H.R. G R ~ M M MARCH 7-6
RcKvch Reports provide the f o r d record of resarch conducted by the I n t u n a t i o d Institute for Applied Systems Analysis. They are carefully reviewed before publication and represent, in the Institute's best judgment.
competent scientific work. Views or opinions expressed herein, however, do not n e c d y reflect those of the N a t i o d Member Organizations support- ing the Institute or of the Institute itself.
International lnatikrte for Applied Systems Anaiymis
2361 Laxerrburg, Austria
PREFACE
One of t h e most s t r i k i n g examples of c o l l a b o r a t i o n between p r o j e c t s a t IIASA was t h e i n f o r m a l " R e s i l i e n c e Group," made up of members of t h e Ecology, Energy and Methodology P r o j e c t s . H o l l i n g ' s o r i g i n a l i d e a o f r e s i l i e n c e a s a p r o p e r t y o r measure o f a n e c o l o g i c a l system found unexpected a p p l i c a t i o n s t o s o c i e t y models c o n s i d e r e d by t h e Energy P r o j e c t ; a t t h e same time, t h e mathematical i d e n t i f i c a t i o n of t h e r e s i l i e n c e c o n c e p t gave r i s e t o s e v e r a l i n t e r e s t i n g methodological problems. The c o l l a b o r a t i o n a r i s i n g o u t of t h e s e common i n t e r e s t s proved e x t r e m e l y f r u i t f u l t o a l l p a r t i c i p a n t s of t h i s i n f o r m a l group. From t h e group d i s - c u s s i o n s , t h e a u t h o r has d i s t i l l e d p r e c i s e mathematical d e f i n i - t i o n s f o r t h e many f a c e t s of t h e r e s i l i e n c e c o n c e p t . T h i s paper shows t h a t t h e language of d i f f e r e n t i a l t o p o l o g y i s r i c h enough t o e x p r e s s a l l the--sometimes d i v e r q i n g - - i d e a s a b o u t r e s i l i e n c e t h a t came up a t IIASA. By n e c e s s i t y , t h e d i s c u s s i o n i n t h i s paper t a k e s p l a c e on a somewhat t e c h n i c a l l e v e l ; a n appendix summarizes t h e n e c e s s a r y mathematical terms.
SUMMARY
During t h e p a s t y e a r , s e v e r a l r e s e a r c h e f f o r t s a t I I A S A have t r i e d t o develop a p r e c i s e mathematical d e f i n i t i o n of H o l l i n g ' s v e r y g e n e r a l and r i c h r e s i l i e n c e concept. T h i s paper d e v e l o p s a mathematical language f o r r e s i l i e n c e , u s i n g t h e terms and c o n c e p t s of d i f f e r e n t i a l topology. C e n t r a l t o t h i s t r e a t m e n t is t h e d i v i s i o n of t h e s t a t e s p a c e of a system- i n t o b a s i n s , each c o n t a i n i n g an a t t r a c t o r . The t r a n s l a t i o n of Hol- l i n g ' s c o n c e p t i n t o t h i s language r e a d s roughly a s f o l l o w s : a system is r e s i l i e n t i f , a f t e r p e r t u r b a t i o n , i t w i l l s t i l l t e n d t o t h e same a t t r a c t o r a s b e f o r e ( o r t o an "only s l i g h t l y changedn a t t r a c t o r ) . The r e a s o n f o r t r e a t i n g changes of s t a t e v a r i a b l e s and changes of p a r a m e t e r s s e p a r a t e l y i s e x p l a i n e d . A l l r e s i l i e n c e measures conceived up t o now, a s d e f i n e d w i t h i n t h i s language, a r e l i s t e d a s w e l l . The v a r i o u s d e f i n i t i o n s o f r e s i l i e n c e a r e t h e n compared t o t h e well-known c o n c e p t s of s t r u c t u r a l s t a b i l i t y and of Thorn's c a t a s t r o p h e t h e o r y . F i n a l l y , t h e a u t h o r i n d i c a t e s some--in h i s opinion--important d i r e c t i o n s f o r f u r t h e r r e s e a r c h i n t o t h e g e n e r a l r e s i l i e n c e c o n c e p t .
