Solution manifolds for systems of differential equations

John F. Kennison

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.

Keywords: smooth topos, differential equation.

2000 MSC: 18B25, 58F14, 26E35.

Theory and Applications of Categories, Vol. 7, 2000, No. 13, pp 239-262.

TAC Home