Stretched exponential dynamics of coupled logistic maps on a small-world network

TL;DR

This paper studies the transition to chaos in coupled logistic maps on small-world networks, revealing stretched exponential decay of persistence and the effects of network topology and coupling on relaxation dynamics.

Contribution

It introduces the use of persistence as an order parameter to analyze phase transitions and uncovers the role of network structure and coupling in anomalous relaxation behaviors.

Findings

Persistence decays as a stretched exponential on one critical line.

Crossover from nonexponential to exponential decay as rewiring probability approaches 1.

Distribution of trap times exhibits a long tail, indicating complex relaxation dynamics.

Abstract

We investigate the dynamic phase transition from partially or fully arrested state to spatiotemporal chaos in coupled logistic maps on a small-world network. Persistence of local variables in coarse grained sense acts as an excellent order parameter to study this transition. We investigate the phase diagram by varying coupling strength and small-world rewiring probability p of nonlocal connections. The persistent region is a compact region bounded by two critical lines where band-merging crisis occurs. On one critical line, the persistent sites shows a nonexponential (stretched exponential) decay for all p while for another one, it shows crossover from nonexponential to exponential behavior as p tends to 1. With an effectively antiferromagnetic coupling, coupling to two neighbors on either side leads to exchange frustration. Apart from exchange frustration, non-bipartite topology and nonlocal couplings in a small-world networks could be reason for anomalous relaxation. The distribution of trap times in asymptotic regime has a long tail as well. The dependence of temporal evolution of persistence on initial conditions is studied and a scaling form for persistence after waiting time is proposed. We present a simple possible model for this behavior.

Equations

$$f_{\beta }(\frac{t}{{\tau }_0})=\exp [-(\frac{t}{{\tau }_0}{)}^{\beta }]$$

$$x_{i,t+1}$$

$$x^{\ast }=1-1/\mu$$

$$f_{\beta }(\frac{t}{{\tau }_0})=P(t)\sim \exp (-t^{\beta })$$

$$P(t_w,t_w+t)=\frac{\exp (-r(t+t_w{)}^{\beta })}{\exp (-r{t}_w^{\beta })}$$

$$P(t_w,t_w+t)=-\exp (r{t}_w^{\beta })\exp ((1+t/t_w{)}^{\beta }-1)$$

$$\frac{\log (P(t_w,t_w+t))}{t_w^{\beta }}=-r((1+t/t_w{)}^{\beta }-1)$$

$$\frac{\log (P(t_w,t_w+t))}{t_w^{\beta }+C}=-r((1+t/t_w{)}^{\beta }-1)$$

Full text

• June 2017

Keywords: persistence, non-equilibrium phase transition, small-world network

1 Introduction

The study of spatially extended systems has progressed very well from last few decades. Since such a system is also a many body system, the entire toolbox of statistical mechanics is applicable to it. However, the properties are not so well studied from the statistical mechanics viewpoint. Spatiotemporal dynamical system is a non-equilibrium systems and dynamical phases and phase transitions therein, usually (but not always) have no equivalent analogue in equilibrium systems. They neither have a Hamiltonian nor obey detailed balance. It may not even happen that some coarse-grained quantities obey Langevin dynamics. Thus a reduced free energy description may not be derived for these systems [1].

Coupled map lattice (CML) is one of the most popular model to explore the dynamics of spatially extended systems [2]. It displays wide range of spatiotemporal patterns [3, 4, 5]. Networks is another emerging field which has received wide attention in recent times and CML studies have been extended to them. Connectivity patterns in a wide range of systems in nature as well as in society are described by complex networks. Examples include neural networks, food webs or communication media. They have a complex architecture and do not have d-dimensional lattice as underlying topology. Several dynamical phenomena ranging from the spread of diseases to the spread of rumors or the extinction of species can occur on them. Random network [6], small-world (SW) network [7] and scale-free network [8] are the most popular and extensively studied models. They have different degree distributions and are useful in different contexts[9]. In past few years, we have witnessed several studies on spatially extended dynamical systems on complex networks [10, 11, 12]. In SW networks, one starts with a regular network where each site is connected to $k$ nearest neighbors. Each link is rewired randomly with probability $p$ and connected to a randomly chosen site on lattice. This network has a high clustering coefficient like that of regular networks and has a low characteristic path length like random networks for small $p$. SW networks extrapolate between regular graph for $p=0$ to random graph for $p=1$. Equilibrium and few non-equilibrium systems are studied on SW network. It has been established that equilibrium transitions on small-world network fall in the universality class of mean-field models for $p\neq 0$ while situation is not so clear in non-equilibrium systems[13].

We study coupled logistic maps on a SW network where each site is coupled with four other sites. Studies on coupled logistic maps in 1-dimension show the transition to an antiferromagnetic order for certain coupling strength and at the critical point, decay of persistence matches with Glauber-Ising model[14]. We argue that coupled logistic maps have an effective antiferromagnetic coupling since $f(x)$ is decreasing function near nonzero fixed point. If we couple each site to the nearest as well as next-nearest neighbors, it leads to exchange-frustration for antiferromagnetic coupling. Besides, SW network does not even have bipartite topology and any long-range order with effectively antiferromagnetic couplings cannot exist. For small coupling, we observe dynamically arrested states which do not have any apparent spatial order but are frozen temporally in a coarse grained sense. Usual order parameters such magnetization or sublattice magnetization do not work for this state. We study the temporal properties of correlations for such transition.

Frustrated systems can have anomalous relaxation properties.The Kohlrausch-Williams-Watts function (KWW or stretched exponential)[15, 16] is defined as :

[TABLE]

for $t>0$ and $0<\beta<1$ with $\tau_{0}>0$ an effective time constant, is usually used as a phenomenological description of relaxation in a variety of complex materials, disordered, frustrated systems such as spin-glasses, polymers, coupled oscillators, jamming transitions, driven temporal networks, coupled nonlinear systems, molecular systems, glass soft matter, porous and noncrystals silicon [17, 18, 19, 20, 21, 22, 23, 24]. In first passage times, stretched exponential behavior has been observed in fluctuation barrier problem[25] and currency exchange rates[26]. Such behavior is essentially a signature of long-time confinement of dynamics to some regions in phase space. In these systems, frozen disorder leads to slow dynamics and various theories have been proposed to explain these.

We conduct a detailed study of phase transition from partially or fully arrested state to spatiotemporal chaos as a dynamics phase transition. We consider local persistence as an appropriate order parameter to quantify this transition. Persistence quantifies the fraction of sites in system which have not deviated even once from its initial state. This quantifier is analogous to, but much stronger than Edwards-Anderson order parameter used in studies on spin glass. These studies are also similar to branch of statistics called ‘survival analysis’ which studies durations till a given event such as death in living systems or failure in mechanical systems happens[27]. The most popular fitting function in this analysis is Weibull distribution which has stretched exponential survival function for some parameters. This survival function is expected when hazard rate is nonstationary, it decreases in time and there are multiple causes of failure. On the other hand, we expect exponential survival function when hazard rate is constant and mortality is independent of age. We also study other quantifiers such as spin flipping probability and trap time distribution. Spin flip probability is analogous to (but not same as) hazard rate in mortality studies since spins can flip multiple times. We also present a simple model for this behavior.

2 Model

We construct coupled map lattice on a small-world network. Time evolution of the variable $x_{i}(t)$ at site $i$ at time $t$ depends on itself and $4$ nearest neighbors ($1\leq i\leq N$ where $N$ is system size). We assume periodic boundary conditions and $x_{i}(0)$ are random variables in the interval $(0,1)$.

[TABLE]

where $\epsilon$ is the coupling strength. Using the above definition of SW network, $\xi_{1}(i)=i+1,\xi_{2}(i)=i+2,\xi_{3}(i)=i-1,\xi_{4}(i)=i-2$ chosen with probability $1-p$ and any randomly chosen lattice site with probability $p$. This topology does not change during evolution. The topological properties such as clustering coefficient and average path length change with $p$. Question is how the dynamical properties and dynamical transitions are affected as we change structural properties.

