ODE’s

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 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)

The Lorenz equations

The Lorenz equations are

Where c= Prandtl number, a = (normalised) Rayleigh number and b is a geometrical factor.

In MathGrapher you may import these equations by going to File=> Open=>Function=>ODE and selecting Lorenz.fct.

Time Series

Below the result is shown of an integration of the Lorenz oscillator (see Examples). All 3 Functions are selected in the combo box on the right. Different colors are chosen for the Functions using the Edit button in the main button bar.

Power Spectrum

The normalized power spectrum is calculated from the results of the Integration. Note that only the first 2k points are used in the calculation of the Power spectrum, where k is the largest integer for which 2k is smaller than, or equal to N. The example below is again for an Integration of the Lorenz oscillator showing chaotic behavior in spite of the nice regular orbit (see 2D and 3D projection).

Projection in 2D

Choose a pair of F’s from the Functions that are integrated and push Draw to make a graph where the X-axis represents the first and the Y-axis the second Function. Below the results of an integration of the Lorenz oscillator is given.

Erase the Graph and Push the Show Evolution button in the Integration and Analysis panel of the Prepare / Draw window to see the orbit in slow motion. When more than 4 equations are integrated multiple orbits may be drawn by selecting more than one pair of Functions in the combo box of the Analysis panel.

Projection in 3D

Choose three F’s from the Functions that are integrated and push Draw to make a graph where the X-axis represents the first, the Y-axis the second Function and the Z-axis the third Function. Push the Edit button in the main button bar to edit the Graph You may draw the orbit in 3D as well as the projections of the orbit on the three planes.

Below the results of an integration of the Lorenz oscillator are shown. Erase the Graph and Push the Show Evolution button in the Integration and Analysis panel of the Prepare / Draw window to see the orbit evolve in time.