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# Modules

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 (non-linear) ODE's. A well-known example is the Lorenz atrractor, illustrating the "Butterfly effect": small causes can have large effects. Mathgrapher uses an accurate Adams-Bashforth variable order, variable step predictor-corrector algorithm to integrate systems of up to 20 coupled differential equations.

Several analytical tools are available for ODE's such as:
 Time series Power spectra 2D and 3D projections Phase portraits Poincare section Equilibrium points Lyapunov exponents

Examples:

# Rössler equation

Is given by:

We have adopted a=0.2, b=0.2 and c=4.2 with initial values F1=X=0.8, F2=Y=0.5 and F3=Z=0.25.
The integration was done from T=0 to 1024 and output was generated at 1024 intermediate points.
The graph shows X = F1 versus Y = F2

The same parameters were used to calculate the power spectrum of Z = F3 below. It is interesting to study this system for different values of c. As c is increased from about 2.5 to 5 the orbit undergoes a sequence of period-doubling bifurcations (as in the logistic map; see Iterations) until a strange attractorlike behaviour is attained.

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