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 focused on the often chaotic, i.e. unpredictable behavior of (non-linear) ODE’s. A well-known example is the Lorenz atractor, 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 projection | 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 cylindrical symmetric galaxy (it also provides a simple model for a pair of non-linearly coupled oscillators), yields regular orbits for some initial conditions and irregular, chaotic orbits for other initial conditions. This behavior is nicely illustrated in the so-called 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.