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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 set of ODE's has an equilibrium point when dF/dt=0 for all F's.
There are several types of equilibrium points depending on the behaviour of the
orbits in their neighbourhood: stable and unstable nodes, saddle points, stable
and unstable spiral points. The behaviour of the orbits near an equilibrium point
can be found by studying the linearized equations
(see below). Suppose we have a system of 2 coupled ODE's:

dx/dt=x(4-x-y)

dy/dt=y(x-2)

The eigenvalues of the stability matrix J (see below) determine the type of equilibrium
point and its stability properties. They are the roots of the equation

where I is the identity matrix. In the 2-dimensional case discussed here this
generally yields 2 solutions. Stable point have negative eigenvalues,
unstable points have positive eigenvalues, saddle points have one negative
and one positive eigenvalue and spiral points have complex eigenvalues.
They may be stable or unstable depending on their position in the complex
plain.

The fixed points are (0.0), (2,2) and (4,0). To find the eigenvalues we have
to solve the equation:

For (0,0) these are -2 and 4 => hyperbolic point, for (2,2) we find -1+i*sqrt(3)
and -1-i*sqrt(3) i.e. stable spiral point, and for (4,0) -4 and 2, i.e. hyperbolic point.
Check these results with Mathgrapher. To find an equilibrium point (and its eigenvlaues)
you have to give initial values somewhere in its neighbourhood.