1reduct1.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 one differential equation depending on one independent variable. 6The de has the form y'' + f(y,y',alpha) = 0. y = y(x) is defined on a 7real interval with dirichlet or neumann boundary conditions and f 8depends only linearly on alpha. 9 10The example is from p168. maxima-5.9.0 cvs reproduces the 11results from the book. 12 13 14(C1) load("./reduct1.mac"); 15(D1) ./reduct1.mac 16(C2) reduction1(); 17ENTER DEPENDENT VARIABLE 18Y; 19USE X AS THE INDEPENDENT VARIABLE AND ALPHA AS A PARAMETER TO VARY 20ENTER THE CRITICAL BIFURCATION VALUE ALPHA 21%PI^2; 22 2 23WE DEFINE LAM = ALPHA - %PI 24ENTER THE CRITICAL EIGENFUNCTION 25COS(%PI*X); 26WHAT IS THE LENGTH OF THE X-INTERVAL 271; 28SPECIFY THE BOUNDARY CONDITIONS 29YOUR CHOICE FOR THE B.C. ON Y AT X=0 AND X= 1 30ENTER 1 FOR Y=0, 2 FOR Y'=0 31B.C. AT 0? 322; 33B.C. AT 1 ? 342; 35THE D.E. IS OF THE FORM Y'' + F(Y,Y',ALPHA) = 0,ENTER F 36ALPHA*SIN(Y); 37 2 38d Y 2 39--- + (LAM + %PI ) SIN(Y) 40 2 41dX 42DO YOU KNOW APRIORI THAT SOME TAYLOR COEFFICENTS ARE ZERO, Y/N 43Y; 44TO WHICH ORDER DO YOU WANT TO CALCULATE 455; 46IS DIFF(W,AMP, 2 ,LAM, 0 ) IDENTICALLY ZERO 47 48, Y/N 49Y; 50IS DIFF(W,AMP, 3 ,LAM, 0 ) IDENTICALLY ZERO 51 52, Y/N 53N; 54 55Dependent equations eliminated: (2) 56 3 57 d W COS(3 %PI X) 58[----- = - ------------] 59 3 32 60 dAMP 61IS DIFF(W,AMP, 4 ,LAM, 0 ) IDENTICALLY ZERO 62 63, Y/N 64Y; 65IS DIFF(W,AMP, 1 ,LAM, 1 ) IDENTICALLY ZERO 66 67, Y/N 68N; 69 70Dependent equations eliminated: (2) 71 2 72 d W 73[--------- = 0] 74 dAMP dLAM 75IS DIFF(W,AMP, 2 ,LAM, 1 ) IDENTICALLY ZERO 76 77, Y/N 78Y; 79IS DIFF(W,AMP, 3 ,LAM, 1 ) IDENTICALLY ZERO 80 81, Y/N 82N; 83 84Dependent equations eliminated: (2) 85 4 86 d W 9 COS(3 %PI X) 87[---------- = - --------------] 88 3 2 89 dAMP dLAM 256 %PI 90IS G_POLY( 1 , 0 ) IDENTICALLY 91 92ZERO, Y/N 93Y; 94IS G_POLY( 2 , 0 ) IDENTICALLY 95 96ZERO, Y/N 97Y; 98IS G_POLY( 3 , 0 ) IDENTICALLY 99 100ZERO, Y/N 101N; 102IS G_POLY( 4 , 0 ) IDENTICALLY 103 104ZERO, Y/N 105Y; 106IS G_POLY( 5 , 0 ) IDENTICALLY 107 108ZERO, Y/N 109N; 110IS G_POLY( 1 , 1 ) IDENTICALLY 111 112ZERO, Y/N 113N; 114IS G_POLY( 2 , 1 ) IDENTICALLY 115 116ZERO, Y/N 117Y; 118IS G_POLY( 3 , 1 ) IDENTICALLY 119 120ZERO, Y/N 121N; 122IS G_POLY( 4 , 1 ) IDENTICALLY 123 124ZERO, Y/N 125Y; 126 3 2 5 2 3 127 AMP LAM AMP LAM 3 %PI AMP %PI AMP 128(D2) - -------- + ------- + ----------- - --------- 129 16 2 1024 16 130(C3) solve(%,lam); 131 2 4 2 2 132 3 %PI AMP - 64 %PI AMP 133(D3) [LAM = --------------------------] 134 2 135 64 AMP - 512 136(C4) taylor(%,amp,0,4); 137 2 2 2 4 138 %PI AMP (5 %PI ) AMP 139(D4)/T/ [LAM + . . . = --------- + ------------- + . . .] 140 8 512 141 142 143Local Variables: *** 144mode: Text *** 145End: ***