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: ***