^{1,a)}, Jürgen Jost

^{1,b)}and Fatihcan M. Atay

^{1,c)}

### Abstract

We study coupled dynamics on networks using symbolic dynamics. The symbolic dynamics is defined by dividing the state space into a small number of regions (typically 2), and considering the relative frequencies of the transitions between those regions. It turns out that the global qualitative properties of the coupled dynamics can be classified into three different phases based on the synchronization of the variables and the homogeneity of the symbolic dynamics. Of particular interest is the homogeneous unsynchronized phase, where the coupled dynamics is in a chaotic unsynchronized state, but exhibits qualitative similar symbolic dynamics at all the nodes in the network. We refer to this dynamical behavior as symbolic synchronization. In this phase, the local symbolic dynamics of any arbitrarily selected node reflects global properties of the coupled dynamics, such as qualitative behavior of the largest Lyapunov exponent and phase synchronization. This phase depends mainly on the network architecture, and only to a smaller extent on the local chaotic dynamical function. We present results for two model dynamics, iterations of the one-dimensional logistic map and the two-dimensional Hénon map, as local dynamical function.

Nonlinear dynamical elements interacting with each other can lead to synchronization or other types of coherent behavior at the system scale. Coupled map models are one of the most widely accepted models to understand these behaviors in systems from many diverse fields such as physics, biology, ecology, etc. Their important feature is that the individual elements can already exhibit some complex behavior, for example, chaotic dynamics. The question then is how to detect coordination at larger scales beyond the simplest one, synchronization. An important tool in the analysis of dynamical systems are symbolic dynamics. We develop a new scheme of symbolic dynamics that is based on the special partitions of the phase space that prevent the occurrence of certain symbol sequences related to the characteristics of the dynamics. In particular, we report a new behavior of coupled dynamics, which we refer to as symbolic synchronization, i.e., synchronization of the nodes at the coarse grained level, whereas microscopically all elements behave differently. Through the framework of this symbolic dynamics, we detect various global properties of coupled dynamics on networks by using a scalar time series of any randomly selected node. A decisive advantage of our method is that the global properties are inferred by using a short time series, hence the method is computationally fast, does not depend on the size of the network, and is reasonably robust against external noise.

We acknowledge Dr. Wenlian Lu for useful discussions.

I. INTRODUCTION

II. MODEL AND DEFINITION OF SYMBOLIC DYNAMICS

III. DIFFERENT STATES OF THE COUPLED DYNAMICS

IV. HOMOGENEOUS PHASES AND COUPLED DYNAMICS ON NETWORK

A. Homogeneous phase and networkproperties

B. Symbolic synchronized phase and global properties of coupled dynamics

V. RELATION BETWEEN DYNAMICAL PHASES AND NETWORKPROPERTIES

VI. PHASE SYNCHRONIZATION: SYMBOLIC SYNCHRONIZATION

VII. COUPLED HÉNON MAPS

VIII. CONCLUSION

### Key Topics

- Synchronization
- 32.0
- Networks
- 31.0
- Chaotic dynamics
- 16.0
- Time series analysis
- 16.0
- Thermodynamic properties
- 9.0

## Figures

Examples of coupled networks showing all three phases. All figures are plotted for scale-free networks, generated by using the BA algorithm (Ref. ^{ 19 } ) of size and (a) average degree 2 (phase 1), (b) average degree 6, (c) average degree 10, (d) average degree 20. The axis represents the coupling strength and the axis gives the synchronization measure (엯) and the homogeneity measure (●) for the whole network. The largest Lyapunov exponent is plotted as a function of the coupling strength (see inset). Panels (a'), (b'), (c'), and (d') show exact values of the transition probability for different nodes as a function of the coupling strength. For clarity we plot only a few arbitrarily selected nodes.

Examples of coupled networks showing all three phases. All figures are plotted for scale-free networks, generated by using the BA algorithm (Ref. ^{ 19 } ) of size and (a) average degree 2 (phase 1), (b) average degree 6, (c) average degree 10, (d) average degree 20. The axis represents the coupling strength and the axis gives the synchronization measure (엯) and the homogeneity measure (●) for the whole network. The largest Lyapunov exponent is plotted as a function of the coupling strength (see inset). Panels (a'), (b'), (c'), and (d') show exact values of the transition probability for different nodes as a function of the coupling strength. For clarity we plot only a few arbitrarily selected nodes.

