1reduct2.mac is from the book "Perturbation Methods, Bifurcation Theory 2and Computer Algebra" by Rand & Armbruster (Springer 1987) 3 4It performs a Liapunov-Schmidt reduction for steady state bifurcations 5in systems of ordinary differential equations. 6 7The example is from p178. maxima-5.9.0 cvs reproduces the 8results from the book. 9 10The 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 16It is know that for rho=1 one of the eigenvalues is zero with critical 17eigenvector [1,1,0] and the adjoint critical eigenvector [1/sigma,1,0] 18The following run determines the bifurcation equation for the 19instability. 20 21(C1) load("./reduct2.mac"); 22(D1) ./reduct2.mac 23(C2) reduction2(); 24NUMBER OF EQUATIONS 253; 26ENTER VARIABLE NUMBER 1 27x1; 28ENTER VARIABLE NUMBER 2 29x2; 30ENTER VARIABLE NUMBER 3 31x3; 32ENTER THE BIFURCATION PARAMETER 33rho; 34ENTER THE CRITICAL BIFURCATION VALUE RHO 351; 36WE DEFINE LAM = RHO - 1 37ENTER THE CRITICAL EIGENVECTOR AS A LIST 38[1,1,0]; 39ENTER THE ADJOINT CRITICAL EIGENVECTOR 40[1/sigma,1,0]; 41ENTER THE DIFFERENTIAL EQUATION 42DIFF( x1 ,T)= 43sigma*(x2-x1); 44DIFF( x2 ,T)= 45-x1*x3+rho*x1-x2; 46DIFF( x3 ,T)= 47x1*x2-beta*x3; 48[SIGMA (x2 - x1), - x1 x3 - x2 + (LAM + 1) x1, x1 x2 - BETA x3] 49DO YOU KNOW APRIORI THAT SOME TAYLOR COEFFICENTS 50 ARE ZERO, Y/N 51N; 52TO WHICH ORDER DO YOU WANT TO CALCULATE 533; 54 55Dependent 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 62Dependent 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 80Local Variables: *** 81mode: Text *** 82End: ***