Synchronization of extended chaotic systems

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Alessandro Torcini and Massimo Cencini (2013), Scholarpedia, 8(6):30650. doi:10.4249/scholarpedia.30650 revision #134083 [link to/cite this article]
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Curator: Alessandro Torcini

Synchronization is ubiquitous in Nature: neuronal populations, cardiac pacemakers, Josephson circuits, power-grid networks, lasers and even coupled chaotic systems can synchronize during their activity. Remarkably, all these different phenomena can be described within the common framework of nonlinear dynamics (Pikovsky et al, 2001).

Synchronization of spatially extended chaotic systems is very interesting as it bridges statistical mechanics and nonlinear dynamics. In particular, the phenomenology of the synchronization transition (ST) can be put in direct correspondence with non-equilibrium critical phenomena. This parallel grounds on the erratic nature of the synchronized state, where chaos plays the role of thermal noise typical of statistical mechanics systems. In the last decade, an ongoing research activity has been devoted to relate chaotic STs to non-equilibrium phase transitions.



Figure 1: Spatiotemporal evolution of the synchronization error (i.e. the local difference between two replicas) in a 1d diffusive CMLs: (left) Tent Map and (right) Bernoulli Shift, corresponding to MN and DP universality classes, respectively. Color coding: the synchronization error decreases from yellow to violet; black means complete synchronization.

Following preliminary suggestions reported in (Grassberger, 1999), the first evidence of a direct connection between the synchronization of spatially extended chaotic systems and the critical properties of non-equilibrium phase transitions has been established in (Baroni et al., 2001) and in (Ahlers and Pikovsky, 2002) for stochastically driven and deterministically coupled replicas of diffusively coupled extended systems, respectively.

Noticeably, the synchronization transition (ST) of diffusively coupled chaotic systems has been found to belong to only two universality classes -- Multiplicative Noise (MN) and Directed Percolation (DP). These two universality classes of non-equilibrium phase transitions were originally identified in completely different contexts, namely epidemics spreading and pinning/depinning of interfaces to/from a substrate (Hinrichsen, 2000; Muñoz, 2004). So far no experimental evidence of the MN class has been reported, while only recently DP critical exponents have been experimentally measured in one (Rupp et al. 2003) and two spatial dimensions (Takeuchi et al. 2007).

The critical properties of STs have been mainly studied for diffusively coupled map lattices (CML) (Kaneko, 1993), which are prototype models for systems exhibiting spatio-temporal chaos. Analogously to discrete-time maps (such as the logistic or the Hénon map), which somehow reproduce continuous flows, CMLs can be considered as time and space discrete versions of partial differential equations. In particular, depending on the prevalence of linear or nonlinear instabilities in the system (see e.g. Cencini and Torcini (2001)), the synchronization transition belongs to the MN or the DP universality class, respectively. An example of the different dynamical evolution leading to synchronization in the two cases is shown Fig. 1. These results originally obtained for identical replicas of one-dimensional CMLs (Alhers and Pikovsky, 2002) were further confirmed in two-dimensions (Ginelli et al, 2009).

A possible objection in studying identical replicas of coupled extended systems is that in realistic experimental settings it would be impossible to have two identical copies of a system. This situation can be theoretically mimicked by studying coupled imperfect replicas of a system (e.g. by imposing a parameter mismatch in the single elements composing the extended systems). In this framework it has been shown that the presence of a mismatch is akin to the effect of an external field and, thus, can be still accounted in the framework of non-equilibrium phase transitions (Ginelli et al, 2009).

The link between non-equilibrium phase transitions and the ST has been extended to the case of spatially chaotic systems with long range interactions (Tessone et al, 2006; Cencini et al, 2008). In this case, depending on the prevalence of linear vs. nonlinear instabilities, two continuous families of universality classes whose critical properties depends on the interaction range have been identified.

Linear versus nonlinear propagation mechanisms

Figure 2: Spatiotemporal evolution of a disturbance initially localized in the center of an one-dimensional diffusively coupled CML of Bernoulli maps $f(x)=r x$ mod $1$ with $r=1.03$. When the perturbation becomes nonlinear (i.e. $E_i(t)\sim {\cal O}(1)$), it starts propagating with a nonlinear propagation speed $V_{NL}$ larger than the velocity $V_L$, predicted by linear stability analysis. (see Cencini and Torcini, 2001)
Figure 3: Summary of critical indices $\delta,\beta,z$ characterizing the synchronization transition in 1d CML of Tent and Bernoulli Shift maps. The exponents well agree with the best estimate of the MN (Tent Maps) and DP (Bernoulli Shift) critical exponents.