DEFINITIONS OF RESILIENCE
1
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INTRODUCTIONThe r e s i l i e n c e c o n c e p t , p i o n e e r e d by C . S . H o l l i n g [ I ] , h a s been t h e s u b j e c t o f a p r o l o n g e d d i s c u s s i o n a t IIASA. V a r i o u s e x p r e s s i o n s f o r a r e s i l i e n c e v a l u e h a v e been proposed by Hol- l i n g ' s g r o u p 121
,
and, i n t h e c o u r s e of t h e s t u d y of t h e "New S o c i e t a l E q u a t i o n s , " a n o t h e r p o s s i b l e e x p r e s s i o n f o r r e s i l i e n c e w a s computed e x p l i c i t l y [ 3 ] and w i l l be used a s i n p u t t o a n o p t i - m i z a t i o n problem. N e v e r t h e l e s s , a p a r t from [ U l , no g e n e r a li n v e s t i g a t i o n h a s been made of t h e r e s i l i e n c e c o n c e p t i n t h e con- t e x t o f t h e t h e o r y of d i f f e r e n t i a b l e dynamical s y s t e m s . I t i s t h e p u r p o s e of t h i s p a p e r t o g i v e p r e c i s e , workable d e f i n i t i o n s o f r e s i l i e n c e i n t h e l a n g u a g e of t h i s t h e o r y . T h i s l a n g u a g e
h i l l
t u r n o u t t o be r i c h enough t o e x p r e s s a l l d i f f e r e n t f a c e t s o f t h e
" r e s i l i e n c e c o n c e p t . " A s we s h a l l s e e , H o l l i n g ' s o r i g i n a l c o n c e p t h a s t o be s p l i t i n t o two e s s e n t i a l l y d i f f e r e n t p r o p e r t i e s . T h i s d i s t i n c t i o n h a s been n o t e d i n t h e r e s i l i e n c e d i s c u s s i o n f o r some t i m e u n d e r t h e l a b e l s " r e s i l i e n c e i n s t a t e s p a c e " v s . " r e s i l i e n c e o f s t a t e s p a c e . " The p r e s e n t work i s r e s t r i c t e d t o d e t e r m i n i s t i c systems. The problem i s f i r s t d i s c u s s e d i n a n i n f o r m a l c o n c e p t u a l way, a s a m o t i v a t i o n f o r t h e r i g o r o u s m a t h e m a t i c s c o n t a i n e d i n t h e second p a r t . An a p p e n d i x summarizes some m a t h e m a t i c a l d e f i - n i t i o n s u s e d .
2. The MATHEMATICAL STAGE : DIFFERENTIABLE DYElAMICAL SYSTEMS
Although some a t t e m p t s have been made t o l i n k r e s i l i e n c e t o " l o n g e x i t t i m e e x p e c t a t i o n s " i n a s t o c h a s t i c a p p r o a c h [ 5 ]
,
t h e d i s c u s s i o n s o f a r h a s been c e n t e r e d on d e t e r m i n i s t i c s y s t e m s . One n a t u r a l m a t h e m a t i c a l l a n g u a g e f o r t h e d e s c r i p t i o n of s u c h s y s t e m s i s t h e t h e o r y of d i f f e r e n t i a l dynamical s y s t e m s , i . e . , of d i f f e r e n t i a b l e maps and f l o w s on a m a n i f o l d . (An i n f o r m a l c o l l e c t i o n of c o n c e p t s and r e s u l t s of t h i s t h e o r y i s g i v e n i n
t h e a p p e n d i x . ) For t h e d e f i n i t i o n s o f r e s i l i e n c e , we t h e r e f o r e assume t h a t we have a b s t r a c t e d a m a t h e m a t i c a l model of t h e g i v e n
system i n t h e f o l l o w i n g form:
M d e n o t e s t h e s t a t e s p a c e of t h e s y s t e m , assumed t o be a m a n i f o l d ;
$ t g i v e s t h e t o t a l dynamic e v o l u t i o n of t h e s y s t e m o v e r time, e i t h e r d i s c r e t e ( a s i n v a r i o u s e c o l o g i c a l models, where t h e a v e r a g e r e p r o d u c t i o n t i m e of a s p e c i e s g i v e s a n a t u r a l t i m e s t e p ) o r c o n t i n u o u s . By. c h o o s i n g a p a r t i c u l a r f u n c t i o n a l form f o r
@--or f o r t h e d i f f e r e n t i a l e q u a t i o n d e f i n i n g i t - - w e perform t h e
" f i r s t c u t " f o r o u r model: w e s e p a r a t e t h e " s y s t e m " from t h e
" r e s t o f t h e r e a l w o r l d , " which w i l l be r e g a r d e d a s unknown p e r t u r b a t i o n s . The second c u t i n v o l v e s t h e s e p a r a t i o n of s t a t e v a r i a b l e s and p a r a m e t e r s ; t h e d i m e n s i o n a l i t y and i n t e r p r e t a t i o n o f t h e s t a t e s p a c e e n t e r h e r e . T h i s s e p a r a t i o n c a n be performed o n l y v i a a s i g n i f i c a n t d i f f e r e n c e i n time s c a l e s ; i n t h e l a n g u a g e of [ l U ] , s t a t e v a r i a b l e s and p a r a m e t e r s must b e l o n g t o d i f f e r e n t
