| blob | ba07e7ee0ae19dbdeb2cc5324d23f28da385a3e3 |
1 twovar.mac is from the book "Perturbation Methods, Bifurcation Theory
2 and Computer Algebra" by Rand & Armbruster (Springer 1987)
4 This maxima routine applies the two variable expansion method to a
5 non-autonomous (forced) system of n differential equations. This
6 sample run from p 93 applies the method to the van der Pol equation.
8 The routine is case sensitive. When I enter the inputs in lower case
9 I get different (wrong) answers.
11 (C1) load("./twovar.mac");
12 Warning - you are redefining the MACSYMA function SETIFY
13 (D1) ./twovar.mac
14 (C2) twovar();
15 DO YOU WANT TO ENTER NEW DATA (Y/N)
16 Y;
17 NUMBER OF D.E.'S
18 1;
19 THE 1 D.E.'S WILL BE IN THE FORM:
20 X[I]'' + W[I]^2 X[I] = E F[I](X[1],...,X[ 1 ],T)
21 ENTER SYMBOL FOR X[ 1 ]
22 X;
23 ENTER W[ 1 ]
24 1;
25 ENTER F[ 1 ]
26 (1-X^2)*'DIFF(X,T);
27 THE D.E.'S ARE ENTERED AS:
28 2 dX
29 X '' + X = E (1 - X ) --
30 dT
31 THE METHOD ASSUMES A SOLUTION IN THE FORM:
32 X[I] = X0[I] + E X1[I]
33 WHERE X0[I] = A[I](ETA) COS W[I] XI + B[I](ETA) SIN W[I] XI
34 WHERE XI = T AND ETA = E T
35 REMOVAL OF SECULAR TERMS IN THE X1[I] EQS. GIVES:
36 2 3
37 A B A
38 d 1 1 1
39 2 (---- (A )) + ----- + -- - A = 0
40 dETA 1 4 4 1
41 3 2
42 B A B
43 d 1 1 1
44 - 2 (---- (B )) - -- - ----- + B = 0
45 dETA 1 4 4 1
46 DO YOU WANT TO TRANSFORM TO POLAR COORDINATES (Y/N)
47 Y;
48 3
49 R R
50 d 1 1 d
51 [[---- (R ) = -- - --, ---- (THETA ) = 0]]
52 dETA 1 2 8 dETA 1
53 DO YOU WANT TO SEARCH FOR RESONANT PARAMETER VALUES (Y/N)
54 N;
57 The second example is from pp 100-103. The system of equations is
59 x'' + (1+e*delx) x + e nu x^3 = e k y
61 y'' + (1+e*dely) y + e mu cos(w*t) = 0
63 First we use the two variable method to find a list of resonant
64 frequencies w, then we set W to one of the frequencies and determine
65 the slow flow equations for W:2.
