Samples for haverage.m, ver. 2.1.

If version 2.1 of haverage.m is used, the examples of

	K. Yagasaki and T. Ichikawa, Higher-order averaging for periodically
	forced, weakly nonlinear systems, International Journal of Bifurcation
	and Chaos, 9 (1999), 519--531        

should be changed as follows:


Section 5.1

<<haverage.m

F[1]=Expand[e*(-beta*x[1]^2)+e^2*(-delta*y[1]-alpha*x[1]^3+gamma*Cos[psi])];

k={1};l={1};j=2;n=1;

OnlyTrig=True;

g=VanderPol[F,x,y,u,v,psi,omega,{e^j*Omega},k,l,n];

Do[f[i]=Coefficient[i!*g,e,i]/.psi->theta,{i,1,5}]

n2=2;m0=5;

haverage[f,barf,{u[1],v[1]},theta,omega,n2,m0]


Section 5.2

<<haverage.m

F[1]=Expand[-e*beta*(x[1]-gamma0*Cos[psi])^2
  -e^2*alpha*(x[1]-gamma0*Cos[psi])^3
  -e^3*delta*(y[1]+omega*gamma0*Sin[psi])];

OnlyTrig=True;

g=VanderPol[F,x,y,u,v,psi,omega,{e^2*Omega},{3},{2},1];

Do[f[i]=Coefficient[i!*g,e,i]/.psi->3*theta,{i,1,4}]

haverage[f,barf,{u[1],v[1]},theta,omega/3,2,4]

(If you already typed "<<haverage.m" and "OnlyTrig=True;", then you do not need
to input these commands.)
 

Section 5.3

<<haverage.m

F[1]=Expand[-e*(gamma*Cos[psi]-delta*y[1]-3*x[1]^2-x[2]^2)
  -e^2*(x[1]^2+x[2]^2)*x[1]];

F[2]=Expand[e*((nu-2*x[1])*x[2]-delta*y[2])-e^2*(x[1]^2+x[2]^2)*x[2]];

OnlyTrig=True;

g=VanderPol[F,x,y,u,v,psi,omega,{e*Omega,e*Omega/4},{1,2},{1,1},2];

Do[f[i]=Coefficient[i!*g,e,i]/.psi->2*theta,{i,1,2}]

haverage[f,barf,{u[1],v[1],u[2],v[2]},theta,omega/2,4,2]