The `xrk` program comes with six models: a general linear model, the
nonlinear pendulum, the van der Pol oscillator, the Gray-Scott model of
autocatalytic chemical behavior, the
Lotka-Volterra predator-prey model, a generalized population model for
two species, and a model of forced oscillations
of a nolinear shock absorber. This section describes these
models, and gives the mapping between parameters in the mathematical
equations used for the models and the parameter names used in
`xrk`. As you will discover when you use `xrk`, you can interactively
change the values of model parameters. Each parameter is represented
by a label identifying the parameter and an editable text field
displaying the value of the parameter. The mapping referred to above
relates the parameters in a model to the labels used to identify them
in `xrk`. This mapping is necessary because `xrk` is not able to display
the Greek letters often used to denote parameters.

This is just a general two-dimensional linear model described by the following system of equations.

There are four parameters, and they are represented in `xrk` by the
four letters a, b, c, and d.

The model equation for a damped nonlinear pendulum is usually written as the second order equation

where *m* is the mass, *l* is the length, and *g* is the acceleration
of gravity. However, for `xrk` we need to write this model as a system of first
order differential equations. Doing so, and doing some rescaling, we
get the following system.

This model has only two parameters, and they are represented in `xrk`\
as a and b.

This is a famous model of nonlinear oscillations. If it is written as the second order equation

we can recognize it as a modification of the linear oscillator, where the damping term has been made nonlinear.

Written as a first order system, the van der Pol oscillator is

This model has only a single parameter, . Because it is not
easy to mix symbols from different alphabets in the `xrk` main window,
this parameter is represented in `xrk` by the text string
`alpha`.

Gray and Scott are two English chemists, who came up with a simple model of autocatalytic behavior in chemical systems. Autocatalysis is thought to be important in many chemical systems which exhibit oscillations or other exotic behavior. The model equations appear below.

Where , , , , , and are positive constants.

The model as written above has three components, but adding the three equations together gives the linear equation

which can be solved analytically to show that asymptotically the solution of the Gray-Scott model always satisfies

This equation can be solved for *w* and the result substituted in the
Gray-Scott model equations to obtain the two-dimensional system

which is the one used in `xrk`.
This model has six parameters. They are represented in `xrk` as shown
in the table below.

This well-known model deals with the interaction of two populations, a predator and a prey. The model equations are given by

where *a*, *b*, *m*, and *n* are positive constants, *x* is the
population of prey and *y* is the population of predators.
The four parameters in this model are represented simply in `xrk` by
the letters a, b, m, and n.

This is a generalization of the Lotka-Volterra model to modeling interactions of two species, which may be either competing species or a predator-prey pair. The model equations are given by

where *x* and *y* are the populations and , , , ,
, are constants. The constants and must be
positive; the other four constants can have either sign.

The six parameters in this model are represented in `xrk` as follows.

This another nonlinear oscillation model, but for forced oscillations. It is a modification of a model of a shock absorber for automobiles. The model equations are given by

where , *a*, *b*, , and *A* are positive constants
and and are constants that can have any value.

Note that the third and fourth equations can be decoupled from the
first two. Using polar coordinates, it is not hard to show that the
asymptotic solutions for *z*(*t*) and *w*(*t*) are given by *z*(*t*) =0,
*w*(*t*) = 0 if and, for positive values of , by

where is a phase angle that depends on the initial conditions
of *z* and *w*.

Given this, it is clear that this model is a forced oscillator with nonlinear damping.

The six parameters in this model are represented in `xrk` as follows.

Fri Oct 25 13:53:45 EDT 1996