Global interactions, information flow, and chaos synchronization

G. Paredes, O. Alvarez-Llamoza, and M. G. Cosenza

Phys. Rev. E 88, 042920 – Published 22 October 2013

Abstract

We investigate the relationship between the emergence of chaos synchronization and the information flow in dynamical systems possessing homogeneous or heterogeneous global interactions whose origin can be external (driven systems) or internal (autonomous systems). By employing general models of coupled chaotic maps for such systems, we show that the presence of a homogeneous global field, either external or internal, for all times is not indispensable for achieving complete or generalized synchronization in a system of chaotic elements. Complete synchronization can also appear with heterogeneous global fields; it does not requires the simultaneous sharing of the field by all the elements in a system. We use the normalized mutual information and the information transfer between global and local variables to characterize complete and generalized synchronization. We show that these information measures can characterize both types of synchronized states and also allow us to discern the origin of a global interaction field. A synchronization state emerges when a sufficient amount of information provided by a field is shared by all the elements in the system, on the average over long times. Thus, the maximum value of the top-down information transfer can be used as a predictor of synchronization in a system, as a parameter is varied.

Received 6 July 2013

DOI:https://doi.org/10.1103/PhysRevE.88.042920

Authors & Affiliations

G. Paredes¹, O. Alvarez-Llamoza², and M. G. Cosenza³

¹LFAC, Universidad Nacional Experimental del Táchira, San Cristóbal, Venezuela
²Departamento de Física, FACYT, Universidad de Carabobo, Valencia, Venezuela
³Centro de Física Fundamental, Universidad de los Andes, Mérida, Venezuela

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 88, Iss. 4 — October 2013

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Top panels: homogeneous global interactions. (a) External field $g (y_{t})$ acting with probability $p$ on all elements. (b) Internal field $h (x_{t}^{j} | j \in Q_{t})$ acting with probability $p$ on all elements. Bottom panels: heterogeneous global interactions. (c) External field $g (y_{t})$ acting on a fraction $p$ of elements chosen at random at every time. (d) Internal field $h (x_{t}^{j} | j \in Q_{t})$ acting on a fraction $p$ of elements chosen at random at every time.Reuse & Permissions
Figure 2
Complete chaos synchronization in systems with homogeneous global fields. (a) $〈 δ 〉$ vs $ε$ , (b) mutual information $M_{y_{t}, x_{t}^{i}}$ vs $ε$ , and (c) information transfer $T_{y_{t}, x_{t}^{i}}$ vs $ε$ . On each panel, the continuous line corresponds to the homogeneous driven system Eq. (11) with $g = f$ and the dashed line corresponds to the homogeneous autonomous system Eqs. (12) and (13). Both information measures are calculated with $2 \times 10^{5}$ points in the time series, after discarding transients, and averaged over 50 randomly chosen maps. The number of states used to calculate the corresponding probability distributions is 100. The same conditions are used in Figs. 3 and 5. Fixed parameters: $p = 0.8$ , $N = 10^{4}$ , $q / N = 0.4$ .Reuse & Permissions
Figure 3
Generalized chaos synchronization in systems with homogeneous global fields. (a) $〈 σ 〉$ vs $ε$ (inset: $〈 δ 〉$ vs $ε$ ); (b) $M_{y_{t}, x_{t}^{i}}$ vs $ε$ ; and (c) $T_{y_{t}, x_{t}^{i}}$ vs $ε$ . On each panel, the continuous line corresponds to a homogeneously driven system Eq. (11) with $g$ given in Eq. (14) and the dashed line corresponds to the homogeneous autonomous system Eqs. (12) and (15). Fixed parameters are $p = 0.8$ , $N = 10^{4}$ , $q / N = 0.4$ .Reuse & Permissions
Figure 4
Regions for chaos synchronization on the plane $(p, ε)$ for systems with homogeneous global interactions. (a) Complete synchronization (CS) for both the homogeneous driven system Eq. (11) with $g = f = 4 x (1 - x)$ , and the homogeneous autonomous system Eqs. (12) and (13). The boundary of the region where complete synchronization takes place is given by $ε = 1 - e^{- λ_{f} / p}$ , with $λ_{f} = \ln 2$ for the map $f$ . (b) Generalized synchronization (GS) for both the homogeneously driven system Eqs. (11) and (14) (continuous line) and for the homogeneous autonomous system Eqs. (12) and (15) (dashed line), with $N = 10^{4}$ , $q / N = 0.4$ .Reuse & Permissions
Figure 5
Complete chaos synchronization in systems with heterogeneous global fields. (a) $〈 δ 〉$ vs $ε$ ; (b) $M_{y_{t}, x_{t}^{i}}$ vs $ε$ ; and (c) $T_{y_{t}, x_{t}^{i}}$ vs $ε$ . On each panel, the continuous line corresponds to the heterogeneously driven system Eq. (18) with $g = f$ , and the dashed line corresponds to the heterogeneous autonomous system Eqs. (19) and (13). Fixed parameters: $p = 0.8$ , $N = 10^{4}$ , $q / N = 0.4$ .Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics