by Jean-Marc Ginoux (ginoux@univ-tln.fr)

© Copyright Jean-Marc Ginoux,2005

http://ginoux.univ-tln.fr

This notebook contains an applet elaborated in order to provide the slow manifold analytical equation of any three-dimensional dynamical system. It just needs Mathematica (version 3, 4 or 5).

A. Dynamical system

Let's consider the following dynamical system :

((dx)/(dt))=(Subscript[x, 1])=f(x, y, z, λ)
((dy)/(dt))=(Subscript[y, 1])=g (x, y, z, μ)
((dz)/(dt))=(Subscript[z, 1])=h(x,y,z,ρ)

where λ, μ and ρ are parameters.

B. Functional Jacobian matrix elements

a = Simplify[D[(Subscript[x, 1]), x]]

b = Simplify[D[(Subscript[x, 1]), y]]

c = Simplify[D[(Subscript[x, 1]), z]]

d = Simplify[D[(Subscript[y, 1]), x]]

e = Simplify[D[(Subscript[y, 1]), y]]

f = Simplify[D[(Subscript[y, 1]), z]]

g = Simplify[D[(Subscript[z, 1]), x]]

h = Simplify[D[(Subscript[z, 1]), y]]

i = Simplify[D[(Subscript[z, 1]), z]]

C. Slow Manifold Analytical Equation

For more details see ThesisCorpus

K = (-((((c^2) d + c (-a + e) f - b (f^2)))^2) (-f (g^2) + (d^2) h + 
                    d g (-e + 
                          i))) (Subscript[x, 1]^3) + (-((((c^2) d + c (-a + e) f - b (f^2)))^2) (-(b^2) g + c (h^2) + 
                    b h (a - 
                          i))) (Subscript[y, 1]^3) + (-((((c^2) d + c (-a + e) f - b (f^2)))^2) ((b^2) d - a b e - 2 b c g + a c h - 
                    2 c e h + b f h + a b i + b e i + c h i - 
                    b (i^2))) (Subscript[y, 1]^2) (Subscript[z, 1]) + (-((((c^2) d + c (-a + e) f - b (f^2)))^2) (2 b c d - a c e + c (e^2) - 
                    a b f - b e f - (c^2) g - c f h + a c i - 
                    c e i + 
                    2 b f i)) (Subscript[y, 1]) (Subscript[z, 1]^2) + (-((((c^2) d + c (-a + e) f - b (f^2)))^2) (-b (f^2) + (c^2) d  - 
                    c f (a - e))) (Subscript[z, 1]^3) + 
          (Subscript[x, 1]^2) ((-((((c^2) d + c (-a + e) f - b (f^2)))^2) (b d g + a e g - (e^2) g + c (g^2) - 2 a d h + 
                          d e h - 2 f g h - a g i + e g i + 
                          d h i)) (Subscript[y, 1]) + (-((((c^2) d + c (-a + e) f - b (f^2)))^2) (-b (d^2) + a d e - c d g + 
                          2 a f g - e f g + 2 d f h - a d i - 
                          d e i - f g i + d (i^2))) (Subscript[z, 1])) + 
          (Subscript[x, 1]) ((-((((c^2) d + c (-a + e) f - b (f^2)))^2) (-a b g + 2 b e g + (a^2) h - b d h - a e h + 
                          2 c g h - f (h^2) - b g i - a h i + 
                          e h i)) (Subscript[y, 1]^2) + (-((((c^2) d + c (-a + e) f - b (f^2)))^2) (a b d - (a^2) e - b d e + 
                          a (e^2) - a c g + 3 b f g - 3 c d h + 
                          e f h + (a^2) i - (e^2) i + c g i - f h i - 
                          a (i^2) + 
                          e (i^2))) (Subscript[y, 1]) (Subscript[z, 1]) + (-((((c^2) d + c (-a + e) f - b (f^2)))^2) (a c d + c d e - (a^2) f - 
                          2 b d f + a e f + c f g + (f^2) h - 
                          2 c d i + a f i - 
                          e f i)) (Subscript[z, 1]^2));

<< Graphics`ContourPlot3D`

plot1 = 
      ContourPlot3D[K, {x, -30, 30}, {y, -30, 30}, {z, 0, 50}, 
        PlotPoints -> {10, 10}, Axes -> True, 
        AxesLabel -> {"x", "y", "z"}]

D. Example

This example provides the slow manifold analyticla equation of the dynamical system elaborated by E.N. Lorenz [1963].

(Subscript[x, 1]) = 10 (y -  x);
(Subscript[y, 1]) = -x z + 28  x - y;
(Subscript[z, 1]) = x y - (8/3) z;

a = -10;
b = 10;
c = 0;
d = 28 - z;
e = -1;
f = -x;
g = y;
h = x;
i = -(8/3);

K = -1524096 (x^4) + 21168 (x^6) + 5166504 (x^3) y - 
          2772 (x^5) y + 1818126 (x^2) (y^2) - 1332 (x^4) (y^2) + 
          27 (x^6) (y^2) - 2195100 x (y^3) + 6030 (x^3) (y^3) + 
          194400 (y^4) + 3420 (x^2) (y^4) - 2700 x (y^5) + 
          15906240 (x^2) z + 71904 (x^4) z - 1512 (x^6) z + 
          5112720 x y z - 539550 (x^3) y z - 45 (x^5) y z - 
          5680800 (y^2) z - 168420 (x^2) (y^2) z + 168120 x (y^3) z - 
          270 (x^3) (y^3) z - 7200 (y^4) z - 1550864 (x^2) (z^2) - 
          432 (x^4) (z^2) + 27 (x^6) (z^2) - 485600 x y (z^2) + 
          20700 (x^3) y (z^2) + 404800 (y^2) (z^2) + 
          3150 (x^2) (y^2) (z^2) - 2700 x (y^3) (z^2) + 55200 (x^2) (z^3) + 
          10320 x y (z^3) - 270 (x^3) y (z^3) - 7200 (y^2) (z^3) - 
          720 (x^2) (z^4);

<< Graphics`ContourPlot3D`

plot1 = 
      ContourPlot3D[K, {x, -30, 30}, {y, -30, 30}, {z, 0, 50}, 
        PlotPoints -> {10, 10}, Axes -> True, 
        AxesLabel -> {"x", "y", "z"}]; 
    Show[plot0, plot1, ImageSize -> {400, 400}]