From chaotic itinerancy to intermittent synchronization in complex networks

TL;DR

The paper addresses how synchronization emerges in complex networks when global order parameters obscure multiscale dynamics. It introduces a nodal symbolic-dynamics framework that yields a Chaotic Itinerancy index $\Phi_\sigma$ and an Intermittent Synchronization index $\Phi_E$, uncovering a two-stage route in which CI dominates at weak coupling and IS arises after CI collapses, with a robust crossover at $d/d_c \approx 2/3$ across models. A Lyapunov-spectrum analysis links these regimes to a hierarchical dimensional reduction of the attractor as coupling increases, with residence-time distributions displaying power-law behavior indicative of criticality. Experimental validation with electronic oscillators confirms the theoretical sequence and highlights the role of network structure in shaping itinerancy. The results offer a unified framework for route-to-synchronization dynamics with implications for brain information routing and reservoir computing.

Abstract

Although synchronization has been extensively studied, important processes underlying its emergence have remained hidden by the use of global order parameters. Here, we uncover how the route unfolds through a sequential transition between two well-known but previously unconnected phenomena: chaotic itinerancy (CI) and intermittent synchronization (IS). Using a new symbolic dynamics, we show that CI emerges as a collective yet unsynchronized exploration of different domains of the high-dimensional attractor, whose dimension is reduced as the coupling increases, ultimately collapsing back into the reference chaotic attractor of an individual unit. At this stage, the IS can emerge as irregular alternations between synchronous and asynchronous phases. The two phenomena are therefore mutually exclusive, each dominating a distinct coupling interval and governed by different mechanisms. Network structural heterogeneity enhances itinerant behavior since access to different domains of the attractor depends on the nodes' topological roles. The CI--IS crossover occurs within a consistent coupling interval across models and topologies. Experiments on electronic oscillator networks confirm this two-step process, establishing a unified framework for the route to synchronization in complex systems.

Figures (8)

