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 partial differential equations depending on one 6independent space variable. 7 8The example is from p187. maxima-5.9.0 cvs reproduces the 9results from the book. 10 11 12 13(C1) load("reduct3.mac"); 14Warning - you are redefining the MACSYMA function SETIFY 15(D1) reduct3.mac 16(C2) reduction3(); 17ENTER THE NUMBER OF DIFFERENTIAL EQUATIONS 182; 19ENTER THE DEPENDENT VARIABLES AS A LIST 20[y1,y2]; 21ENTER THE SPATIAL COORDINATE 22x; 23ENTER THE BIFURCATION PARAMETER 24alpha; 25ENTER THE CRITICAL BIFURCATION VALUE 261; 27WE DEFINE LAM = ALPHA - 1 28ENTER THE CRITICAL EIGENFUNCTION AS A LIST 29sin(x)*[1,0]; 30ENTER THE ADJOINT CRITICAL EIGENFUNCTION AS A LIST 312/%pi*sin(x)*[1,1]; 32ENTER THE DIFFERENTIAL EQUATION NUMBER 1 33y2; 34ENTER THE DIFFERENTIAL EQUATION NUMBER 2 35'diff(y1,x,2)+alpha*y1-y2-y1^3-a*y1^5; 36 2 37 d Y1 5 3 38[Y2, - Y2 + ---- - a Y1 - Y1 + (LAM + 1) Y1] 39 2 40 dx 41WHAT IS THE LENGTH OF THE SPACE INTERVAL 42%pi; 43DO YOU KNOW APRIORI THAT SOME TAYLOR COEFFICIENTS ARE 0 44Y,N 45Y; 46TO WHICH ORDER DO YOU WANT TO CALCULATE 475; 48IS DIFF(W(AMP, 2 ,LAM, 0 ) IDENTICALLY ZERO, Y/N 49Y; 50IS DIFF(W(AMP, 3 ,LAM, 0 ) IDENTICALLY ZERO, Y/N 51N; 52 53Dependent equations eliminated: (1) 54 3 3 55 d W1 3 SIN(3 x) + 72 SIN(x) d W2 9 SIN(x) 56[----- = ----------------------, ----- = - --------] 57 3 16 3 2 58 dAMP dAMP 59IS DIFF(W(AMP, 4 ,LAM, 0 ) IDENTICALLY ZERO, Y/N 60Y; 61IS DIFF(W(AMP, 1 ,LAM, 1 ) IDENTICALLY ZERO, Y/N 62N; 63 64Dependent equations eliminated: (2) 65 2 2 66 d W1 d W2 67[--------- = - SIN(x), --------- = SIN(x)] 68 dAMP dLAM dAMP dLAM 69IS DIFF(W(AMP, 2 ,LAM, 1 ) IDENTICALLY ZERO, Y/N 70Y; 71IS DIFF(W(AMP, 3 ,LAM, 1 ) IDENTICALLY ZERO, Y/N 72N; 73 74Dependent equations eliminated: (1) 75 4 4 76 d W1 69 SIN(3 x) + 2304 SIN(x) d W2 77[---------- = - -------------------------, ---------- = 18 SIN(x)] 78 3 128 3 79 dAMP dLAM dAMP dLAM 80IS G_POLY( 1 , 0 )IDENTICALLY ZERO, Y/N 81Y; 82IS G_POLY( 2 , 0 )IDENTICALLY ZERO, Y/N 83Y; 84IS G_POLY( 3 , 0 )IDENTICALLY ZERO, Y/N 85N; 86IS G_POLY( 4 , 0 )IDENTICALLY ZERO, Y/N 87Y; 88IS G_POLY( 5 , 0 )IDENTICALLY ZERO, Y/N 89N; 90IS G_POLY( 1 , 1 )IDENTICALLY ZERO, Y/N 91N; 92IS G_POLY( 2 , 1 )IDENTICALLY ZERO, Y/N 93Y; 94IS G_POLY( 3 , 1 )IDENTICALLY ZERO, Y/N 95N; 96IS G_POLY( 4 , 1 )IDENTICALLY ZERO, Y/N 97Y; 98 5 3 99 3 (- 1200 %PI a - 3195 %PI) AMP 3 AMP 100(D2) 3 AMP LAM + AMP LAM + ------------------------------ - ------ 101 1920 %PI 4 102(C3) solve(%,lam); 103 4 2 104 (80 a + 213) AMP + 96 AMP 105(D3) [LAM = ---------------------------] 106 2 107 384 AMP + 128 108(C4) taylor(%,amp,0,4); 109 2 4 110 3 AMP (80 a - 75) AMP 111(D4)/T/ [LAM + . . . = ------ + ---------------- + . . .] 112 4 128 113 114 115Local Variables: *** 116mode: Text *** 117End: ***