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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.

In the second half of the last century it was discovered that most (non-linear) dynamical
systems exhibit choatic behaviour. This came as a surprise. It was thought that
deterministic systems would behave in a predictable way and that choatic behaviour
in such systems would be due to many different external influences on a system. It
turned out, however, that non-linear systems which are fully deterministic and quite
simple and straightforward in mathematical sense often show chaotic behaviour. This
means that a very small change in the initial conditions of the system may may lead
to completely different end results thereby making the outcome completely unpredictable.
This was surprising because it occurs in systems whose behaviour is governed by
well-understood physical principles described by exact mathematical formulae. The
example that is often mentioned is the butterfly in Alaska which may cause a
thunderstorm in Texas just by flapping its wings.
Scientist may be surprised by this finding, however every pinball player knows
that a very slight change in the velocity of the ball at some point along its
trajectory (for instance by kicking the machine) may change the end result completely.

The examples contain two well-known examples of systems that exhibit chaotic behaviour.
Both are incorporated in the Demonstrations, so you just have to choose Demonstrations
from Mathgrapher's menu to see how you may
calculate and analyse such systems. The first example is a simple iterative
algorithm: the logistic map. The second example is a well known system of
Ordinary Differential Equations (ODE's): the Lorenz equations.