" s t r a t a . I' Loosely s p e a k i n g , p a r a m e t e r s a r e a l l o w e d t o v a r y i n o n l y two ways: e i t h e r by sudden jumps,' s u c h t h a t t h e s t a t e v a r i a b l e s c a n be approximated t o be c o n s t a n t d u r i n g t h e c h a n g e , o r by s l o w " a d i a b a t i c " c h a n g e s , s u c h t h a t t h e s y s t e m c a n be assumed t o be a l r e a d y i n i t s a s y m p t o t i c s t a t e , a s i n t h e a p p l i - c a t i o n s of c a t a s t r o p h e t h e o r y [ 1 3 ] . T h i s d i s t i n c t i o n w i l l a p p e a r
i n t h e t r e a t m e n t o f r e s i l i e n c e of t h e s t a t e s p a c e ; it r e m i n d s one o f t h e sudden and t h e a d i a b a t i c a p p r o x i m a t i o n i n time-dependent quan tun-mec h a n i c a l p e r t u r b a t i o n t h e o r y .
I n t h i s l a n g u a g e , a p a r a m e t e r i s t h e r e f o r e a v a r i a b l e c o n t a i n e d i n $, w i t h o u t b e i n g a f u n c t i o n on t h e s t a t e s p a c e M.
L a t e r , it w i l l be v a r i e d o v e r a p a r a m e t e r m a n i f o l d P . The semi-group p r o p e r t y @t+s = @ @ t s t h e n e x p r e s s e s t h e autonomy of o u r system. I n t h e d i s c r e t e c a s e , t h i s i m p l i e s immediately On = f n ( n
f
N) w i t h a s i n g l e map f . W e assume a l l maps t o be'".Suddenn and "slow" w i t h r e s p e c t t o a t y p i c a l time s c a l e of t h e s y s t e m i t s e l f .
o n c e d i f f e r e n t i a b l e (C 1 )
.
F u r t h e r d i f f e r e n t i a b i l i t y r e q u i r e - ments c a n n o t be made i n g e n e r a l : t h r e s h o l d b e h a v i o r s o f r e p r o - d u c t i o n r a t e s , f o r example, imply d i s c o n t i n u i t i e s i n t h e f i r s t o r h i g h e r d e r i v a t i v e s a s i n [ 6 ].
I n t h e c o n t i n u o u s c a s e , $ w i l l o f c o u r s e n o t be g i v e n
e x p l i c i t l y , b u t o n l y t h r o u g h a d i f f e r e n t i a l e q u a t i o n x ( t ) = F ( x ) . The " i n t e g r a t e d form" ( 2 . 1 ) i s i n t r o d u c e d f o r c o n v e n i e n c e i n t h e d e f i n i t i o n s . We n e g l e c t f o r t h e moment d i f f i c u l t i e s d u e t o i n c o m p l e t e n e s s o f F ( i - e . , n o t a l l t r a j e c t o r i e s c a n be e x t e n d e d t o a r b i t r a r y l a r g e t i m e s ) . For a n example o f t h i s problem, s e e [7 ]
.
3. EXPRESSING THE RESILIENCE CONCEPT
I n a v e r y c r u d e way, t h e d e s i r e d d e f i n i t i o n c o u l d be g i v e n i n t h e form: " t h e s y s t e m c a n a b s o r b c h a n g e s . " Those c h a n g e s , t o be s u r e , a r e assumed t o be sudden and e x t e r n a l ( c o n t r o l l a b l e o r u n c o n t r o l l a b l e , p r e d i c t a b l e o r random) and t h e r e f o r e n o t t o be i n c l u d e d i n t h e m a t h e m a t i c a l d e s c r i p t i o n v i a $ t . But a q u e s t i o n s u g g e s t s i t s e l f immediately: "How l a r g e c a n t h o s e c h a n g e s b e l w 2 Thus r e s i l i e n c e , a s we s e e i t , w i l l be a two- s t a g e c o n c e p t . F i r s t , on t h e q u a l i t a t i v e s i d e , we t r y t o a n s w e r t h e q u e s t i o n , " I s a s y s t e m r e s i l i e n t ? " I f t h e answer i s y e s , t h e n t h e q u a n t i t a t i v e s i d e a s k s , "How r e s i l i e n t i s i t ? " By no means w i l l t h i s q u e s t i o n h a v e a u n i q u e a n s w e r ; t h e c o n s e n s u s s t r e s s e s t h e n e c e s s i t y o f s e v e r a l " r e s i l i e n c e m e a s u r e s . " One s y s t e m c o u l d v e r y w e l l be more r e s i l i e n t t h a n a n o t h e r w i t h
r e s p e c t t o o n e measure, and l e s s s o w i t h r e s p e c t t o a s e c o n d one.
