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  • Ordinary differential equations
    You are here: Modules > ODE's
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    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

    Lorenz Equations Hénon-Heiles potential Rössler Equation
    van der Pol oscillator    Duffing oscillator Predator-Prey equation (Voltera)

    Predator-Prey equation (Voltera)

    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.

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