reduct2.mac is from the book "Perturbation Methods, Bifurcation Theory
and Computer Algebra" by Rand & Armbruster (Springer 1987)

It performs a Liapunov-Schmidt reduction for steady state bifurcations
in systems of ordinary differential equations. 

The example is from p178.  maxima-5.9.0 cvs reproduces the
results from the book.

The system of equations is the Lorenz system

    x1' = sigma (x2-x1)
    x2' = rho x1 - x2 - x1 x3
    x3' = -beta x3 + x1 x2

It is know that for rho=1 one of the eigenvalues is zero with critical
eigenvector [1,1,0] and the adjoint critical eigenvector [1/sigma,1,0]
The following run determines the bifurcation equation for the
instability.

(C1) load("./reduct2.mac");
(D1)                             ./reduct2.mac
(C2) reduction2();
NUMBER OF EQUATIONS
3;
ENTER VARIABLE NUMBER 1
x1;
ENTER VARIABLE NUMBER 2
x2;
ENTER VARIABLE NUMBER 3
x3;
ENTER THE BIFURCATION PARAMETER
rho;
ENTER THE CRITICAL BIFURCATION VALUE RHO
1;
WE DEFINE LAM = RHO - 1
ENTER THE CRITICAL EIGENVECTOR AS A LIST
[1,1,0];
ENTER THE ADJOINT CRITICAL EIGENVECTOR
[1/sigma,1,0];
ENTER THE DIFFERENTIAL EQUATION
DIFF( x1 ,T)=
sigma*(x2-x1);
DIFF( x2 ,T)=
-x1*x3+rho*x1-x2;
DIFF( x3 ,T)=
x1*x2-beta*x3;
[SIGMA (x2 - x1), - x1 x3 - x2 + (LAM + 1) x1, x1 x2 - BETA x3]
DO YOU KNOW APRIORI THAT SOME TAYLOR COEFFICIENTS
 ARE ZERO, Y/N
N;
TO WHICH ORDER DO YOU WANT TO CALCULATE
3;

Dependent equations eliminated:  (1)
  2          2          2
 d W1       d W2       d W3     2
[----- = 0, ----- = 0, ----- = ----]
     2          2          2   BETA
 dAMP       dAMP       dAMP

Dependent equations eliminated:  (1)
    2                                   2
   d W1               SIGMA            d W2               1
[--------- = - --------------------, --------- = --------------------,
 dAMP dLAM          2                dAMP dLAM        2
               SIGMA  + 2 SIGMA + 1              SIGMA  + 2 SIGMA + 1

                                                                   2
                                                                  d W3
                                                                --------- = 0]
                                                                dAMP dLAM
                                             3
                                          AMP
(D2)                            AMP LAM - ----
                                          BETA



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