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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 focussed on the often chaotic, 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:

Time series

Power spectra

2D and 3D projections

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)

ODE's - Introduction

An ordinary differential equation is a differential equation that contains only one variable, for instance Time. Such an equation may contain derivatives and derivatives of derivatives. The highest derivative determines the order of the ODE. For example, Newton's law for the acceleration of a particle in a gravitational field (in one dimension) is described by a second order ODE:

Many dynamical systems can be discribed by such an higher order ODE. The above 2nd order equation is equivalent to the following set of first order ODE's

This example is an example of the following general rule: Any nth order ordinary differential equation can be reduced to a set of n first order differential equations. Mathgrapher allows you to integrate a set of 20 coupled first-order equations. So, first you have to write your differential equation in an equivalent set of first order equations as in the example given above. In the example given above you would define F1=x(t), F2=v(t) and dF1/dt=F2 , dF2/dt=c/x^2. Set the initial values of F1 and F2, the initial and final values of the time and start the integration. When the integration is finished you may draw the orbit. It is straightforward to extend this example to 2 or 3 dimensions and calculate the orbits of a particle in a central force (gravitational) field, and to add frictional terms to study, for example, the orbit of a satellite in the atmosphere of the earth.