Coupled maps

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Kunihiko Kaneko and Tatsuo Yanagita (2014), Scholarpedia, 9(5):4085. doi:10.4249/scholarpedia.4085 revision #149460 [link to/cite this article]
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Curator: Kunihiko Kaneko



A coupled map consists of an ensemble of elements of given discrete-time dynamics ("map") that interact ("couple") with other elements from a suitably chosen set. The dynamics of each element is given by a map. As a consequence, the coupled map is a discrete-time multi-dimensional dynamical system. In most coupled maps, all elements have identical map dynamics; however, coupled maps can also contain heterogeneous elements.

The first, and most thoroughly studied, type of coupled map is the coupled map lattice (CML), in which each element is set on a lattice of a given dimension—resulting in a dynamical system with discrete time ("map"), discrete space ("lattice"), and continuous state. CML was originally introduced to facilitate the study of spatiotemporal chaos, i.e., chaotic dynamics in a spatially extended system [Kaneko, 1984, Kapral, 1985, Crutchfield and Kaneko, 1987, Kaneko, 1992, 1993]. CML is comparable with three other standard models for spatially extended dynamical systems, namely, coupled ordinary differential equations, in terms of discretization of time, partial differential equations (PDEs), in terms of their discretization of space and time and cellular automata (CA), in terms of their continuation of state. The three models are classified according to whether state, space, and time are continuous (C) or discrete (D) (see Table I).

Classification of spatially extended dynamical systems
Model Space Time State
Cellular Automata D D D
Coupled Map Lattice D D C
Coupled Ordinary Differential Eqn. D C C
Partial Differential Eqn. C C C

The following canonical procedure can be used to construct a CML:

(A) Choose a (set of) field variable(s) on a lattice. This set of variable(s) is not at the microscopic but the macroscopic level: It is not a variable for a microscopic element such as the velocity of a molecule, a spin, voltage of a neuron, but a coarse grained-variable such as the velocity of fluid, magnetization or average electric activity over a certain spatial range.

(B) Decompose the phenomena concerned into independent processes. In CML, processes are represented by procedures, with each procedure representing a specific process, such as convection, reaction, or diffusion, or an abstract process such as "local chaos."

(C) Replace each process with simple parallel dynamics ("procedure") on a lattice. This represents nonlinear transformation of state variable(s) at each lattice point and/or a coupling term among suitably chosen neighbors.

(D) Execute each unit dynamics ("procedure") in succession.

The first simple CML was proposed for the study of spatiotemporal chaos: Consider a phenomenon generated by a local chaotic process and a spatial diffusion process. Let us take a state variable $x_n(i)$ for discrete time $n = 0, 1, 2, \cdots$ over a one-dimensional lattice with sites $i = 1, 2, \cdots, N$. Next, take a one-dimensional map as the simplest representative chaos and a discrete Laplacian operator for the diffusion.

The first, simple example of CML that was proposed for the study of spatiotemporal chaos. Here, consider a phenomenon, generated by a local chaotic process and spatial diffusion process. Let us take a state variable $x_n(i)$ for discrete time $n=0,1,2,\cdots$ over a one-dimensional lattice with sites $i=1,2,\cdots ,N$. Now take a one-dimensional map as a simplest representative of chaos and a discrete Laplacian operator for the diffusion. The former process is given by $ x'_n(i) = f(x_n(i))$, where $x'_n(i)$ is introduced as a virtual variable for an intermediate step. The discrete Laplacian operator for diffusion is given by \[ x_{n+1}(i) = (1-\epsilon ) x'_n(i) + \frac{\epsilon }{2} \{ x'_n(i+1) + x'_n(i-1) \}. \]

Combining the above two processes results in CML given as

\[ \tag{1} x_{n+1}(i) = (1-\epsilon)f(x_{n}(i)) + \frac{\epsilon}{2} \{ f(x_{n}(i+1)) + f(x_{n}(i-1)) \}. \]

The mapping function $f(x)$ is chosen according to the type of local chaos. For example, the logistic map ($f(x) = rx(1-x)$) can be chosen as a typical model for chaos. Because "local chaotic dynamics" by one-dimensional mapping are well understood, the above CML is extensively studied as a canonical model for spatiotemporal chaos.

