Ordinary differential equations
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ODE's
Many dynamical systems in physics, astronomy, chemistry, physiology, meteorology, economics, population
dynamics can be described by
Ordinary Differential Equations.
In the second half of the 20th century much attention has been focussed on the
often
chaotic,
i.e. unpredictable behaviour of (nonlinear) ODE's. A wellknown example is the
Lorenz atrractor, illustrating the "Butterfly effect": small causes can have large effects.
Mathgrapher uses an accurate AdamsBashforth variable order, variable step predictorcorrector
algorithm to integrate systems of up to 20 coupled differential equations.
Several analytical tools
are available for ODE's such as:
Examples:
The Lorenz equations
The Lorenz equations are
Where c= Prandtl number, a = (normalised) Rayleigh number and b is a geometrical factor.
In MathGrapher you may import these equations by going to File=> Open=>Function=>ODE and selecting Lorenz.fct.
Time Series
Below the result is shown of an integration of the Lorenz oscillator (see Examples).
All 3 Functions are selected in the combo box on the right. Different colors are chosen for the
Functions using the Edit button in the main button bar.
Power Spectrum
The normalised power spectrum is calculated from the results of the Integration.
Note that only the first 2**k points are used in the calculation of the Power spectrum,
where k is the largest integer for which 2**k is smaller than, or equal to N.
The example below is again for an Integration of the Lorenz oscillator showing
chaotic behaviour in spite of the nice regular orbit (see 2D and 3D projection).
Projection in 2D
Choose a pair of F's from the Functions that are integrated and push Draw to make
a graph where the Xaxis represents the first and the Yaxis the second Function.
Below the results of an integration of the Lorenz oscillator is given.
Erase the Graph and Push the Show Evolution button in the Integration and Analysis
panel of the Prepare / Draw window to see the orbit in slow motion.
When more than 4 equations are integrated multiple orbits may be drawn by
selecting more than one pair of Functions in the combo box of the Analysis panel.
Projection in 3D
Choose three F's from the Functions that are integrated and push Draw to make a graph where the
Xaxis represents the first, the Yaxis the second Function and the Zaxis the third Function.
Push the Edit button in the main button bar to edit the Graph
You may draw the orbit in 3D as well as the projections of the orbit on the three planes.
Below the results of an integration of the Lorenz oscillator are shown.
Erase the Graph and Push the Show Evolution button in the Integration and Analysis panel of
the Prepare / Draw window to see the orbit evolve in time.