The on-site map is logistic map $f(x)=\mu x(1-x)$. where $\mu$ is the map parameter. We study the system in chaotic regime by keeping map parameter $\mu=3.9$ constant by varying $\epsilon$ and $p$ from $[0,1]$.

The nonzero fixed point of the logistic map is given by

[TABLE]

To explore the dynamical phases and transitions therein, we need to simulate the system for larger time over large configurations and an appropriate order parameter has to be defined. We consider local persistence as an order parameter. Persistence has been helpful in studying transitions to a dynamically arrested state [11, 28]. It is an analogue of first passage times. In spin systems, persistence $P(t)$ is number of sites which did not change their initial value even once till time $t$. This is a non-Markovian quantity which requires knowledge of complete time evolution of the system for its computation. We coarse-grain the dynamics by labeling the sites with $x_{i}^{\prime}s$ values greater than $x^{*}$ as ‘up’ spins and rest as ‘down’ spins. For logistic map, this definition needs to be modified as the slope of the map at $x^{*}$ is negative for $\mu>2$. We expect the sites with value greater than $x^{*}$ returns a lower value than $x^{*}$ and vice versa. Hence we use the generalized definition and check for persistence at all even time steps as in ref. [14]. The spin variable $s_{i}(t)=1$ if $x_{i}(t)\geq x^{*}$ otherwise $s_{i}(t)=-1$. The local persistence $P(\tau)$ is defined as follows- $P(\tau)$ is a fraction of sites for which $s_{i}(0)=s_{i}(2\times t)$ for all $0\leq t\leq\tau$.

3 Phase Diagram and Dynamic behavior

We examine the system by varying parameters mentioned above. In $\epsilon-p$ space, a significant part is occupied by dynamically arrested state. They retain the memory of initial state for indefinite time leading to a non-zero asymptotic value of persistence. The persistent region ($PR$) and non-persistent region ($NPR$) are plotted in Fig.1. The persistent region is compact and is bounded by two critical lines. The dynamic behavior on the critical lines is different at different points. The phase diagram remains qualitatively same for larger values of $N$ and for longer time. This diagram is obtained only after waiting for very long times (for $N=10^{4}$ and wait for $5X10^{7}$ transient), otherwise spurious persistent points can appear in non-persistent region.

$PR$ region is overall the same for all values of $p$. The nonlocal connections seem to make only a marginal difference. The dynamical origin of $PR$ region can be inferred from the bifurcation diagram. Fig.2 shows the bifurcation diagram for $p=0.0$ and $p=0.1$. In $PR$ region, we find a two-band attractor while in $NPR$ region the values of $x_{i}$ are spread over entire interval. Thus there is a attractor-widening crisis on either border. For large $\epsilon$ and $p>0.7$, we observe spatial synchronization. This transition has been investigated in detail by several authors and we do not study transition to chaotic synchronization in this work. We study the transition between $NPR-PR-NPR$, and explore the dynamics on critical lines. It is expected that the dynamics is slower for lower values of coupling. We denote the border on left side for $NPR-PR$ transition as border $I$ and the one on right side for $PR-NPR$ transition as border $II$.

We investigate persistence and model it by a stretched exponential function defined in eq.(1) which is an excellent fit at critical point for almost all points on both borders. It is defined by:

[TABLE]

