Coupled ordinary differential equation

The coupled ordinary differential equations systems (ode) are generally obtained from ode's of order equal or higher than two (this can be done simply adding one more variable, for instance, making $y=\dot{x}$). Usually it is not possible to obtain an analytic solution to these systems (as in the case of pde's, and, if it is possible, the solution ``is so complicated that is difficult to interpretate and understand it'' (Ferrara, 17).

The study of such edo is concerned with the qualitative behavior of the solutions - i.e., try to identify important features of it without resolving the equations. These method is used both to linear and to non-linear equations. A powerful way of analysis in such cases is the study of the configuration space behavior.

One example of such systems is the $d$-dimensional lattice of Rössler system. Here we have a dynamics where the space is discrete, but the time and the field are continuous. The discrete space feature springs from the fact that in a lattice we can have a finite number of neighbors that are able to interact with one specific site. And, since the dynamics of each site (usually called local dynamics )is ruled by an ordinary differential equation, the time and the field are continuous. When we couple the sites, i.e., we make the value of each site to depend on its local dynamics and, for example, on the average value of its neighborhood. This is the way we introduce correlation among different sites. It is not hard to show that this averaging on the neighborhood is equivalent to the laplacean term in a pde.

For simplicity, let's consider a $2$-dimensional square lattice. We write:

$$X_{i,j}^{t+\tau} = (1-\varepsilon) {\cal F}^{\tau}(X_{i,j}^... ... \sum_{i',j' \in {\cal V}_{i,j}} {\cal F}^{\tau}(X_{i',j'}^t)$$ (3)

where $X_{i,j}^t$ denotes the site value (value of the field associated to that site, to bo more exact) at time $t$, the site is localized at a position on the lattice given by the pair $(i,j)$ - $i$ = row, $j$ = column; $\tau$ is the interval between coupling steps; ${\cal N}$ is the number of neighbors in its neighborhood ${\cal V}_{i',j'}$ that interact with the site $(i,j)$, ${\cal F}$ is a generic function of $X$ (to be more precise, ${\cal F}$ is a function related to the ode that rules the time development of $X$); and $\varepsilon$ is the coupling parameter, one of the fundamental parameters of the system, usually taken as control parameter. Notice that $\varepsilon$ is a measure of the relative weight given to the site ${i,j}$ and its neighborhood. When $\epsilon \rightarrow 0$ the evolution of each site depends on only itself, i.e., the sites become decoupled, disappearing spatial correlations.

In the specif case of Rössler system, $X$ is a three component vector (omiting the subscripts):

$$\dot{X}=\pmatrix{\dot{x} \cr \dot{y} \cr \dot{z} \cr} = \pmatrix{-y-z \cr x+ay \cr b+xz-cz \cr } \equiv {\cal R}(X).$$ (4)

and ${\cal F}$ is given by:

$${\cal F}^{\tau}(X^t) = X^t + \int_{t}^{t+\tau} \! dt' \: \dot{X}$$ (5)

These system has been well studying and can be used to model a lot of different problems related to spatial pattern formation. As an example, we may claim that this kind of lattice formed from Rössler system is suitable to model a set of coupled chemical reactors, since each site - which we can think as a reactor - is ruled by a system of ordinary differential equations that evolves continuously in time, each reactor (site) is point in the lattice interacting with its neighborhood.

Figure 9: Dynamics on a $2$-dimensional lattice of the Rössler system (Brunnet et al, 1994)