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Models distributed with xrk

  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.

Nonlinear Pendulum

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.

Van der Pol oscillator

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, tex2html_wrap_inline1143 . 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-Scott model

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 tex2html_wrap_inline1149 , tex2html_wrap_inline1151 , tex2html_wrap_inline1153 , tex2html_wrap_inline1155 , tex2html_wrap_inline1157 , and tex2html_wrap_inline1159 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.


Lotka-Volterra model

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.

Population model

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 tex2html_wrap_inline1191 , tex2html_wrap_inline1193 , tex2html_wrap_inline1195 , tex2html_wrap_inline1197 , tex2html_wrap_inline1199 , tex2html_wrap_inline1201 are constants. The constants tex2html_wrap_inline1195 and tex2html_wrap_inline1201 must be positive; the other four constants can have either sign.

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


Shock model

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 tex2html_wrap_inline1201 , a, b, tex2html_wrap_inline1225 , and A are positive constants and tex2html_wrap_inline1199 and tex2html_wrap_inline1149 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 tex2html_wrap_inline1241 and, for positive values of tex2html_wrap_inline1149 , by


where tex2html_wrap_inline1245 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.


next up previous
Next: Tutorial Previous: Introduction

William W. Farr
Fri Oct 25 13:53:45 EDT 1996