## Abstract

In this chapter we consider a special class of autonomous systems, *x′ = X (x)*, on open sets *O* ⊆ ℝ ^{ n }, whose integral curves are completely “determined” by *n –* 1 functions, *F* ^{1}, *F* ^{2},*…*, *F* ^{ n−1}: *U* ⊆ *O* → *ℝ*, defined on an open dense subset *U* of *O*. These functions are called first integrals, or constants of the motion, and have, by definition, constant values along each integral curve of *X.* In addition, there are conditions on *F* ^{1}, *F* ^{2},…, *F* ^{ n } ^{−1}, so that the level sets *F* ^{ i } *(x) = k* _{ i }, *i = 1*,*…*, *n − 1*, intersect to give 1-dimensional submanifolds or curves in *ℝ* ^{ n } and these curves coincide, in a sense, with the integral curves of *X.* Such systems are called *integrable systems* and will be defined more precisely below.

## Keywords

Vector Field Integrable System Phase Portrait Level Surface Integral Curve## Preview

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