Dynamical systems, vector fields

Vector fields typically represent flows on discrete locations in space. Various grid structures (regular grid, curvilinear grid, etc.) are in use. Dynamical systems on the other hand are usually defined analytically, for example, by a set of differential equations.

In the previous section various fields of applications of visualization briefly have been presented. The work presented here closely fits into the flow visualization area, since flow data and dynamical systems match up quite good with respect to visualization - many techniques developed for flow visualization are useful for dynamical system visualization and vica versa.

Dynamical systems are a description of the evolution of some (usually inter-dependent) entities within a common system. A food chain, for instance, describing the who-eats-whom relation between several species, sharing some common place of living, is modeled as a dynamical system. The predator-prey model by Lotka and Volterra [72] is an example of such a food chain. It describes the evolution of a system consisting of one species of consumers (predators) and another one of resource (prey). Basically, it consists of two numbers representing the amount of both species present in the system at a certain time, and a description of the temporal change of these numbers due to the given setting of the system. More general, a dynamical system is a set of n numbers  x[i] - usually n is called the dimensionality of the system - that vary according to specific set of rules. These system variables  x[i] build up the state  $\mathbf{x}\in\Omega\!\subseteq\!\mathbf{R}^n$ of the system, where $\Omega$ usually is called the phase space of the dynamical system. Some specific value  x(t) represents the actual configuration of the system at a specific point in time t. In addition to system variables and time, usually parameters  p[j] are part of the rules of evolution. Their different values span a class of dynamical systems over $\Pi$$\subseteq$ R^m, called parameter space.

A continuous dynamical system usually is given by a set of ordinary differential equations (ODEs) [4], whereas a discrete dynamical system is specified by difference equations:

$$\begin{array}{cccclr} \dot{\mathbf{x}}(t) &\!=\!& \mathrm{d... ...,t) & \textrm{(discrete case, $t\in\mathbf{Z}$ )} \end{array}$$ (1.1)

There are other possibilities to describe the dynamics of a dynamical system, for instance, discrete dynamical systems are sometimes written as  $\mathbf{x}(t\!+\!1)=\mathbf{f}_\mathbf{p}(\mathbf{x}(t),t)$. Usually most of the alternatives are either compatible to the notation presented above, or can be transformed such that they match the above definition.

A dynamical system is called time-dependent, if the rules determining the dynamics depend on time, i.e., f_p itself depends on time t (see Eqs. 1.1). If, on the other hand, these rules are static over time, a steady, i.e., a time-independent system is given. In this case f_p only depends on the present state of the system  x(t) and parameters  p.

In the case of the Lotka and Volterra model, a two-dimensional, continuous, and steady dynamical system is given: the state  $\mathbf{x}\in\Omega\!=\!\left[0,\infty\right)^2$ of the system is composed of x (amount of prey) and y (predators), and two ODEs including four parameters that represent the rules of evolution:

$$\begin{array}{rcll} \dot{x} & = & r\cdot{}x - p\cdot{}x\c... ... - m\cdot{}y & \textrm{(evolution of predators)} \end{array}$$ (1.2)

In this rather simple model prey is assumed to grow exponentially at a rate r ( $\dot{x} = r\cdot{}x$ ) if no predators are present. Predators hunt a certain percentage p of prey, thereby decreasing the amount of prey proportionally ( $\dot{x} = \ldots - p\cdot{}x\cdot{}y$). Hunted prey is `used' for reproduction of predators on the basis of a certain efficiency e ( $\dot{y} = e\cdot{}{}p\cdot{}x\cdot{}y - \ldots$). Opposed to reproduction of predators there is a certain rate of mortality m ( $\dot{y} = \ldots - m\cdot{}y$).

Solutions of a dynamical system, i.e., solutions to the differential or difference equations, are called trajectories or orbits. For continuous and steady dynamical systems a trajectory  $\mathcal{T}_\mathbf{s}(t)$ starts at a specific seed value  s and evolves over time according to the following equation:

$$\mathcal{T}_\mathbf{s}(t) = \mathbf{s} + \int_{u=0}^{t} \mathbf{f}_\mathbf{p}(\mathcal{T}_\mathbf{s}(u))\,\mathrm{d}u$$

  

Figure 1.2: (a) Cycles of evolution in 2D phase space, and  (b) evolution of variable x over time t (both computed for a predator-prey model by Lotka/Volterra).

Dynamical systems usually are depicted in phase space $\Omega$. Sometimes other spaces, e.g., $\Omega$$\times$ R - time is added as an additional dimension - or $\Omega$$\times$$\Pi$, i.e., investigating the dynamics of an entire class of dynamical systems, are used. If the dimensionality of such a space gets too large, sub-spaces are examined instead. Returning to the Lotka and Volterra example again, one could investigate the dynamics of this model in the phase plane assuming a fixed set of parameter values (see Fig. 1.2(a)). Another possibility is to plot one of the state variables against time, again in the plane - in Fig. 1.2(b) the amount of prey (x) is plotted over time t.

mailto:helwig@cg.tuwien.ac.at.