with $\tau_{0}=1$ and $0<\beta<1$. Stretched exponential relaxation essentially is a signature of long-time confinement of dynamics to some regions in phase space. On border $I$, we obtained stretched exponential behavior for all values of $p$. In Fig.3 $P(t)$ is plotted as function of $t^{\beta}$ on semi-log scale for $p=0.1$ and $p=0.7$. On border $II$, we observe a stretched exponential behavior for small $p$ followed by exponential for large values of $p$ (Fig.4). Apart from visual fit, we have used standard programs such as Origin to find the value of $\beta$ which gives the best fit. In literature, $\beta$ is often approximated by closest rational and the values we obtain are very close to simple rationals such as $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\frac{2}{5}$ or 1. However, we have plotted the unbiased estimators of $\beta$ in the figure. On border $I$, the $\beta$ is bounded by $\frac{1}{2}$ while it tends to 1 as $p\rightarrow 1$ on border $II$. We also note that at early times, persistence drops faster than stretched exponential. Initial conditions are chosen randomly from unit interval. Initially, there are more negative spins since $x^{*}>1/2$. In a short time interval, number of $+$ and $-$ spins becomes approximately equal. This leads to fast drop in persistence. Asymptotic behavior is very well described by stretched exponential. For $p=0$ on border $II$, the value of $\beta$ obtained is very close to zero, indicating that dynamics could even be slower. We obtain equally good power law $1/t$ for persistence at this point.

Fig.5 show the value of $\beta$ as a function of $p$ for border $I$ and $II$. As expected, the dynamics is slower for smaller values of couplings for both borders. The nonzero asymptotic value of persistence indicates that some spin values never change from their initial conditions.

We introduce another parameter $F(t)$ to demonstrate the nonstationarity of dynamics. $F(t)$ is the probability that spin value at time $2t$ is different from its value at time $2t-2$. If this parameter goes to zero asymptotically, the dynamics is completely frozen . If $F(t)$ and $P(t)$ are both nonzero asymptotically, it indicates partially frozen dynamics. Here spins keep flipping, but some spins never flip. $F(t)$ is similar to hazard rate in survival statistics. (The difference is that spin flips can occur multiple times unlike death.) It decays as power-law and goes to a very small nonzero value on border $I$. On border $II$, we observe a logarithmic decay followed by saturation. This indicates that though persistence decays as stretched exponential on either border, detailed dynamical behavior is different. Within our computational limitations, it indicates that spin state is not completely frozen but partially arrested on both borders.

We also study trap time distribution. Trap time is the time period to which particular site stays in same state (not just initial state). If a site changes state at time $\tau$ and change it again at time $\tau+\tau_{1}$, it is trapped for $\tau_{1}$ period. We compute the probability distribution $T(\tau_{1})$ for trap time $\tau_{1}$ The persistence is a generalization of first passage time studies in stochastic systems while the trap times are similar to return times in stochastic systems. They usually do not obey same statistics. Since the system is highly nonstationary, this distribution is expected to change in time. We wait for $10^{7}$ time-steps so that steady-state properties can be observed. Fig.7 shows that this distribution is well described by power-law tail on border $II$. The exponent depends on value of $p$. The behavior on border $I$ could not be approximated by a function with closed form. But it is clearly slower than exponential and longer traps are more probable than border $II$ since the $F(t)$ is smaller asymptotically.

In glasses, stretched exponential relaxation is often accompanied with aging. The flipping probability $F(t)$ clearly depends on time elapsed in our system as well. The properties change in time. If we cool the system below glass transition temperature for time $t_{w}$ and bring it back to critical temperature, relaxation depends on the history of the system and time for which sample is cooled and is a function of $t/t_{w}$ in general[29, 30]. If we carry out evolution inside the persistent regime for time $t_{w}$ and then bring system to critical coupling value on either border, we do not observe any marked dependence on history of the system. However, if we allow system to equilibrate for time $t_{w}$ (waiting time) at any point on border $I$ or border $II$ and compute persistence after this time, we observe interesting results. Similar studies have been conducted in survival statistics. We may study the survival of machine/ animal provided it has already survived $t_{w}$ years. (These two studies are not exactly equivalent since even sites which have not persisted till time $t_{w}$ are also part of our statistics.) If we assume that probability of persistence with respect to initial conditions at $t_{w}$ time steps has same form as survival probability for Weibull distribution after $t+t_{w}$, given it has already survived till time $t_{w}$, the following form for persistence after waiting time $t_{w}$ can be proposed:

[TABLE]

Thus

[TABLE]

[TABLE]

