# Chaos in ODEs - Equations

## Lorenz Equation

The Lorenz model is defined by three differential equations giving the time evolution of the variables X(t), Y(t), Z(t):

 dX/dt = -c(X - Y) dY/dt = aX - Y - XZ dZ/dt = b(XY - Z)

## Rossler Equation

The Rossler equation was written down as a caricature of a system of chemical reactions.
 dX/dt = Y - Z dY/dt = bY - X dZ/dt = c + Z(X - a)

## Duffing Equation

Perhaps the simplest nonlinear oscillator - the motion of a particle in a quartic potential - driven harmonically:

d2x/dt2 + b dx/dt - x + x3 = a cos(ct)

This can be written in autonomous form with X=x, Y=dx/dt, Z=ct:

 dX/dt = Y dY/dt = -bY + X - X3 + a cosZ dZ/dt = c

## Pendulum

Non-linear and driven:

d2x/dt2 + b dx/dt + sin x = a cos(ct) + d

Again in autonomous form with X=x, Y=dx/dt, Z=ct:

 dX/dt = Y dY/dt = -bY - sinX + a cosZ + d dZ/dt = c

## Van der Pohl Oscillator

In this oscillator the nonlinearity is in the damping (which is negative for small amplitudes).

d2x/dt2 - b (1 - x2) dx/dt + x = a cos(ct)

In autonomous form with X=x, Y=dx/dt, Z=ct:

 dX/dt = Y dY/dt = b(1 - X2)Y - X + a cosZ dZ/dt = c

## Chua's Circuit

These are equations for an electronic circuit that shows chaos. The equations are rather complicated:

 dX/dt = a(Y -X ) - f(X) dY/dt = b[a(X - Y) + Z] dZ/dt = -c(Y + d Z)

where f is a nonlinear function that is odd in X and for positive X is defined by:

 f(X) = -X for X < 1 f(X) = -1 - 0.636(X - 1) for 1 < X < 10 f(X) = 10(X - 10) - 6.724 for X > 10

It is easier to look at the function. The number 0.636 is a particular choice of the ratio g1/g2 there.

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