By adopting different procedures, models for different types of spatially extended dynamic phenomena can be constructed. For example, for problems of phase transition dynamics, it is useful to adopt a map with bistable fixed points (e.g., $f(x) = \tanh x$ ) as local dynamics, which is used for phase transition kinetics.

Several extensions are possible by adopting different procedures for local dynamics and coupling. For example, to discuss open-flow (such as pipe flow), unidirectional coupling is relevant, as given by

\[ x_{n+1}(i) = (1-\epsilon)f(x_{n}(i)) + \epsilon f(x_{n}(i-1)). \]

Another choice of coupling is a mean-field-type global coupling, \[ x_{n+1}(i) = (1-\epsilon)f(x_{n}(i)) + \frac{\epsilon}{N}\sum_j f(x_n(j)), \] which has been extensively studied as a prototype of collective chaos, while chaos networks are investigated by using coupling on given networks. Several other choices of coupling forms, such as derivative coupling with the form \( F(x_n (i)-x_n (j)) \), and inclusion of a conservation law have also been discussed.

The Lyapunov spectrum of the model (1) is computed from the Jacobian matrix, which is successive multiplication of the diagonal matrix given by local derivative $f'(x_n(i))\delta_{ij}$ and the tridiagonal matrix of the discrete Laplacian operator \( (1-\epsilon)\delta_{i, j} + (\epsilon/2)(\delta_{i, j+1} + \delta_{i, j-1}) \). Indeed, from this form, one can directly show the stability of a spatially uniform, temporally periodic cycle, if the cycle is stable in the local one-dimensional map \( x_{n+1} = f(x_n) \), i.e., $|f'(x_1)f'(x_2)..f'(x_p)|<1$ .

Universality class of the spatiotemporal chaos revealed by CML

CML has uncovered typical salient behaviors in spatiotemporal chaos that form a universality class common to diverse spatially extended systems. The canonical CML, with a logistic map $f(x) = 1-ax^2$, provides a rich variety of such classes according to the parameter values, i.e., $a$ in the logistic map and $\epsilon$ for the coupling (see Figures 1–12). They include (I) frozen random patterns with spatial bifurcation and localized chaos, (II) pattern selection with suppression of chaos, (III) spatiotemporal intermittency, (IV) Brownian motion of chaotic defects, and (V) global traveling wave by local phase slips [Kaneko, 1989, Kaneko and Tsuda, 2000].

Frozen random patterns

As parameter $a$ of the isolated logistic map is increased to show a period-doubling route to chaos, a spatially homogeneous state becomes unstable because of the sensitive dependence on initial conditions that is characteristic of chaos. Through the time evolution, domains of different sizes, with different phases of oscillations are formed as attractors. Depending on the domain size, the motion of $x(i)$ in each domain is approximately period-2, 4, 8, ..., and chaotic. Indeed, in a large domain, the motion is chaotic, whereas it is virtually period-$2^k$ in smaller domains, with $k$ decreasing with the decrease in domain size, down to period-2 for the smallest domain of size one. Depending on the initial condition, there is a large number of attractors, whose numbers increase exponentially (at least) with system size $N$ (Figure 1, Figure 2).  It is also noted that the Feigenbaum's scaling in the period-doubling bifurcations is extended to include the spatial scaling in CML (Kuznetsov and Pikovsky, 1986)

Figure 1: Space-amplitude plot: Time evolution of $x_n(i)$ plotted against lattice site $i$, $a = 1.45, \epsilon = 0.2$
Figure 2: Space-time diagram: Time evolution of $x_n(i)$ plotted with horizontal axis $i$ and vertical axis $x_n(i)$. The patterns in different time steps, $x_n(i), n = t, t+1, \cdots, t+7$ are represented in color and are plotted per eight steps.

Pattern selection with suppression of chaos

As the parameter $a$ controlling the non-linearity is further increased, larger domains with chaotic dynamics start to become unstable and split into smaller domains. At an attractor, domains of a few sizes in particular (for example, three or four lattice points) are selected. Domain sizes are selected such that the dynamics within are less chaotic (Figure 3, Figure 4).