In deterministic chaos, sensitive dependence on initial conditions leads to an exponential amplification of infinitesimally small errors, thus inducing a flow of information in bit space from "insignificant" towards "significant" digits. In spatially extended chaotic systems, information can flow also in real space mediated by the spatial coupling (Grassberger,1989). Bit-space information flow reflects into the exponential growth rate of infinitesimal perturbations, quantified by the maximal Lyapunov exponent $\lambda$ and the Kolmogorov-Sinai entropy. The spatial information flow, instead, can be quantified in terms of the maximal propagation velocity of disturbances $V_I$ (i.e. the velocity of propagation of an initially localized perturbation), in analogy to damage spreading in cellular automata and epidemics spreading. An alternative description can be obtained in terms of the comoving Lyapunov Exponents (Deissler and Kaneko, 1987), which generalize the usual LEs to a comoving reference frame. When the information flow is driven by linear instabilities there is a tight link between $V_I$ and the comoving LEs: $V_I$ coincides with the velocity $V_L$ of the reference frame where an infinitesimal perturbation is seen neither growing nor decreasing, i.e. characterized by a vanishing comoving LE.

However, there are relevant systems for which this identification dramatically fails. For instance, a rather intriguing phenomenon -- termed stable chaos (Politi et al, 1993) -- has been observed where even linearly stable systems ($\lambda < 0$) display an erratic behavior with finite propagation velocity ($V_I > 0$). The origin of stable chaos has been traced back to the response of the system to finite perturbations. Indeed, the interplay between diffusive coupling and amplification of finite disturbances can lead to the prevalence of nonlinear mechanisms over linear ones, so that infinitesimal disturbances propagate slower than finite amplitude ones (Cencini and Torcini, 2001). In particular, a necessary condition to observe such phenomena is that the local dynamics should be almost discontinuous, this amounts for one dimensional maps to exhibit a large first derivative. An example of this kind of map is the well renowned Bernoulli Map.

The importance of nonlinear propagation mechanisms is demonstrated in Fig.2: when the perturbation front is pushed by nonlinear effects, associated to ${\cal O}(1)$ perturbations, the induced propagation velocity $V_{NL}$ becomes larger than that associated to the linear evolution of infinitesimal perturbations $V_L$, corresponding to vanishing comoving LE. Stable chaos is an extreme instance of this phenomenon, in which $V_{NL}>V_L\equiv 0$. While for typical systems, dominated by linear instabilities, $V_{NL}\equiv V_{L}$.

Remarkably, the above described phenomenology can be put in correspondence with the propagation of fronts separating different phases as observed in liquid crystals, epidemics, crystal growth, chemical reactions, and biological aggregations (Van Saarloos, 2003): disturbance propagation dominated by linear or nonlinear instabilities corresponds to the so-called pulled or pushed front dynamics, respectively (Torcini et al., 1995; Cencini and Torcini, 2001).

Models and Indicators to characterize the Synchronization Transition

A simplified setting to study chaotic STs is represented by a system of two coupled replicas of one-dimensional CMLs: \begin{eqnarray} x_i(t+1) &=& (1-\gamma) F(\tilde{x_i}(t))+\gamma F(\tilde{y_i}(t))\quad, \nonumber\\ y_i(t+1) &=& (1-\gamma) F(\tilde{y_i}(t))+\gamma F(\tilde{x_i}(t)) \quad , \tag{1} \end{eqnarray} with $i = 1, \dots, L$ and where $\tilde{z_i}=\{\tilde{x_i}, \tilde{y_i}\}$ represents the spatial coupling, e.g. for diffusively coupled maps $\tilde{z_i}=(1-2\varepsilon) z_i+\varepsilon(z_{i-1}+z_{i+1})$ with $\varepsilon$ controlling the coupling intensity (typically $\varepsilon=1/3$); $t$ and $i$ are the discrete temporal and spatial indexes, respectively; $x_i(t),y_i(t) \in [0:1]$ are the state variables; $L$ is the lattice size, and periodic boundary conditions are assumed (i.e. $z_{i+L}(t)=z_i(t)$). The chosen local dynamics $F(x)$ is typically a chaotic map of the interval. As representative of maps with prevalence of linear versus nonlinear instabilities we consider the Tent map $F(x) \!=\! 1 - 2|x -1/2|$ and the Bernoulli Shift $F(x) \!=\! 2 x$ (mod 1), respectively. The parameter $\gamma$ controls the strength of the site-wise interaction between the two replicas. A synchronized state ($x_i(t) = y_i(t)$ for long time $t$) is achieved whenever the coupling exceeds a critical value $\gamma_c$.

As a matter of fact, the first studies of chaotic STs have been performed by considering two replicas of one-dimensional CMLs driven by the same realization of spatio-temporal noise, namely: \begin{eqnarray} x_i(t+1) &=& F(\tilde{x_i}(t))+\sigma \xi_i(t)\quad, \nonumber\\ y_i(t+1) &=& F(\tilde{y_i}(t))+\sigma \xi_i(t) \quad , \tag{2} \end{eqnarray} where $\xi_i(t)$ is a stochastic variable zero average and unitary distributed in the interval $[-1;1]$, for $\sigma > \sigma_c$ the two replicas eventually synchronizes. The universality classes describing the observed STs are the same identified for model (1) (Baroni et al., 2001). Therefore in the following we will limit to the latter model.

