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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:

# Chaos

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.

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