4

CHARLES CONLE Y

solution curv e whic h is contained (completely ) i n K Thi s last conditio n i s obviously "stable "

under smal l change s in th e equation . Therefor e i f TV is an isolating neighborhood o f some

equation, the n N wil l b e a n isolating neighborhood fo r al l equations near th e give n one . Th e

isolated invarian t set s thu s determine d b y N ar e th e continuations . Specifically , i f N i s an

isolating neighborhood fo r a connected se t o f equations, then th e correspondin g isolated in-

variant set s are said t o b e related b y continuation .

Having the abov e relatio n betwee n th e isolate d invarian t set s o f nearby equations , a

relation betwee n thos e o f no t a t al l nearby equation s i s obtained b y making th e local on e

transitive. A solution se t is in par t "described " b y listing (some of ) th e equivalenc e classe s

of its isolated invarian t sets . Suc h a description i s not very refined , bu t i s is "stable. "

The collection o f isolated invariant set s is closed under finite intersection s an d finite

disjoint union s an d is ordered b y inclusion. Thes e se t theoreti c relation s als o "continue, " so

can b e considered part o f th e description . However , such relations ca n vary wit h the equa -

tion; in fact , ne w invariant set s ca n appea r o r ol d one s disappea r a s the equatio n changes .

2. A n example. I n Figur e 2 , phase portrait s ar e indicated fo r th e followin g on e param -

eter (X ) family o f equations o n R

1:

dx/dt = x(l - x

2

) - X .

The cubic curv e is the zer o set o f th e functio n f(x, X ) = x(l - x 2) - X and meets each horiz-

ontal line in th e se t o f critica l point s o f th e equatio n wit h th e correspondin g value o f X. I n

each o f thre e suc h lines an interva l

FIGURE 2

is marked off ; i n eac h case th e interval i s an isolating neighborhood fo r th e equatio n wit h th e

corresponding value o f X - becaus e neithe r boundar y poin t i s on a solution whic h stay s in

the interval . (Eac h interval shoul d b e considered a s lying in th e sam e rea l line; this is not

done in th e pictur e i n orde r tha t th e differen t equation s ca n be shown. )

The res t point s o f thes e equation s provid e example s o f isolated invarian t sets . Mor e

generally, an y interval eac h o f whos e en d point s i s a rest poin t i s an isolated invarian t se t - a

slightly larger interva l serve s as an isolating neighborhood. Recallin g that th e disjoin t unio n

of isolated invarian t set s is also isolated, it i s found tha t th e top , middle an d botto m phas e