The global measure of coupled dynamics from the local symbolic dynamics. We take various networks having coupled dynamics in the phase two. The axis gives the coupling strength and the axis depicts (-) (the largest Lyapunov exponent for the coupled dynamics) as well as ( ) (transition probability for a randomly selected node). (a) For a nearest neighbor coupled network of size ; (b) for a three-nearest neighbor coupled network, ; (c) and (d) are for random and scale-free networks, respectively, with average degree 10 and .

The global measure of coupled dynamics from the local symbolic dynamics. We take various networks having coupled dynamics in the phase two. The axis gives the coupling strength and the axis depicts (-) (the largest Lyapunov exponent for the coupled dynamics) as well as ( ) (transition probability for a randomly selected node). (a) For a nearest neighbor coupled network of size ; (b) for a three-nearest neighbor coupled network, ; (c) and (d) are for random and scale-free networks, respectively, with average degree 10 and .

The measure of homogeneity as a function of the connectivity ratio .

The measure of homogeneity as a function of the connectivity ratio .

The ratio of the number phase synchronized clusters to the maximum possible clusters, and the transition probability for the coupled logistic map as a function of the coupling strength , (a) for a nearest neighbor coupled network with average degree 20 and , (b) for a scale-free network with average degree 10 and , (c) for a random network with average degree 10 and , (d) for a nearest neighbor coupled network with average degree 6 and , and the tent map as the local chaotic function.

The ratio of the number phase synchronized clusters to the maximum possible clusters, and the transition probability for the coupled logistic map as a function of the coupling strength , (a) for a nearest neighbor coupled network with average degree 20 and , (b) for a scale-free network with average degree 10 and , (c) for a random network with average degree 10 and , (d) for a nearest neighbor coupled network with average degree 6 and , and the tent map as the local chaotic function.

An illustration of the choice of the threshold as the point where sharply drops to near zero.

An illustration of the choice of the threshold as the point where sharply drops to near zero.

The transition probability measure for coupled Hénon maps. The axis displays the coupling strength and the axis shows the different transition probabilities and the measure of synchronization . (a) is for two coupled nodes and plots for the symbolic sequence of length 2. (b) and (c) are plotted for globally coupled networks with . (b) plots and (c) plots transition probabilities for symbolic sequences of length 3, namely (◻), (●), and (엯). The synchronized state is detected when all the transition probabilities are equal to those of the uncoupled map; i.e., the transition probabilities at the zero coupling strength. (d), (e), and (f) show the standard deviation (solid thick line) and (vertical dashed line) for these three transition probabilities of an arbitrary selected node with respect to the transition probabilities of the uncoupled function (solid line), i.e., for . is calculated for 20 simulations for the dynamics with different sets of random initial conditions. (d) For a globally connected network with ; (e) and (f) for a random network with , average degrees 10 and 2, respectively. The last panel is plotted to show the behavior of the transition probabilities when we do not get global synchrony even at large coupling strengths.

The transition probability measure for coupled Hénon maps. The axis displays the coupling strength and the axis shows the different transition probabilities and the measure of synchronization . (a) is for two coupled nodes and plots for the symbolic sequence of length 2. (b) and (c) are plotted for globally coupled networks with . (b) plots and (c) plots transition probabilities for symbolic sequences of length 3, namely (◻), (●), and (엯). The synchronized state is detected when all the transition probabilities are equal to those of the uncoupled map; i.e., the transition probabilities at the zero coupling strength. (d), (e), and (f) show the standard deviation (solid thick line) and (vertical dashed line) for these three transition probabilities of an arbitrary selected node with respect to the transition probabilities of the uncoupled function (solid line), i.e., for . is calculated for 20 simulations for the dynamics with different sets of random initial conditions. (d) For a globally connected network with ; (e) and (f) for a random network with , average degrees 10 and 2, respectively. The last panel is plotted to show the behavior of the transition probabilities when we do not get global synchrony even at large coupling strengths.

The measure of homogeneity as a function of connection ratio , with the Hénon map as the local dynamical function.

The measure of homogeneity as a function of connection ratio , with the Hénon map as the local dynamical function.

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