Figure 3: Space-amplitude plot: Time evolution of $x_n(i)$ plotted against lattice site $i$, $a = 1.71, \epsilon = 0.4$
Figure 4: Corresponding space-time diagram: $a = 1.71, \epsilon = 0.4$

Spatiotemporal Intermittency

Transition from an ordered pattern to fully developed spatiotemporal chaos (FDSTC) occurs via spatiotemporal intermittency (STI) (Kaneko 1984, 1985, 1989, Chaté and Manneville, 1988). In STI, there are both laminar motion and turbulent bursts in space-time. Each space-time pixel can be classified into the two states accordingly. Laminar motion is characterized by periodic or weakly chaotic dynamics with spatially regular structures, whereas turbulent bursts have no spatiotemporally regular structure. The introduction of STI by CML has been followed by the presentation of extensive experimental reports on Bénard convection with a large aspect ratio, Faraday instability of waves, two-dimensional electric convection of liquid crystals, viscous rotating fluids, and so forth.

Two types of STI are known to exist. In the first type (type-I STI), there is no spontaneous creation of bursts. If a site and its neighbors are laminar, the site remains laminar in the next step. It also has an absorbing state with a temporally periodic, spatially homogeneous attractor. In type-I STI, a site changes from laminar to burst only if at least one site in neighbors is in a burst state. If this propagation rate of burst exceeds some threshold, the burst states percolate in space-time, to form STI. This transition belongs to a class of directed percolation. A typical example is the CML of a logistics map with the parameter $a$ at the period-3 window (Figure 5, Figure 6) (see also Grassbeger and Schreiber).

In the second type of STI (type-II STI), spontaneous creation of turbulent bursts occurs, even if all the states of the sites and its neighbors are laminar. Type-II STI is observed with the transition under a spatial pattern. By taking advantage of a continuous change in the state variable, the pattern is distorted continuously, producing a burst, which then propagates in space. In some cases, this transition, when viewed locally in space, can be associated with on-off intermittency (Figure 7, Figure 8).

Figure 5: Space-amplitude plot: Time evolution of $x_n(i)$ plotted against lattice site $i$, $a = 1.7522, \epsilon = 0.00115$ and plotted per 12 steps.
Figure 6: Corresponding space-time diagram: The patterns in different time steps $x_n(i), n = t, t+k, \cdots, t+8k$ are represented in color with $k = 30$.

Figure 7: Space-amplitude plot: Time evolution of $x_n(i)$ plotted against lattice site $i$, $a = 1.75, \epsilon = 0.3$
Figure 8: Corresponding space-time diagram

Traveling Wave

When the coupling $\epsilon$ is large (e.g., larger than 0.45), the domain structures in (I) and (II) are no longer fixed, but can move in space. At the parameter corresponding to the frozen state (I), the motion of a domain is rather irregular, whereas a regular traveling wave is seen in the pattern selection regime (Figure 9, Figure 10). We also note that for a given parameter value and coefficient, attractors with different wave velocities coexist, according to the spatial asymmetry of the selected wave pattern.

Figure 9: Space-amplitude plot: Time evolution of $x_n(i)$ plotted against lattice site $i$, $a = 1.67, \epsilon = 0.5$ and plotted per 2000 steps.
Figure 10: Corresponding space-time diagram: Plotted per 1000 steps

Chaotic travelling-wave

Figure 11: Space-amplitude plot: Time evolution of $x_n(i)$ plotted against lattice site $i$, $a = 1.69, \epsilon = 0.5$ and plotted per 5000 steps.
Figure 12: Corresponding space-time diagram: Plotted per 2000 steps.

Quasi-stationary super-transients

In some CMLs, very long transients exist. The transient time before approaching an attractor scales exponentially with the system size, $N$. In the transient regime, the dynamics are quasi-stationary, which is virtually indistinguishable from the dynamics on an attractor.

For example, consider a one-dimensional mapping $x_{n+1} = f(x_n)$, with a periodic attractor and topological chaos, such as the period-3 window of the logistic map. Then, corresponding to the periodic attractor reached after transient chaos in the one-dimensional map, the CML has a spatially uniform, temporally periodic attractor for this period. However, if the initial condition is spatially inhomogeneous, ranging over state values with different phases of the cycle, the coupling induces chaotic dynamics, which travel in space, leading to the quasi-stationary super-transient when the coupling is sufficiently large. Such super-transients are observed in fluid turbulence in pipe flow (Hof et al., 2008).

