# 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:
d^{2}x/dt^{2} + b dx/dt - x + x^{3} = 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 - X^{3} + a cosZ |

dZ/dt |
= |
c |

## Pendulum

Non-linear and driven:
d^{2}x/dt^{2} + 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).
d^{2}x/dt^{2} - b (1 - x^{2}) dx/dt + x = a cos(ct)
In autonomous form with *X=x, Y=dx/dt, Z=ct*:
dX/dt |
= |
Y |

dY/dt |
= |
b(1 - X^{2})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.

[Diagnostics]
[Instructions]

Last modified Tuesday, December 30, 1997

Michael Cross