To d e f i n e r e s i l i e n c e v a l u e s , we must assume t h a t i n t h e s t a t e s p a c e M we have a n o t i o n o f d i s t a n c e , ( a m e t r i c d ) i n a c c o r d a n c e w i t h one w e l l - e s t a b l i s h e d i d e a o f r e s i l i e n c e a s " d i s t a n c e t o
'TO i l l u s t r a t e t h i s i d e a , i f t h e c h a n g e i n s t a t e v a r i a b l e s o c c u r s a t t i m e to, t h e e v a l u a t i o n g i v e n by Q t c o n t i n u e s on from a new x i i n s t e a d o f t h e xt r e a c h e d t h r o u g h e v o l u t i o n b e f o r e to.
0 0
To t a l k a b o u t t h e magnitude o f t h e c h a n g e , we need some d i s t a n c e between x i and xt
.
0 0
t h e n e x t p o i n t o f c a t a s t r o p h i c b e h a v i o r " ; o r a n o t i o n of volume ( a measure v ) , e x p r e s s i n g t h e o t h e r n a i n i d e a of r e s i l i e n c e a s
" s i z e o f b a s i n s . " The i d e n t i f i c a t i o n of t h i s d i s t a n c e o r volume w i l l be a n o n - t r i v i a l problem t o be s o l v e d f o r e a c h s y s t e m i n d i - v i d u a l l y ; any i d e n t i f i c a t i o n o f d i s t a n c e i n v o l v e s i m p l i c i t assump- t i o n s a b o u t t h e s t r u c t u r e of p o s s i b l y o c c u r r i n g d i s t u r b a n c e s , any i d e n t i f i c a t i o n o f volume i n v o l v e s a s s u m p t i o n s of a v e r a g e d i s t r i - b u t i o n s of p o i n t s d e s c r i b i n g t h e s y s t e m o v e r t h e b a s i n . Here we m e n t i o n o n l y two q u e s t i o n s a l l u d e d t o i n [ 2 ] and [ 3 ] : l o g a r i t h m i c
s c a l e v s - l i n e a r s c a l e o f t h e p e r t u r b a t i o n s , and t h e problem o f
" n a t u r a l u n i t s " i f t h e c o o r d i n a t e s i n M ( t h e s t a t e v a r i a b l e s ) have d i f f e r e n t d i m e n s i o n s .
I f no c o h e r e n t s c a l i n g of d i f f e r e n t s t a t e v a r i a b l e s i s p o s s i b l e , w e might even have t o r e p l a c e d by a m u l t i d i m e n s i o n a l d i s t a n c e n o t i o n : a f a m i l y o f s e m i - m e t r i c s ( d l
. .
. d i ) m e a s u r i n g t h e s i z e o f jumps i n d i f f e r e n t d i r e c t i o n s o f s t a t e s p a c e . The v a r i o u s d e f i n i t i o n s o f r e s i l i e n c e m e a s u r e s c a n t h e n b e a d j u s t e d a c c o r d i n g l y .A p a r t from t h i s t w o - s t a g e c o n c e p t , t h e r e i s a q u a l i t a t i v e d i s t i n c t i o n when w e t r y t o answer t h e q u e s t i o n : "What k i n d of c h a n g e s c a n t h e s y s t e m a b s o r b ? " Although i n [ I ] H o l l i n g t r e a t e d
" c h a n g e s i n s t a t e v a r i a b l e s " and " c h a n g e s i n p a r a m e t e r s " on a n e q u a l f o o t i n g , f o r a r i g o r o u s m a t h e m a t i c a l t r e a t m e n t w e w i l l h a v e t o make a d i s t i n c t i o n . Changing t h e s t a t e o f t h e s y s t e m a t o n e p a r t i c u l a r t i m e c h a n g e s one p a r t i c u l a r o r b i t , w h i l e c h a n g i n g t h e f u n c t i o n a l form o f t h e f l o w o r map t h r o u g h a c h a n g e
i n p a r a m e t e r s , f o r i n s t a n c e , i n v o l v e s t h e whole p h a s e p o r t r a i t . Given a p r e c i s e f o r m u l a t i o n of t h e s t a t e m e n t " t h e s y s t e m c a n a b s o r b , " we w i l l t h e n have two c o n c e p t s o f r e s i l i e n c e , d e p e n d i n g on t h e n a t u r e o f t h e c h a n g e s .
