## 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)

## Phase portrait.

It is very easy to generate a phase portrait like the one shown below. Just define the differential equations. In this case these are the equations for a predator-prey model:

Push the Prepare / Draw graph button to open the graph window and the prepare window, select the Analysis tab, choose phase portrait and push the Draw button.

The other analytical tools allow you to draw extra orbits (orbit in 2D), find equilibrium points and calculate their stability properties (eigenvalues).