The degree of synchronization among the two replicas can be studied by considering the local synchronization error $e_i(t) = |x_i(t) - y_i(t)|$ and its spatial average \begin{equation} \rho_\gamma(t) = \frac{1}{L} \sum_{i=1}^L e_i(t) \end{equation} When the coupling $\gamma$ is larger than the critical value $\gamma_c$, for sufficiently long times $\rho_\gamma(t) \to 0$ and the two replicas fully synchronize. For $\gamma<\gamma_c$ a finite asymptotic value $\lim_{t \to \infty} \rho_\gamma(t) \equiv \rho^\ast_\gamma>0$ indicates a lack of synchronization. Therefore $\rho$ is a natural order parameter to characterize the ST, and the synchronized state represents the adsorbing state. As typical for adsorbing out-of-equilibrium phase transitions starting with a homogeneously active (completely desynchronized) initial state (Hinrichsen, 2000, Muñoz, 2004), the transition can be characterized in terms of the critical exponents $\delta$, $\beta$ and $z$ defined as follows \begin{equation} \rho^\ast_\gamma \sim (\gamma_c-\gamma)^\beta \quad (\mathrm{for}\; \gamma \le \gamma_c)\,, \qquad \rho_{\gamma_c}(t) \sim t^{-\delta} \,, \qquad \rho_{\gamma_c}(t) \sim L^{-\delta z} f(t/L^z)\,, \tag{3} \end{equation} where $f$ is a system-dependent function which controls the finite-size scaling behavior.

These scaling laws are summarized in Fig. 3 for one-dimensional diffusive coupled CMLs of Tent and Bernoulli maps.

Synchronization Transitions in Systems with Diffusive Coupling

Possible the best understood instance is the synchronization transition for diffusively coupled one-dimensional CMLs with spatial coupling restricted to nearest neighbors sites (Ahlers and Pikovsky, 2002; Droz and Lipowski, 2003; Bagnoli and Rechtman 2006), i.e. with reference to Eq. (1) this corresponds to $\tilde{z_i}=(1-2\varepsilon) z_i+\varepsilon(z_{i-1}+z_{i+1})$.

A dynamical order parameter able to locate the STs is the propagation velocity $V_I$ which is zero (finite) in the synchronized (desynchronized) phase. In continuous maps (like the Tent Map), whose dynamics is dominated by linear effects, $V_I$ vanishes together with the transverse Lyapunov exponent measuring the stability of the synchronized state with respect to infinitesimal perturbations (Pikovsky et al, 2001), meaning that the synchronized state is only a marginally adsorbing state for the dynamics. In this case, the ST is characterized by the critical indexes (3) of the Multiplicative Noise universality class (Tu et al, 1997; Kissinger et al. 2005), as shown in Fig. 3 and the table below. Conversely, in discontinuous (or quasi-discontinuous ) maps (like the Bernoulli shift), nonlinear effects prevail and the replicas synchronize for coupling $\gamma$ larger than that predicted by the linear analysis, i.e. for a definitely negative transverse Lyapunov exponent thus indicating that the synchronized state becomes a true adsorbing state for the dynamics (Ginelli et al, 2003a). For these maps the ST cannot be anymore described within a linear framework and the critical properties of the ST belong to the Directed Percolation universality class (see Fig. 3 and table below). A similar phenomenology is observed in the synchronization of stable chaotic maps and cellular automata (Bagnoli and Cecconi, 2001). These findings on CMLs have been also confirmed by the analysis of stochastic models (the so-called Random Multiplier (RM) model) mimicking the synchronization of continuous/discontinuous maps (Ginelli et al, 2003a; Cencini et al., 2008) and further validated in two-dimensional diffusively coupled CMLs (Ginelli et al, 2009, see also table below).

Remarkably, Muñoz and Pastor-Satorras (2003) proposed a Langevin equation mimicking a Kardar-Parisi-Zhang equation (KPZ) with an attractive wall, able to encompass both the linear and nonlinear phenomenologies. This represents a highly plausible conjecture for an unified theoretical framework for the analysis of the ST in diffusively coupled chaotic extended systems. Furthermore, a microscopic growth model, the single-step plus-wall (SSW) model, describing the depinning transition of an interface from an attractive hard wall, has been shown to be able to display a continuous transition from DP to MN critical behavior by varying a single control parameter (Ginelli et al., 2003b and 2009).

