Ordinary differential equations
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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
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:
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.