67 (C3) twovar();
68 DO YOU WANT TO ENTER NEW DATA (Y/N)
69 Y;
70 NUMBER OF D.E.'S
71 2;
72 THE 2 D.E.'S WILL BE IN THE FORM:
73 X[I]'' + W[I]^2 X[I] = E F[I](X[1],...,X[ 2 ],T)
74 ENTER SYMBOL FOR X[ 1 ]
75 X;
76 ENTER SYMBOL FOR X[ 2 ]
77 Y;
78 ENTER W[ 1 ]
79 1;
80 ENTER W[ 2 ]
81 1;
82 ENTER F[ 1 ]
83 -DELX*X-NU*X^3+K*Y;
84 ENTER F[ 2 ]
85 -DELY*Y-MU*Y*COS(W*T);
86 THE D.E.'S ARE ENTERED AS:
87 3
88 X '' + X = E (K Y - NU X - DELX X)
89 Y '' + Y = E (- MU COS(T W) Y - DELY Y)
90 THE METHOD ASSUMES A SOLUTION IN THE FORM:
91 X[I] = X0[I] + E X1[I]
92 WHERE X0[I] = A[I](ETA) COS W[I] XI + B[I](ETA) SIN W[I] XI
93 WHERE XI = T AND ETA = E T
94 REMOVAL OF SECULAR TERMS IN THE X1[I] EQS. GIVES:
95 3 2
96 3 B NU 3 A B NU
97 1 1 1 d
98 - ------- - ---------- + B K - B DELX + 2 (---- (A )) = 0
99 4 4 2 1 dETA 1
100 2 3
101 3 A B NU 3 A NU
102 1 1 1 d
103 - ---------- - ------- + A K - A DELX - 2 (---- (B )) = 0
104 4 4 2 1 dETA 1
105 d
106 2 (---- (A )) - B DELY = 0
107 dETA 2 2
108 d
109 - A DELY - 2 (---- (B )) = 0
110 2 dETA 2
111 DO YOU WANT TO TRANSFORM TO POLAR COORDINATES (Y/N)
112 N;
113 DO YOU WANT TO SEARCH FOR RESONANT PARAMETER VALUES (Y/N)
114 Y;
115 X EQ'S RESONANT FREQ = 1
116 FREQS ON RHS = [1, 3]
117 Y EQ'S RESONANT FREQ = 1
118 FREQS ON RHS = [1, W - 1, W + 1]
119 WHICH PARAMETER TO SEARCH FOR ?
120 W;
121 [W = - 2, W = 0, W = 2]
122 DO YOU WANT TO SEARCH FOR ANOTHER PARAMETER (Y/N) ?
123 N;
124 (D3)
125 (C4) W:2;
126 (D4) 2
127 (C5) twovar();
128 DO YOU WANT TO ENTER NEW DATA (Y/N)
129 N;
130 THE D.E.'S ARE ENTERED AS:
131 3
132 X '' + X = E (K Y - NU X - DELX X)
133 Y '' + Y = E (- MU COS(2 T) Y - DELY Y)
134 THE METHOD ASSUMES A SOLUTION IN THE FORM:
135 X[I] = X0[I] + E X1[I]
136 WHERE X0[I] = A[I](ETA) COS W[I] XI + B[I](ETA) SIN W[I] XI
137 WHERE XI = T AND ETA = E T
138 REMOVAL OF SECULAR TERMS IN THE X1[I] EQS. GIVES:
139 3 2
140 3 B NU 3 A B NU
141 1 1 1 d
142 - ------- - ---------- + B K - B DELX + 2 (---- (A )) = 0
143 4 4 2 1 dETA 1
144 2 3
145 3 A B NU 3 A NU
146 1 1 1 d
147 - ---------- - ------- + A K - A DELX - 2 (---- (B )) = 0
148 4 4 2 1 dETA 1
149 B MU
150 2 d
151 ----- - B DELY + 2 (---- (A )) = 0
152 2 2 dETA 2
153 A MU
154 2 d
155 - ----- - A DELY - 2 (---- (B )) = 0
156 2 2 dETA 2
157 DO YOU WANT TO TRANSFORM TO POLAR COORDINATES (Y/N)
158 Y;
159 R SIN(THETA - THETA ) K R SIN(2 THETA ) MU
160 d 2 2 1 d 2 2
161 [[---- (R ) = - -------------------------, ---- (R ) = - -------------------,
162 dETA 1 2 dETA 2 4
164 2
165 3 R NU R COS(THETA - THETA ) K
166 d 1 2 2 1 DELX
167 ---- (THETA ) = - ------- + ------------------------- - ----,
168 dETA 1 8 2 R 2
169 1
171 COS(2 THETA ) MU
172 d 2 DELY
173 ---- (THETA ) = - ---------------- - ----]]
174 dETA 2 4 2
175 DO YOU WANT TO SEARCH FOR RESONANT PARAMETER VALUES (Y/N)
176 N;
179 Local Variables: ***
180 mode: Text ***
181 End: ***