Dynamical Systems

Joachim Köppen Kiel/Strasbourg/Illkirch Jan.98

In the Applet we display the behaviour of several examples of dynamical systems, i.e. systems of first-order time-dependent non-linear ordinary differential equations. Such systems arise when one considers the time evolution of physical, biological, economic systems. At each time, the states of the various components of the system are specified by a number of state variables; the differential equations describe how the variables of the state change per unit of time.

Depending on the structure of the equations and the number of the variables, one can have a wide range of behaviour:

• continuous growth or decay
• approach to a steady state, an equilibrium where the state no longer changes
• approach to a partial steady state, where a certain number of state variables do not change or change very slowly
• the system may perform oscillations - growing, decaying, constant amplitude, or with rapid switching - about an equilibrium state
• there may be more complicated oscillation patterns, where the system comes back to the initial state only after a large number of oscillations
• the state changes continously, but never comes back to any state - a chaotic solution

The Volterra equations provide a very simple description of what happens when two (biological) species are in competition: One assumes that the number of sheep grows by natural reproduction in proportion to the number already there, but deceases when sheep and wolves meet. The number of wolves is increased by that process, as their reproduction is enhanced when they have food, but they die out if no food is available:

$s\text{'} = s - s*w$

$w\text{'} = s*w - w$

This results in the following cyclic behaviour: When the wolves are numerous, they decimate the population of the sheep, but if they are too numerous, they die of hunger. Being less disturbed by the few wolves, the sheep population can recover by normal growth, so the small number of wolves thrives better and they become more numerous.... There exists an equilibrium point (s=w=1 in our schematised equations) where the two populations could co-exist. But if one or the other population become larger or smaller - say by external effects - oscillations will commence about the equilibrium point. The very simplified Volterra-system neglects many effects that are present in nature. So the oscillations occur always with constant amplitude.

The VanderPol equations

$x\text{'} = y + ax\left(1 - x*x/3\right)$

$y\text{'} = - x$

describe an electronic oscillator, which is unstable against small perturbations about the zero-amplitude state, so the amplitude of oscillations quickly grow until they are limited by the nonlinearity in the electronic circuit (essentially the vacuum tube which is the essential active but nonlinear element). In the simulations, the limiting solution of a constant amplitude (but) nonlinear oscillation is represented by a closed-loop in the x-y plot. Such as feature is called a limit cycle.

Rössler's system is also a (nonlinear) oscillator, but in three dimensions:

$x\text{'} = - x - z$

$y\text{'} = x + ay$

$z\text{'} = a - bz + xz$

This permits the chaotic flipping between a simple oscillation - represented by a circle in the X-Y-plane - and an excursion to the right. This system is the simplest to show genuinely chaotic solutions

A drastically simplified model for the flow of air and heat in the terrestial atmosphere is Lorentz 's system:

$x\text{'} = - a \left(x - y\right)$

$y\text{'} = - x \left(z - r\right) - y$

$z\text{'} = xy - bz$

It forms two coupled oscillators, and the system chaotically jumps between them. Changing the parameters from the default values may completely cut-off the chaotic behaviour.

The movement of stars in the combined gravitational field of all the other stars in a galaxy lead to the equations of Hénon-Heiles. (under construction)

The controls:
START HERE ......
at first, one clicks here to get to the first model. Thereafter, the button selects the model: Volterra, Rössler, VanderPol, Lorentz, H;eacute;non.
Plot on X-axis:
select the variable to be shown on the x-axis. If this is Time, the plot will be like that of an oscilloscope, the y-values are plotted with time running from left to right, and then jumping back to the left edge
......
the two fields show the current range of the x-axis shown. Here, one may enter the range manually. The next click of Start or Clear will show the plot with the new ranges
Stand.X-range
click it to restore the default plotting range
Plot on Y-axis:
ditto, except that Time cannot be selected
......
ditto
Stand.Y-range
ditto
Start
starts the model from the initial situation
Stop
halts the simulation
Resume
carries on with the same simulation, but with the current value for the Accuracy
Clear
wipes the plot, draws the currently requested plot with the ranges as specified. During a simulation, the plot is wiped before continuing.
StartPoint
click here (also during a running simulation), then click on the plot area where you want the simulation to start with different x-y-values. The values of all the other state variables are not affected.
DragZoom
first click here, then drag a rectangle across the area which one wants to view enlarged. The next Start or Clear will plot that area; this wipes the previous tracks, of course
Accuracy
making this value smaller will increase the accuracy of the simulation
current state variables
when the calculation is stopped, the current values of all the state variables are shown
get Init
clicking this will display the variables' values for the initial situation
put Init
store the state - as it is displayed - as the initial situation. Permits to start the simulation from the same arbitrary point, when changing the parameters
Parameters
this window shows all the parameters of the model; they can be changed by entering the new values

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