### Abstract

Standard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise, The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time-1 Poincare map. Then we add a small amount of noise, so that on each iteration the Poincare map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x(0) and an initial time t(0). We discuss when the set of initial points at a time t(0) whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)>x(0) for all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with noise. (C) 1997 American Institute of Physics.

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

Pages (from-to) | 125-138 |

Number of pages | 14 |

Journal | Chaos |

Volume | 7 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1997 |

### Keywords

- open hydrodynamical flows

### Cite this

*Chaos*,

*7*(1), 125-138. https://doi.org/10.1063/1.166244

**Indecomposable continua in dynamical systems with noise : Fluid flow past an array of cylinders.** / Sanjuan, Miguel A. F.; Kennedy, Judy; Grebogi, Celso; Yorke, James A.

Research output: Contribution to journal › Article

*Chaos*, vol. 7, no. 1, pp. 125-138. https://doi.org/10.1063/1.166244

}

TY - JOUR

T1 - Indecomposable continua in dynamical systems with noise

T2 - Fluid flow past an array of cylinders

AU - Sanjuan, Miguel A. F.

AU - Kennedy, Judy

AU - Grebogi, Celso

AU - Yorke, James A.

PY - 1997/3

Y1 - 1997/3

N2 - Standard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise, The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time-1 Poincare map. Then we add a small amount of noise, so that on each iteration the Poincare map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x(0) and an initial time t(0). We discuss when the set of initial points at a time t(0) whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)>x(0) for all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with noise. (C) 1997 American Institute of Physics.

AB - Standard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise, The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time-1 Poincare map. Then we add a small amount of noise, so that on each iteration the Poincare map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x(0) and an initial time t(0). We discuss when the set of initial points at a time t(0) whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)>x(0) for all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with noise. (C) 1997 American Institute of Physics.

KW - open hydrodynamical flows

U2 - 10.1063/1.166244

DO - 10.1063/1.166244

M3 - Article

VL - 7

SP - 125

EP - 138

JO - Chaos

JF - Chaos

SN - 1054-1500

IS - 1

ER -