### Abstract

High-dimensional chaos displayed by multi-component systems with a single time-delayed feedback is shown to be accessible to time series analysis of a scalar variable only. The mapping of the original dynamics onto scalar time-delay systems defined on sufficiently high dimensional spaces is thoroughly discussed. The dimension of the "embedding" space turns out to be independent of the delay time and thus of the dimensionality of the attractor dynamics. As a consequence, the procedure described in the present paper turns out to be definitely advantageous with respect to the standard embedding technique in the case of high-dimensional chaos, when the latter is practically unapplicable. The mapping is not exact when delayed maps are used to reproduce the dynamics of time-continuous systems, but the errors can be kept under control. In this context, the approximation of delay-differential equations is discussed with reference to different classes of maps. Appropriate tools to estimate the a priori unknown delay time and the number of hidden components are introduced. The generalized Mackey-Glass system is investigated in detail as a testing ground for the theoretical considerations.

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

Pages (from-to) | 165-176 |

Number of pages | 12 |

Journal | European Physical Journal D |

Volume | 10 |

Issue number | 2 |

Publication status | Published - May 2000 |

### Keywords

- TIME-SERIES DATA
- CHAOTIC DYNAMICS
- ATTRACTORS
- MECHANISM
- RECOVERY
- LASER

### Cite this

*European Physical Journal D*,

*10*(2), 165-176.

**Reconstruction of systems with delayed feedback: I. Theory.** / Bunner, M J ; Ciofini, M ; Giaquinta, A ; Hegger, R ; Kantz, H ; Meucci, R ; Politi, A .

Research output: Contribution to journal › Article

*European Physical Journal D*, vol. 10, no. 2, pp. 165-176.

}

TY - JOUR

T1 - Reconstruction of systems with delayed feedback: I. Theory

AU - Bunner, M J

AU - Ciofini, M

AU - Giaquinta, A

AU - Hegger, R

AU - Kantz, H

AU - Meucci, R

AU - Politi, A

PY - 2000/5

Y1 - 2000/5

N2 - High-dimensional chaos displayed by multi-component systems with a single time-delayed feedback is shown to be accessible to time series analysis of a scalar variable only. The mapping of the original dynamics onto scalar time-delay systems defined on sufficiently high dimensional spaces is thoroughly discussed. The dimension of the "embedding" space turns out to be independent of the delay time and thus of the dimensionality of the attractor dynamics. As a consequence, the procedure described in the present paper turns out to be definitely advantageous with respect to the standard embedding technique in the case of high-dimensional chaos, when the latter is practically unapplicable. The mapping is not exact when delayed maps are used to reproduce the dynamics of time-continuous systems, but the errors can be kept under control. In this context, the approximation of delay-differential equations is discussed with reference to different classes of maps. Appropriate tools to estimate the a priori unknown delay time and the number of hidden components are introduced. The generalized Mackey-Glass system is investigated in detail as a testing ground for the theoretical considerations.

AB - High-dimensional chaos displayed by multi-component systems with a single time-delayed feedback is shown to be accessible to time series analysis of a scalar variable only. The mapping of the original dynamics onto scalar time-delay systems defined on sufficiently high dimensional spaces is thoroughly discussed. The dimension of the "embedding" space turns out to be independent of the delay time and thus of the dimensionality of the attractor dynamics. As a consequence, the procedure described in the present paper turns out to be definitely advantageous with respect to the standard embedding technique in the case of high-dimensional chaos, when the latter is practically unapplicable. The mapping is not exact when delayed maps are used to reproduce the dynamics of time-continuous systems, but the errors can be kept under control. In this context, the approximation of delay-differential equations is discussed with reference to different classes of maps. Appropriate tools to estimate the a priori unknown delay time and the number of hidden components are introduced. The generalized Mackey-Glass system is investigated in detail as a testing ground for the theoretical considerations.

KW - TIME-SERIES DATA

KW - CHAOTIC DYNAMICS

KW - ATTRACTORS

KW - MECHANISM

KW - RECOVERY

KW - LASER

M3 - Article

VL - 10

SP - 165

EP - 176

JO - European Physical Journal D

JF - European Physical Journal D

SN - 1434-6060

IS - 2

ER -