## 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)

## Duffing oscillator

is given by

which we can write as a system of coupled ODEs using

to obtain

For small d simple limit cycle behavior is observed. As d increases the cycles bifurcate into double period cycles and the motion becomes increasingly chaotic. The 2D (F1-F2) graph below is the result of an integration from T=0 to 50 using a=0.25, d=200, g=1.5. Initial values are F1=0.8, F2=0,5 and F3=0.25. Output is generated at 3000 intermediate time steps.

Below the power spectrum of F2. The same parameters were used, but in this case T runs from 0 to 250 and output is generated at 256 equal steps.