Coupled Map

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.

# Coupled map lattices

A coupled map lattice (CML) is a type of coupled map where the spatial dimension is also discretized, hence the lattice structure. CMLs are often used to model spatially extended systems where interactions happen mainly between nearest neighbors. Consequently, this results in a dynamical system with discrete time, space, and continuous state. CMLs were originally introduced to facilitate the study of spatiotemporal chaos, that is, chaotic dynamics in a spatially extended system.

## Classification of spatially extended dynamical systems

Spatially extended dynamical systems can be classified by their space, time, and state characteristics. The following table provides a summary of this classification for various system types:

Cellular Automata discrete discrete discrete
Coupled Map Lattice discrete discrete continuous
Coupled ODEs discrete continuous continuous
Partial Differential Equations continuous continuous continuous

## Canonical Construction

To construct a CML, the following canonical procedure is employed:

• Choose a set of field variables on a lattice, representing a coarse-grained view of the system rather than a microscopic one.
• Decompose the phenomena into independent processes, each represented by procedures such as convection, reaction, diffusion, or "local chaos."
• Replace each process with simple parallel dynamics on a lattice. This represents nonlinear transformation of state variables at each lattice point and/or a coupling term among suitably chosen neighbors.
• Execute each unit dynamics in succession.

Let's take the simplest example of a CML designed for the study of spatiotemporal chaos. Consider a phenomenon generated by a local chaotic process and a spatial diffusion process. Let $x_n(i)$ be a state variable for discrete time $n = 0, 1, 2, \ldots$ over a one-dimensional lattice with sites $i = 1, 2, \ldots, N$. Now take a one-dimensional map as the simplest representation of chaos and a discrete Laplacian operator for the diffusion.

The local chaotic process is expressed by: $x'_n(i) = f(x_n(i))$, where $x'_n(i)$ is a virtual variable for an intermediate step. The discrete Laplacian operator for diffusion is:

$$x_{n+1}(i) = (1-\epsilon) x'_n(i) + \frac{\epsilon}{2} \{ x'_n(i+1) + x'_n(i-1) \}$$

Combining these two processes results in the following CML:

$$x_{n+1}(i) = (1-\epsilon)f(x_{n}(i)) + \frac{\epsilon}{2} \{ f(x_{n}(i+1)) + f(x_{n}(i-1)) \}$$

The function $f(x)$ is chosen based on the type of local chaos. For instance, the logistic map $f(x) = rx(1-x)$ can be selected as a typical model for chaos.

A range of spatially extended dynamic phenomena can be modeled by adopting different procedures. For example, for phase transition dynamics, a map with bistable fixed points (such as $f(x) = \tanh x$) as local dynamics is useful for modeling phase transition kinetics.

The potential extensions of this method include the adoption of different procedures for local dynamics and coupling. For instance, unidirectional coupling is relevant for discussing open-flow (such as pipe flow), as given by:

$$x_{n+1}(i) = (1-\epsilon)f(x_{n}(i)) + \epsilon f(x_{n}(i-1))$$

There is also mean-field-type global coupling, expressed by:

$$x_{n+1}(i) = (1-\epsilon)f(x_{n}(i)) + \frac{\epsilon}{N}\sum_j f(x_n(j))$$

Several other forms of coupling have been discussed, such as derivative coupling and inclusion of a conservation law.

The Lyapunov spectrum of the model is computed from the Jacobian matrix, a successive multiplication of the diagonal matrix given by the local derivative $f'(x_n(i))\delta_{ij}$ and the tridiagonal matrix of the discrete Laplacian operator $(1-\epsilon)\delta_{i, j} + (\epsilon/2)(\delta_{i, j+1} + \delta_{i, j-1})$. From this form, the stability of a spatially uniform, temporally periodic cycle can be shown, if the cycle is stable in the local one-dimensional map given by

$x_{n+1} = f(x_n)$, i.e., $|f'(x_1)f'(x_2)..f'(x_p)|<1$

## Universality classes of the spatiotemporal chaos revealed by CML

CML has uncovered typical 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 variety of classes according to the parameter values, $a$ in the logistic map and $\epsilon$ for the coupling. They include:

• Frozen random patterns with spatial bifurcation and localized chaos
• Pattern selection with suppression of chaos
• Spatiotemporal intermittency
• Brownian motion of chaotic defects
• Global traveling wave by local phase slips [Kaneko, 1989, Kaneko and Tsuda, 2000].

As parameter $a$ of the isolated logistic map is increased to show a period-doubling route to chaos, a spatially homogeneous state becomes unstable due to the sensitive dependence on initial conditions that is characteristic of chaos.

## Frozen random patterns

As the parameter $a$ of the isolated logistic map is increased to demonstrate 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 $\displaystyle x(i)$ 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-$\displaystyle 2^{k}$ 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 $N$.

It is also noteworthy that the Feigenbaum's scaling in the period-doubling bifurcations is extended to include the spatial scaling in CML (Kuznetsov and Pikovsky, 1986).

# Coupled map networks

Coupled map networks generalize the concept of coupled map lattices with an arbitrary graph topology. Coupled map networks are characterized by local dynamics operating at each vertex or node of a graph, with these small constituent systems interacting along the edges of a graph, and are not translationally invariant, (unlike $\mathbb{Z}^d$ translational symmetries assumed in case of CMLs.

In simply coupled CMLs, one observes a cascade of bifurcations in which expanding subspaces systematically switch over to contractive subspaces, that is to say, the system goes from expanding to hyperbolic or partially hyperbolic.

A coupled map network is defined to be a triple $(G, \{f_i\}, A)$ where:

• $G$ is a graph specified by a finite or countable set $V$ of vertices and a collection of edges $E$.
• At each vertex $i \in V$ there is a local space $X_i$ and a local map $f_i : X_i \to X_i$
• Network dynamics are defined by iteration $\Phi : X \to X$ where $X = \prod_{i \in V} X_i$ and $\Phi = A \circ F$ for $F = \prod_{i \in V} f_i$, an independent application of local maps and $A : X \to X$ the spatial interaction (coupling). For $x = (x_i)_{i \in V} \in X$, the $i$-th coordinate of $A(x)$ depends only on $x_i$ and those $x_j$ for which $(j, i) \in E$, that is to say all interactions are local.

## Unit Circle CMNs

Let us consider CMNs of the following type: $G$ is an arbitrary finite graph with $d$ vertices, and the local systems $f_i : X_i \to X_i$ are smooth circle maps, so that global phase space is a hypertorus $X \cong \mathbb{T}^d$. Let $\mathbb{S}^1 \equiv \mathbb{R} / \mathbb{Z}$ under additive notation for the phase angle.

### Stable and unstable filtrations.

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### Existence of stable subbundles.

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### Decomposability dynamics.

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# Coupled maps on cellular sheaves

Main article: Spectral Sheaf Theory

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