share/contrib/rand/reduct2.usg

   1 reduct2.mac is from the book "Perturbation Methods, Bifurcation Theory
   2 and Computer Algebra" by Rand & Armbruster (Springer 1987)
   3
   4 It performs a Liapunov-Schmidt reduction for steady state bifurcations
   5 in systems of ordinary differential equations.
   6
   7 The example is from p178.  maxima-5.9.0 cvs reproduces the
   8 results from the book.
   9
  10 The system of equations is the Lorenz system
  11
  12     x1' = sigma (x2-x1)
  13     x2' = rho x1 - x2 - x1 x3
  14     x3' = -beta x3 + x1 x2
  15
  16 It is know that for rho=1 one of the eigenvalues is zero with critical
  17 eigenvector [1,1,0] and the adjoint critical eigenvector [1/sigma,1,0]
  18 The following run determines the bifurcation equation for the
  19 instability.
  20
  21 (C1) load("./reduct2.mac");
  22 (D1)                             ./reduct2.mac
  23 (C2) reduction2();
  24 NUMBER OF EQUATIONS
  25 3;
  26 ENTER VARIABLE NUMBER 1
  27 x1;
  28 ENTER VARIABLE NUMBER 2
  29 x2;
  30 ENTER VARIABLE NUMBER 3
  31 x3;
  32 ENTER THE BIFURCATION PARAMETER
  33 rho;
  34 ENTER THE CRITICAL BIFURCATION VALUE RHO
  35 1;
  36 WE DEFINE LAM = RHO - 1
  37 ENTER THE CRITICAL EIGENVECTOR AS A LIST
  38 [1,1,0];
  39 ENTER THE ADJOINT CRITICAL EIGENVECTOR
  40 [1/sigma,1,0];
  41 ENTER THE DIFFERENTIAL EQUATION
  42 DIFF( x1 ,T)=
  43 sigma*(x2-x1);
  44 DIFF( x2 ,T)=
  45 -x1*x3+rho*x1-x2;
  46 DIFF( x3 ,T)=
  47 x1*x2-beta*x3;
  48 [SIGMA (x2 - x1), - x1 x3 - x2 + (LAM + 1) x1, x1 x2 - BETA x3]
  49 DO YOU KNOW APRIORI THAT SOME TAYLOR COEFFICIENTS
  50  ARE ZERO, Y/N
  51 N;
  52 TO WHICH ORDER DO YOU WANT TO CALCULATE
  53 3;
  54
  55 Dependent equations eliminated:  (1)
  56   2          2          2
  57  d W1       d W2       d W3     2
  58 [----- = 0, ----- = 0, ----- = ----]
  59      2          2          2   BETA
  60  dAMP       dAMP       dAMP
  61
  62 Dependent equations eliminated:  (1)
  63     2                                   2
  64    d W1               SIGMA            d W2               1
  65 [--------- = - --------------------, --------- = --------------------,
  66  dAMP dLAM          2                dAMP dLAM        2
  67                SIGMA  + 2 SIGMA + 1              SIGMA  + 2 SIGMA + 1
  68
  69                                                                    2
  70                                                                   d W3
  71                                                                 --------- = 0]
  72                                                                 dAMP dLAM
  73                                              3
  74                                           AMP
  75 (D2)                            AMP LAM - ----
  76                                           BETA
  77
  78
  79
  80 Local Variables: ***
  81 mode: Text ***
  82 End: ***