Two s u g g e s t i v e names f o r t h o s e c o n c e p t s have been p r o p o s e d :
" r e s i l i e n c e i n s t a t e s p a c e n ( s h o r t f o r "of a p o i n t i n p h a s e s p a c e " ) , c o r r e s p o n d i n g t o c h a n g e s o f s t a t e v a r i a b l e s , v s .
" r e s i l i e n c e of s t a t e s p a c e ' c o r r e s p o n d i n g t o c h a n g e s o f p a r a m e t e r s The l a t t e r c o n c e p t w i l l have d i f f e r e n t a s p e c t s a s w e d e a l w i t h sudden o r a d i a b a t i c p a r a m e t e r c h a n g e s . I n t h i s p a p e r we c a l l them
R l and R2 f o r s i m p l i c i t y .
4 . THE ROLE OF ATTRACTORS AND BASINS
W e u s e t h e p i c t u r e p r e s e n t e d i n t h e a p p e n d i x : a f i n i t e number o f a t t r a c t o r s Ai a r e l o c a t e d i n b a s i n s B i , s e p a r a t e d by s e p a r a t r i c e s S L e t u s a s s u m e t h a t w e h a v e s i n g l e d o u t o n e
j'
a t t r a c t o r 3 A, a s " d e s i r a b l e n ( t h i s , of c o u r s e , i s a n e x t e r n a l v a l u e j u d g m e n t ) , and t h a t t h e c u r r e n t s t a t e l i e s i n B1; t h e s y s t e m t h u s t e n d s t o w a r d s A, a s t +
+-.
A c h a n g e w i l l o b v i o u s l y b e a b s o r b e d i f , a f t e r t h e c h a n g e , t h e s y s t e m s t i l l t e n d s t o" a l m o s t " t h e same r e g i o n o f s t a t e s p a c e .
C o r r e s p o n d i n g t o t h e t w o k i n d s of r e s i l i e n c e d i s c u s s e d a b o v e , n o n - r e s i l i e n t b e h a v i o r c a n o c c u r i n two ways:
R1: The s u d d e n jump o f t h e s t a t e v a r i a b l e s moves t h e p o i n t d e s c r i b i n g t h e s y s t e m a c r o s s a s e p a r a t r i x i n t o a n o t h e r b a s i n ;
R 2 : The p h a s e p o r t r a i t c h a n g e s i n s u c h a way t h a t t h e s y s t e m (assumed t o h a v e t h e same v a l u e s o f t h e s t a t e v a r i a b l e s a s b e f o r e t h e c h a n g e s ) now l i e s i n a b a s i n whose a t t r a c t o r i s i n a d i f f e r e n t r e g i o n o f s t a t e s p a c e . W e see h e r e , by t h e way, o n e d i f f e r e n c e between s u d d e n a n d a d i a b a t i c c h a n g e s : a s l o w " a d i a b a t i c " c h a n g e w i l l n o t c a u s e t h e s y s t e m t o t e n d t o a d i f f e r e n t a t t r a c t o r ; r a t h e r , it w i l l t e n d t o a n " a d i a b a t i c a l l y c h a n g e d " o n e .
5. MATHEMATICAL DEFINITION
W e f i r s t a s s u m e t h e same s i t u a t i o n a s i n t h e l a s t s e c t i o n and d e f i n e t h e set S a s M
-
UBi ( t h e u n i o n of a l l s e p a r a t r i c e s ) .1
D e f i n i t i o n ( 5 . 1 ) ( R , ) :
Given a s y s t e m { @ t ) i n M w i t h a f i n i t e number o f a t t r a c t o r s Ai and b a s i n s Bi, it i s c a l l e d r e s i l i e n t i n t h e R, s e n s e
'or,
i n g e n e r a l , a s u b s e t o f a t t r a c t o r s !i f M
-
UBi h a s measure z e r o .u
i
T h i s means t h a t a l m o s t a l l i n i t i a l c o n d i t i o n s l e a d t o a t t r a c t o r s and s m a l l s h i f t s d o n o t d i s t u r b t h e a s y m p t o t i c b e h a v i o r s i n c e t h e b a s i n s a r e open s e t s . T h i s c o n d i t i o n i s f u l f i l l e d , e . g . , i f t h e non-wandering s e t c o n s i s t s o n l y of h y p e r b o l i c f i x e d p o i n t s and c l o s e d o r b i t s ( f i n i t e i n n u m b e r ) , o r i f Q s a t i s f i e s Axiom A and i s t w i c e d i f f e r e n t i a b l e 18 ]
.
