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  • Ordinary differential equations
    You are here: Modules > ODE's
    Links in the text refer to the lower part of the page

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

    Hénon-Heiles potential

    Is given by:

    The potential supports bounded motion for E<1/6 which is the triangular central region. The astronomers Michel Hénon and Carl Heiles discovered that this potential, which describes the potential felt by a star moving in a smooth cylindracally symmetric galaxy (it also provides a simple model for a pair of nonlinearly coupled oscillators), yields regular orbits for some initial conditions and irregular, chaotic orbits for other initial conditions. This behaviour is nicely illustrated in the socalled Surfaces of Section (originally introduced by Poincaré, see below)

    The equations of motion can be derived from the Hénon-Heiles Hamiltonian

    where the (non-linear) potential is given by

    The potential supports bounded motion for E<1/6 (see a 3D graph and Contour
    plot of this potential produced by MathGrapher) The Hamiltonian yields the following equations of motion

    import these equations by going to File=> Open=>Function=>ODEs and selecting Henon-Heiles.fct.
    It is interesting to study the braking up of smooth regular orbits into chaotic motion when the energy increases.
    This can be studied using the well known surfaces of section (or Poincaré section).

    Surface of Section (Poincaré section)

    The Henon-Heiles equations with initial values F1=0.2, F2=0.0, F3=0.4, F4=0.0 (E=0.1) are integrated from T=1 to 250 giving output at 2000 intermediate points.
    This yields the following surface of section at F1=0.0
    The regular orbits ly on a torus. The linear structures seen in the graph represent cross-sections of the torus through the plane. Irregular orbits are not confined to a torus and spread out over a region in the graph.

    The graph below gives the section points for another orbit with E=0.1 starting at F1=0.0, F2=0.0, F3=0.4, F5=0.2 yielding the two circular patterns. The other points are for a slightly higher energy (E=1/7) starting at F3=0.5 instead of 0.4. It illustrates how the orbit is no longer confined to a torus, but spreads out over a larger volume in phase space as the energy increases.

     
     
     
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