys. Rep. 400, 67 (2004) ]. For the case of high-dimensional flows, we show that it is more efficient to apply stabilizing transformations directly to the flows without the use of the Poincaré surface of section. We use the proposed approach to find many unstable periodic orbits in the model example of a chaotic spatially extended system-the Kuramoto-Sivashinsky equation. The performance of the proposed method is compared against other methods such as Newton-Armijo and Levenberg-Marquardt algorithms. In the latter case, we also argue that the Levenberg-Marquardt algorithm, or any other optimization-based approach, is more efficient and simpler in implementation when applied directly to the detection of periodic orbits in high-dimensional flows without the use of the Poincaré surface of section or other additional constraints.

KW - chaos

KW - nonlinear dynamical systems

KW - partial differential equations

UR - http://dx.doi.org/10.1063/1.3222860

U2 - 10.1063/1.3222860

DO - 10.1063/1.3222860

M3 - Article

VL - 19

JO - Chaos, Solitons and Fractals

T2 - Chaos, Solitons and Fractals

JF - Chaos, Solitons and Fractals

SN - 0960-0779

IS - 3

ER -