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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.

A simple equation to describe population dynamics was introduced by Voltera. It is given by

where x might describe a rabbit population and y a fox population. The populations
are expected to change in cycles depending on the initial conditions.