• Figure 1: Examples of CI and IS. (a) Mean synchronization error $\langle E \rangle$ of a network of Rössler units, highlighting two points along the route to synchronization: the red dot marking a coupling regime where chaotic itinerancy (CI) takes place, and at higher coupling, the blue square pointing out a regime of intermittent synchronization (IS). (b) Evidence of the CI marked as a red point in (a) ($d=0.08$), showing the maxima of the $x$ variable of two arbitrary nodes in the network (shifted for visualization). (c) Evidence of the IS marked as a blue square in (a) ($d=0.28$) using the instantaneous synchronization error $E(t)$. (d) Example of CI showing the $x$ variable (shifted for clarity) of two arbitrary Lorenz nodes from a larger clique configuration. (e) Globally coupled $N$=100 logistic maps as an example of a discrete dynamical system showing CI. Models used in Eq. \ref{['eq:dynet']}: (a) and (b) $N=100$ Rössler nodes coupled in a scale-free configuration with average node degree $\langle k\rangle=4$, with $\mathbf{f}(\mathbf{x}) = \left[-y-z,x+ a y,z(x-4)+2\right]$ and $\mathbf{h}(\mathbf{x})=[0,y,0]$, with $a=0.432$; (d) a clique graph of $N=5$ Lorenz systems with $\mathbf{f}(\mathbf{x}) = \left[-10(y-x),x(28-z)-y,xy-2z\right]$, $\mathbf{h}(\mathbf{x})=[0,y,0]$. Parameters used in (d) for globally coupled Logistic maps, $y_i(t+1)=f\left((1-d)y_i(t)+\frac{d}{N}\sum_{j=1}^N y_j(t)\right)$ with $f(x)=4(1-x^2)$ and $d=0.1$.
• Figure 2: Symbolic encoding procedure used to characterize chaotic itinerancy, illustrated for two arbitrary nodes in a network of coupled Rössler oscillators. (a, b) Normalized sampled time series $\hat{X}_i$ and $\hat{X}_j$, corresponding to the scalar variables $X_i(t)$ and $X_j(t)$ of two arbitrary nodes, obtained here as the sequence of maxima of the Rössler variable $x$. The data points are distributed over a grid (gray lines) composed of $N_{\tau} \times N_{\beta}$ cells, where each time window $\tau$ contains $D$ consecutive samples. In this example, $D = 100$ and only 20 time windows (out of the total $N_{\tau}$) are shown. (c, d) Binary occupancy maps $\Pi^{(i)}_{\tau,\beta}$ for the nodes shown in panels (a) and (b). Each cell $(\tau, \beta)$ is marked as visited ($\Pi^{(i)}_{\tau,\beta} = 1$, yellow) if it contains more than $\theta=2$ points, and as empty ($\Pi^{(i)}_{\tau,\beta} = 0$, blue) otherwise. (e, f) Symbolic state $\sigma^{(i)}_{\tau}$ for each time window, defined as the number of active cells visited by node $i$ during window $\tau$. In the example, full exploration of the spatial partition corresponds to $\sigma^{(i)}_{\tau} = N_{\beta} = 6$. (g) Symbolic raster plot $\boldsymbol{\sigma} = \{ \sigma^{(i)}_{\tau} \}$ showing the symbolic evolution of all $N$ nodes in the network. Example shown corresponds to a scale-free (SF) network of $N = 100$ coupled Rössler oscillators with coupling strength $d = 0.08$ and parameter $a = 0.432$.
• Figure 3: Synchronization route and emergence of chaotic itinerancy (CI) and intermittent synchronization (IS) in a network of $N=100$ chaotic Rössler oscillators. (a) Global synchronization error $E$ (green), chaotic itinerancy index $\Phi_\sigma$ (red), and intermittent synchronization index $\Phi_E$ (blue) as a function of the coupling strength $d$. The $N$ largest Lyapunov exponents are also plotted with different colors of gray and line styles as indicated in the legend. The inset zooms the coupling interval where CI vanishes and gives rise to IS until the system reaches global synchrony with $\lambda_1>0$, $\lambda_2=0$, $\lambda_3<0$. (b) Log-log probability distribution of the residence duration $n_\tau$ in the most developed chaotic symbolic state $\sigma=N_{\beta}$, at $d$ = 0.16, red circle in (a), showing power-law behavior. The solid straight-line shows the best fit to a power law $P(n_\tau) \sim n_\tau^{-1.73}$, and the dashed straight-lines bounding the data correspond to exponents $2$ (upper) and $1.5$ (bottom). (c) Log-log probability distribution of the coherent phase durations $t_\ell$ during IS, at $d$ = 0.27 (blue square in (a)), following a $P(t_\ell) \sim t_\ell^{-3/2}$ scaling typical of on-off intermittency. Ensemble averages are computed over 30 independent realizations of initial conditions. Rest of parameters: $D = 200$, $N_\beta = 6$, $\theta = 3$. The network structure is an instance of a network with scale-free degree distribution and average degree 4.
• Figure 4: Comparison of the chaotic itinerancy index $\Phi_\sigma$ (red) and intermittent synchronization index $\Phi_E$ (blue) for different combinations of network structures, sizes, and dynamical systems, distinguished by linestyles as indicated in the legend. The corresponding synchronization error $E$ (green) is also shown. In all cases, the coupling strength $d$ is normalized by the critical synchronization threshold $d_c$ of each network. Remarkably, despite the diversity of systems, the crossover point consistently occurs at $d/d_c \approx 2/3$.
• Figure 5: Analysis of the symbolic state realization for the network used in Fig. \ref{['fig:joint_num_SFER']}. (a) Realization probability of symbol $\Omega_\beta$ for each symbol $\beta = 1, \dots, N_\beta = 6$, and (b) average symbolic state per degree class $\langle \sigma \rangle_k$ as a function of coupling $d$.