The critical properties of the ST in locally coupled chaotic systems with time delays have been also subject of investigations revealing the occurrence of first order transitions (Mohanty, 2004) as well as of continuous MN behaviors (Szendro and López, 2005).

Critical exponents for nearest-neighbor coupled Tent and Bernoulli maps as well as for the corresponding stochastic models. The best estimates of the critical indexes for DP in 1d and 2d and MN universality classes in 1d are reported, as well as the exponents derived for the SSW model and the KPZ equation in 2d.
One Dimension $\delta$ $\beta$ $z$ Two Dimensions $\delta$ $\beta$ $z$
Bernoulli 0.159(1) 0.27(1) 1.58(4) Bernoulli 0.449(4) 0.584(9) 1.77(3)
DP (Jensen, 1999) 0.159464(6) 0.276486(6) 1.580745(6) DP (Hinrichsen, 2000) 0.451(6) 0.584(4) 1.76(3)
Tent 1.275(15) 1.70(8) 1.5(1) Tent 1.81(5) 2.19(9) 1.55(8)
MN (Tu et al., 1997) 1.10(5) 1.70(5) 1.53(7) SSW (Ginelli et al., 2009) 1.80(4) 2.36(9) 1.7(1)
MN (Kissinger et al., 2005) 1.184(10) 1.776(15) --- KPZ (Tu et al., 1997) --- --- 1.607(3)

Synchronization Transitions in Systems with Long Range Interactions

Recently, synchronization has been also studied for chaotic systems with long-range interactions, which are relevant to many real contexts, such as disease spread via aviation traffic, neuronal populations, Josephson junctions and cardiac pacemaker cells. In particular, it has been considered the case of CMLs with coupling decaying as a power-law, with reference to Eq. (1) this means choosing (Tessone et al, 2006 and Cencini et al, 2008) \begin{equation} \tilde{z}_i = (1-2\varepsilon) z_i+\frac{\varepsilon}{\eta(\sigma)} \sum_{j=1}^{L} \frac{z_{i-j}+z_{i+j}}{j^{1+\sigma}} \tag{4} \end{equation} where $\sigma$ controls the range of the interactions, $\eta$ is a suitable normalization factor. Periodic boundary conditions are assumed and the sum in ((4)) has to be understood as extended to infinity by winding around the periodic chain. The properties of the ST strongly depends on the interaction range, controlled by $\sigma$, ranging from diffusive-like phenomenology (for large enough $\sigma$) and mean-field-like (see Fig. 4 for an illustration of the evolution to the synchronized state at varying $\sigma$ in the case of local dynamics given by the Bernoulli shift map).

Figure 4: Spatiotemporal evolution of the synchronization error $e_i(t)$ in a 1d CML of Bernoulli Shift at varying the range of interaction $\sigma$.
Figure 5: Critical indices $\beta$ and $\delta$ for in a 1d CML of long range coupled Bernoulli Shift maps as a function of interaction range $\sigma$. Notice that for $\sigma<0.5$ and $\sigma>\sigma_c$ the mean field and diffusive coupling limits are recovered. The black line shows the best known estimates of the critical indices for anomalous DP: they agree well with the numerically computed exponents characterizing the synchronization transition. Previous studies on Anomalous DP suggest $\sigma_c=2.0677(2)$ (Janssen et al, 1999; Hinrichsen and Howard, 1999)

In such a model, when the local dynamics is dominated by nonlinear mechanisms (e.g. with local dynamics given by Bernoulli shift), the STs have been found to belong to a family of universality classes known as Anomalous Directed Percolation (ADP), see Fig. 5. Anomalous DP has been previously identified for epidemic spreading whenever the infective agent can perform unrestricted Levy flights. Such processes, originally introduced in (Mollison, 1977), can be modeled by assuming, e.g. in $d=1$, that the disease propagates from an infected site to any other with a probability $P(r) \sim r^{-(1+\sigma)}$ algebraically decaying with the distance $r$ between the two sites, where $\sigma$ controls the interaction range. Hinrichsen and Howard (1999) have numerically shown for a stochastic lattice model (generalizing directed bond percolation) that the critical exponents vary continuously with $\sigma$. These findings confirm theoretical studies by Janssen et al (1999), indicating that usual DP should be recovered for sufficiently short-ranged coupling ($\sigma > \sigma_c \equiv 2.0677(2)$) and that a mean-field description should become exact for $\sigma < \sigma_m \equiv 0.5$ (for a recent and exhaustive review see Hinrichsen, (2007)).

When the dynamics is ruled by continuous maps, i.e. dominated by linear instabilities, for sufficiently large $\sigma$ the diffusive coupled CMLs result is obtained with ST belonging to the MN universality class, for smaller $\sigma$, when long range interactions are at play, critical exponents continuously varying with $\sigma$ have been measured. However, a theoretical framework for this family of universality classes is still lacking and demands for future studies (Cencini et al, 2008).


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