Chaos and Strange Attractors: Lorentz Equations

Dr. Frank Wang

> restart:

> with(DEtools): with(plots):

Warning, the name changecoords has been redefined

> sigma, b, r := 10, 8/3, 21;

sigma, b, r := 10, 8/3, 21

> Eq1 := diff(x(t),t) = sigma*(-x(t) + y(t));

Eq1 := diff(x(t), t) = -10*x(t)+10*y(t)

> Eq2 := diff(y(t),t) = r*x(t) - y(t) - x(t)*z(t);

Eq2 := diff(y(t), t) = 21*x(t)-y(t)-x(t)*z(t)

> Eq3 := diff(z(t),t) = -b*z(t) + x(t)*y(t);

Eq3 := diff(z(t), t) = -8/3*z(t)+x(t)*y(t)

> DEplot3d({Eq1, Eq2, Eq3}, [x(t), y(t), z(t)], t=0..20, [[x(0)=5, y(0)=5, z(0)=5]], stepsize=0.01, linecolor=t, orientation=[-30,60]);

[Plot]

> Soln1 := dsolve({Eq1, Eq2, Eq3, x(0)=5, y(0)=5, z(0)=5}, {x(t), y(t), z(t)}, type=numeric);

Soln1 := proc (x_rkf45) local res, data, vars, solnproc, outpoint, ndsol, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 14; i...

> Soln2 := dsolve({Eq1, Eq2, Eq3, x(0)=5.01, y(0)=5, z(0)=5}, {x(t), y(t), z(t)}, type=numeric);

Soln2 := proc (x_rkf45) local res, data, vars, solnproc, outpoint, ndsol, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 14; i...

> p1 := odeplot(Soln1, [t, x(t)], t=0..12, numpoints=800):

> p2 := odeplot(Soln2, [t, x(t)], t=0..12, numpoints=800, color=blue):

> display([p1, p2]);

[Plot]

>