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

P

a b

FIGURE 1

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

3

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