Several other classes of phenomenology have been uncovered. In open flow, in particular, spatial bifurcation to down-flow chaos and convective chaos are notable findings. Emergence of order in a higher-dimensional CML is also discussed (Chaté and Manneville, 1992, Kaneko 1993).

Analysis of spatiotemporal chaos by CML

CML can use the analytical tools developed in dynamical systems (Kaneko 1993, Kaneko and Tsuda, 2000). Characterization of dynamics in phase space is extended to include the spatial dimension. These characterizations include Lyapunov analysis, information flow through space, propagation of disturbances through space, and density of attractor dimension or Kolmogorov-Sinai entropy per spatial volume. Typically, Lyapunov spectra are scaled with system size $N$, so that the Kolmogorov-Sinai entropy and dimension of an attractor are proportional to system size $N$. Hence, they are “extensive” quantities and their densities are relevant intensive quantities. The propagation speed of perturbation is computed from the positivity of the co-moving Lyapunov exponent.

Mathematical analysis of the invariant measure for the temporal evolution of CML has been developed over decades. Recall that the invariant measure for low-dimensional chaos is mapped into the statistical mechanics of one-dimensional chain of symbols via symbolic dynamics. The invariant measure of $d$-dimensional CML can be mapped to Gibbsian distribution of $(d+1)$-dimensional lattice. Thus far, however, the rigorous analysis is often limited to hyperbolic dynamical systems, and further developments are required to analyze the distinct phases presented herein.

Applications of Coupled Map Lattices

Various spatially extended dynamical phenomena are modeled by CML. The strategy used in modeling consists of decomposition of the phenomena into several procedures on a lattice and successive execution of them in discrete time, as already mentioned. Indeed, the majority of these dynamical phenomena are described by the combination of some elementary local dynamics and spatial coupling as modeled by CML (Kaneko 1992, 1993, Kaneko and Tsuda 2000). Applications in which CML has been utilized include (i) pattern formation (spinodal decomposition) (Oono and Puri, 1986), (ii) crystal growth (Kessler et al. 1990), (iii) spiral turbulence in excitable media, (iv) boiling (Yanagita, 1992) (Figure 13, Figure 14), (v) thermal convection (Yanagita and Kaneko 1993), (vi) cloud dynamics (Yanagita and Kaneko 1997) (Figure 15, Figure 16, Figure 17), (vi) sand-ripple (Ouchi and Nishimori, 1993), fibrillation in the heart rhythm (Ito and Glass, 1991). Besides the application to these pattern dynamics, possible application to vacuum fluctuation in high energy physics is also discussed (Beck 2002).

For the model of boiling, the temperature is adopted as a field variable on a lattice, while the procedures for the successive maps are chosen as diffusion of heat, floating motion of bubbles, and their generation with latent heat to decrease the temperature. The CML provides a minimal model that exhibits the transition from nucleate to film boiling.
Figure 13: Nucleate boiling phase: The temperature field is color coded with blue indicating the lowest and red indicating the high. White indicates air bubble where the temperature exceeds a threshold value.
Figure 14: Film boiling phase: when the temperature of the bottom plate exceeds a critical value, bubbles are percolated to form a thin film.

As a CML model for cloud dynamics, we assign the velocity field, the mass of the vapor and the condensed phase (i.e., water or ice), and the internal energy as field variables on a two-dimensional lattice. The dynamics consist of the successive operations of the physical processes; buoyancy, diffusion, viscosity, adiabatic expansion, fall of a droplet by gravity, descent flow dragged by the falling droplet, and advection.
Figure 15: Stratus phase (Left: water density; Right: Arrows represent stream lines of the air, while the temperature is color coded with blue indicating the lowest and red indicating the highest. The phase appears when the surface temperature is high, and the amount of water is not high.)
Figure 16: Stratocumulus phase (The phase appears when the amount of water is intermediate between the Stratus and Cumulonimbus phases.)
Figure 17: Cumulonimbus phase (The phase appears when the water amount is high.)


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