Dynamical systems, vector fields

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*
of the
system, where
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
**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:

There are other possibilities to describe the dynamics of a dynamical system, for instance, discrete dynamical systems are sometimes written as . 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
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:

In this rather simple model prey is assumed to grow exponentially at a rate

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
starts at a
specific seed value
**s** and evolves over time according
to the following equation:

Dynamical systems usually are depicted in phase space . Sometimes other spaces, e.g.,

Helwig Löffelmann, November 1998,