1takens.mac is from the paper "Determinacy of Degenerate Equilibria
2with Linear Part x'=y, y'=0 Using MACSYMA", R.H.Rand, W.L.Keith
3Applied Mathematics and Computation 21:1-19 (1987)
4(http://tam.cornell.edu/Rand.html)
5
6The program implements Taken's method of proving the determinacy of a
7flow in the neighbourhood of a equilibrium point by successive blowup
8transformations.
9
10The appendix in the paper is reproduced with maxima-5.9.0-cvs.  Some
11of the inputs are case sensitive - when I entered the equations in
12lower case the answers differed.
13
14(C1) load("takens.mac");
15(D1)                              takens.mac
16(C2) takens();
17 ENTER THE RHS'S TO BE STUDIED
18 USE VARIABLES X,Y, THEY WILL BE CONVERTED TO X1,Y1
19U1 =
20Y+B2*X^2+B3*X^3;
21          3        2
22Y1 + B3 X1  + B2 X1
23V1 =
24A3*X^3+A4*X^4;
25     4        3
26A4 X1  + A3 X1
27          4           3                   4        3
28F1 = A4 X1  Y1 + A3 X1  Y1 + X1 Y1 + B3 X1  + B2 X1
29         2        3           2           5        4
30G1 = - Y1  - B3 X1  Y1 - B2 X1  Y1 + A4 X1  + A3 X1
31 TAKENS' TEST
32 TRUNCATE F AND G TO HOMOGENEOUS POLYNOMIALS
33                    2
34[Y1 X1 + . . ., - Y1  + . . .]
35SOLVING GTRUNC = 0
36TOTAL NO. OF ROOTS = 1
37Y1 = 0
38FTRUNC IS ZERO!
39FAILED TEST
40          4    4                    3    3
41P1 = A4 R1  COS (S1) SIN(S1) + A3 R1  COS (S1) SIN(S1) + R1 COS(S1) SIN(S1)
42
43                                                  3    4            2    3
44                                           + B3 R1  COS (S1) + B2 R1  COS (S1)
45          2            2    3                        2
46Q1 = - SIN (S1) - B3 R1  COS (S1) SIN(S1) - B2 R1 COS (S1) SIN(S1)
47
48                                                  3    5            2    4
49                                           + A4 R1  COS (S1) + A3 R1  COS (S1)
50DIVIDE OUT 1
51NOW SET R1 = 0
52PP1 = 0
53NOTE: PREVIOUS SHOULD BE ZERO!
54           2
55QQ1 = - SIN (S1)
56
57SOLVE is using arc-trig functions to get a solution.
58Some solutions will be lost.
59ROOT NO. 1 , S1 = 0
60THERE ARE 1 ROOTS
61PICK A ROOT NO., OR 0 TO ENTER ONE
621;
63S1 STAR = 0
64KEEP TERMS OF WHAT POWER?
653;
66U2 =
67             2        3
68Y2 X2 + B2 X2  + B3 X2  + . . .
69V2 =
70    2                   2         3           2
71- Y2  - B2 Y2 X2 + A3 X2  + (A4 X2  - B3 Y2 X2 ) + . . .
72         3        2   2           2        3           2        2           4
73F2 = - Y2  - B3 X2  Y2  - B2 X2 Y2  + A4 X2  Y2 + A3 X2  Y2 + X2  Y2 + B3 X2
74
75                                                                             3
76                                                                      + B2 X2
77              2          3             2           4        3
78G2 = - 2 X2 Y2  - 2 B3 X2  Y2 - 2 B2 X2  Y2 + A4 X2  + A3 X2
79 TAKENS' TEST
80 TRUNCATE F AND G TO HOMOGENEOUS POLYNOMIALS
81     3        2                    2        3
82[- Y2  - B2 Y2  X2 + (A3 + 1) Y2 X2  + B2 X2  + . . .,
83
84                                          2                2        3
85                                    - 2 Y2  X2 - 2 B2 Y2 X2  + A3 X2  + . . .]
86SOLVING GTRUNC = 0
87TOTAL NO. OF ROOTS = 5
88              2
89       SQRT(B2  + 2 A3) Y2 - B2 Y2
90X2 = - ---------------------------
91                   A3
92            2
93     SQRT(B2  + 2 A3) Y2 + B2 Y2
94X2 = ---------------------------
95                 A3
96X2 = 0
97              2
98       SQRT(B2  + 2 A3) X2 + B2 X2
99Y2 = - ---------------------------
100                    2
101            2
102     SQRT(B2  + 2 A3) X2 - B2 X2
103Y2 = ---------------------------
104                  2
105PASSED TEST
106(D2)                                 DONE
107
108
109Local Variables: ***
110mode: Text ***
111End: ***