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.
[Diagnostics]
[Instructions]
Last modified Tuesday, December 30, 1997
Michael Cross