https://doi.org/10.1007/s100530050538
Reconstruction of systems with delayed feedback: I. Theory
1
Istituto Nazionale di Ottica Applicata, Largo E. Fermi 6,
50125 Firenze, Italy
2
Instituto Pluridisciplinar, Paseo Juan XXIII 1, 28040
Madrid, Spain
3
Max-Planck-Institut für Physik komplexer Systeme,
Nöthnitzer Str. 38, 01187 Dresden, Germany
4
INFM, Unità di Firenze, 50125 Firenze, Italy
Received:
14
June
1999
Revised:
4
November
1999
Published online: 15 May 2000
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.
PACS: 02.30.Ks – Delay and functional equations / 05.45.Tp – Time series analysis
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2000
