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: ***