"haveraged.m" is a package of the computer algebra system, Mathematica
for performing necessary computations of higher-order averaging in
periodically forced, weakly nonlinear systems. See the following
references for the basic idea about this package and its usage:

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

	K. Yagasaki, Detection of bifurcation structures by higher-order
	averaging in Duffing's equation, to appear in Nonlinear Dynamics.

The preprints of these papers are available from
	"http://mech.gifu-u.ac.jp/~yagasaki/public_e.html".

See also sample.txt for the usage. The following references include the
related matters:

	K. Yagasaki, Higher-order averaging and ultra-subharmonics
	in forced oscillators, Journal of Sound and Vibration, 210
	(1998), 529-553.

	K. Yagasaki, Higher-order averaging of strongly nonlinear
	oscillators using trigonometric approximations, in preparation. 

This package is licenced free of charge and there is absolutely no warranty.


Kazuyuki Yagasaki

Department of Mechanical and Systems Engineering
Gifu University
Gifu, Gifu 501-1193, Japan

email: yagasaki@cc.gifu-u.ac.jp

June 22, 1999