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Solution manifolds for systems of differential equations

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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.

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