1/* Filename vandpol.mac
2
3   ***************************************************************
4   *							         *
5   *                     <package name>                          *
6   *                <functionality description>                  *
7   *                                                             *
8   *          from: Computer Algebra in Applied Math.            *
9   *                   by Rand (Pitman,1984)                     *
10   *                Programmed by Richard Rand                   *
11   *      These files are released to the public domain          *
12   *            						 *
13   ***************************************************************
14*/
15
16
17/*
18(d12) This program computes a perturbation solution for the
19
20
21limit cycle in Van der Pol's equation.  Call it by typing:
22
23
24                     VANDERPOL(N)
25
26
27where N is the order of truncation.
28*/
29
30vanderpol(n):=(setup1(n),setup2(n),
31	  for i thru n do
32	      block(step1(i),step2(i),if i > 1 then output1(i),
33		    if i = n then go(end),step3(i),step4(i),step5(i),end),
34	  output2(n))$
35setup1(n):=(w:1,for i thru n do w:w+k[i]*e^i,x:2*cos(t),
36       for i thru n do x:x+y[i](t)*e^i)$
37setup2(n):=(temp1:diff(x,t,2)+x/w^2-e*(1-x^2)*diff(x,t)/w,
38       temp1:taylor(temp1,e,0,n),for i thru n do eq[i]:coeff(temp1,e,i))$
39step1(i):=temp1
40      :expand(trigreduce(expand(ev(eq[i],makelist([e[j],f[j]],j,1,i-1),
41				   diff))))$
42step2(i):=(
43      if i = 1 then f[i]:solve(coeff(temp1,cos(t)),k[i])
44	  else f[i]:solve([coeff(temp1,cos(t)),coeff(temp1,sin(t))],
45			  [k[i],b[i-1]]),temp1:ev(temp1,f[i]))$
46step3(i):=(temp1:ode2(temp1,y[i](t),t),
47  temp1:subst(a[i],%k1,temp1),temp1:subst(b[i],%k2,temp1))$
48step4(i):=(temp2:rhs(temp1),temp2:diff(temp2,t),
49      temp2:solve(ev(temp2,t:0),a[i]))$
50step5(i):=e[i]:ev(temp1,temp2)$
51output1(i):=(print(expand(ev(e[i-1],f[i]))),print(" "))$
52output2(n):=(print("w=",ev(w,makelist([f[j]],j,1,n))),print(" "))$
53