This paper defines a solution manifold and a stable submanifold for a system of differential equations. Although we eventually work in the smooth topos, the first two sections do not mention topos theory and should be of interest to non-topos theorists. The paper characterizes solutions in terms of barriers to growth and defines solutions in what are called filter rings (characterized as $C^{\infty}$-reduced rings in a paper of Moerdijk and Reyes). We examine standardization, stabilization, perturbation, change of variables, non-standard solutions, strange attractors and cycles at infinity.
Theory and Applications of Categories, Vol. 7, 2000, No. 13, pp 239-262 http://www.tac.mta.ca/tac/volumes/7/n13/n13.dvi http://www.tac.mta.ca/tac/volumes/7/n13/n13.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/7/n13/n13.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/7/n13/n13.psTAC Home