The behavior of $P(t_{w},t_{w}+t)$ as a function of $(1+t/t_{w})^{\beta}$ on semi-log scale is clearly linear along both borders even for smaller $t_{w}$. This behavior is clearly observed in Fig. 8 and in Fig. 9. (Still this fit is surprising due to approximation involved.) This is another evidence that asymptotic behavior is indeed described by stretched exponential. The validity of scaling form in eq.7 depends on whether the persistence at time $t_{w}$ and $t+t_{w}$ is well defined by stretched exponential. For very small and very large $t_{w}$’s the assumption of stretched exponential decay itself is not valid either. There is a large drop in persistence at early times. Inevitable finite size effects as well as error in finding critical point can affect long-time behavior. However, this scaling holds in most cases. Fig. 9 shows the scaling for data in Fig. 8. To account for drop at early times, we propose a modified scaling law

[TABLE]

where $C$ is a small constant. With this modification, above scaling holds for a bigger range of $t_{w}$.

There are few models of stretched exponential relaxation. Models such as resonance energy transfer model are not very relevant for this work. A model that could be of particular interest is random trap model. Grassberger and Procaccia studied diffusion particle density in a $d$-dimensional medium (in which static traps are distributed randomly). Particle density decays as $\exp[-t^{d/d+2}]$ [31]. (In $1-d$, the decay will be $\exp[-t^{1/3}]$ and decay will be exponential as $d\rightarrow\infty$.) For $p=0$ on border $I$, $\beta\sim\frac{1}{3}$ and for $p>0.6$ on border $II$, we obtain $\beta\sim 1$. SW network is an effectively infinite dimensional system because the characteristic distances between nodes grows as logarithm of system size111In a $d$-dimensional Cartesian network with $N$ sites ($N=L^{d}$ ), characteristic length scale is $L=N^{1/d}$ and since logarithm is slower than power-law, it is effectively infinite dimensional network.. Therefore it is expected that systems on a small-world network shows mean-field type behavior. Several equilibrium systems on a small-world network display behavior in the universality class as mean-field models for infinitesimal small values of $p$. However in non-equilibrium case, there are several examples in which mean-field behavior is not reached even for $p=1$ [12, 32, 33]. On border $II$, we observe mean-field type behavior for large rewiring probability. Persistence decays exponentially for $p>0.6$. On border $I$, we do not find such crossover. High coupling strength and small-world effect (involvement of more random connections due to large $p$) might be a reason of crossover on border $II$.

We propose another possible model for stretched exponential relaxation. Let us consider $N$ spins which are assigned value 1 or -1 initially. Each spin is coupled to two nearest neighbors on either side with antiferromagnetic coupling. To generate SW network, we rewire the links with probability $p$ as in previous section. All sites are updated in parallel with following rule. a) If sum of spins of all neighbors is positive, the spin value is set to -1. b) If sum of spins of all neighbors is negative, the spin value is set to 1. b) If sum of spins of all neighbors is zero, the spin value is changed with probability $q$. We set $q=0.05$. With these rules, we obtain stretched exponential relaxation. As shown in fig.10, the exponent gradually increases from $0.36$ to 1 with increasing $p$.

4 Conclusion

We study coupled logistic map on a small-world networks and explore the phase diagram for persistence in $\epsilon-p$ space. The persistent region is a bounded by two critical lines $I$ and $II$. Persistence on both borders wows stretched exponential decay with crossover to exponential on border $II$ for large $p$. We also obtain scaling form for decay of persistence after waiting for time $t_{w}$. It has the same form on both the borders. However, other quantifiers flip probability $F(t)$ and trap time distribution show different behavior along both borders. Trap time distribution has power law decay in time along border $II$ while no such behavior is obtained on border $I$. On the other hand, $F(t)$ shows power-law decay on border $I$ and logarithmic decay on border $II$. We believe that exchange frustration is key to understanding this behavior and present a Ising-type model to explain it.

5 Acknowledgement

AVM acknowledges DST WOS-A scheme for financial support and A. Banpurkar and S. Barve for helpful discussion. PMG thanks RTMNU and DST research schemes for financial assistance.