Chapter I. O n Stable Properties of the Solution Set
of an Ordinary Differential Equatio n
The genera l aim of this chapter is to describ e solutio n set s o f ordinary differentia l
equations a t the topological level . Th e equations ar e always assumed t o be of first order
and th e solution set is pictured a s a set of curves in a "phas e space. " A s indicated in the
preface, the equations ar e considered t o be known onl y in a rough way so the features t o be
described are , by design , shared to some extent b y nearby equations .
The descriptio n come s in terms o f "discrete " invariants, like the index o f a singular
point, whic h are well-defined o n open subset s o f some "space " of equations (wit h a natural
topology). Thes e invariants ar e constant o n connected set s (of equations) wher e they are
defined; consequentl y a kind o f "structura l stability " is built in . Furthermore , a "bifurca -
tion" theory i s implied because , on leaving the domai n o f some invariant, a change in the
structure o f the solution se t results. Suc h changes are also object s t o be described .
This chapter is , itself, purely descriptive ; the actual definition s an d justifications ar e in
the later chapters .
1. Isolate d invariant sets and continuation. Th e basic object s o f study ar e the iso-
lated invariant set s of a differential equation . A set (in the phase space) is called invarian t
if it is the union o f solution curves . I t is isolated i f it is the maximal invarian t se t in some
neighborhood o f itself. A compact suc h neighborhood i s called an isolating neighborhoo d
for th e invariant set . Fo r example, the hyperbolic poin t i n Figure l a is isolated sinc e it is
the maxima l invariant se t in the square; the square is an isolating neighborhood. Fo r con-
trast, th e center i n Figure l b is not isolated sinc e any neighborhood contain s some (periodic )
solution curves .
j ^
* L
a b
Isolated invariant set s are singled ou t because the y ca n be "continued " t o nearby equa -
tions in a natural way ; in this sens e the y ar e "stable" objects. Th e continuation i s define d
in term s o f isolating neighborhoods a s follows. A compact set , N, i s an isolating neighbor-
hood o f the maximal invarian t se t contained i n N i f and only i f that se t is interior t o N.
Equivalently, N is an isolating neighborhood i f and only i f no boundary poin t o f TV is on a
Previous Page Next Page