T h e r e a r ec
I c o u n t e r - e x a m p l e s s a t i s f y i n g Axiom A where t h e s e p a r a t r i c e s have p o s i t i v e measure.I t i s n o t c l e a r what t h i s m a t h e m a t i c a l s t a t e m e n t would mean i n t h e r e a l w o r l d ; i t c o u l d be i n t e r p r e t e d a s s a y i n g t h a t t h e r e i s a non-zero p r o b a b i l i t y t h a t t h e s y s t e m l i e s a r b i t r a r i l y c l o s e t o a b a s i n boundary.
W e g e n e r a l i z e t h i s c o n c e p t t o d e a l w i t h c o n t i n u o u s f a m i l i e s of f i x e d p o i n t s , e . g . :
D e f i n i t i o n 5.1 . a ( g e n e r a l i z e d R , ) :
For a p o i n t x E: M, w e d e n o t e by w(x) t h e f u t u r e l i m i t s e t o f x ( y E: w ( x ) i f t h e r e e x i s t s a s e q u e n c e of r e a l numbers ti t
-
s u c h t h a t Q t ( x ) w y ) . Then { q t } i s c a l l e d g e n e r - a l i z e d R l i f t h e ma$ x ++ u ( x ) i s c o n t i n u o u s w i t h t h e Haus- d o r f f m e t r i c 5 on t h e s p a c e o f compact s u b s e t s of M e x c e p t on a s e t of measure z e r o . T h i s e x p r e s s e s t h e s t a b i l i t y of t h e a s y m p t o t i c b e h a v i o r u n d e r d i s t u r b a n c e s of t h e i n i t i a l c o n d i t i o n . Of c o u r s e (5.1 . a ) c o n t a i n s ( 5 . 1 ) s i n c e w ( x ) : Ai f o r x E: Bi.T ~ r n i n g t o r e s i l i e n c e m e a s u r e s , we d e f i n e r ( x ) = d ( x , S ) f o r e a c h p o i n t i n S. W e d i s t i n g u i s h d i f f e r e n t e x p r e s s i o n s ( d e p e n d e n t on t h e a p p l i c a t i o n i n t e n d e d ) .
he
p r o p e r t y of z e r o measure i s i n d e p e n d e n t o f t h e p a r t i c - u l a r d i s t r i b u t i o n o f s t a t e v a r i a b l e s d e s c r i b e d by, e . q . t h e volume n o t i o n , a s l o n g a s i t h a s a d e n s i t y .5 ~ h e Hausdorff d i s t a n c e between t h e compact s e t s A and B i s d e f i n e d by d(A,B) = max(sup i n f d ( x , y ) , s u p i n f d ( x , y ) ) .
x c A y c B yE:B x f A
( T h e r e is a p o i n t i n B a t d i s t a n c e i ( A , B ) from any p o i n t ' i n A, and v i c e v e r s a . )
i ) Mean r e s i l i e n c e o f t h e b a s i n B, ( H o l l i n g and c o - w o r k e r s ) :
w i t h some p r o b a b i l i t y measure
u .
T h i s c o n c e p t i s u s e f u l i n some e c o l o g i c a l a p p l i c a t i o n s - - e . g . , where o n e i s d e a l i n g w i t h a n ensemble of systems--and h a s been u s e d by H o l l i n g ' s g r o u p .
i i ) T r a j e c t o r y r e s i l i e n c e . Here we f o c u s on o n e p a r t i c u l a r i n i t i a l c o n d i t i o n , s o t h a t t h e r e s i l i e n c e v a l u e i s a f u n c t i o n of x. H a f e l e i n (91 p r o p o s e d " a v e r a g e r e s i l i e n c e : "
and t h e a u t h o r (71 p r o p o s e d "minimal r e s i l i e n c e : "
Rmin(x) = min r ( x t )
, t,o
e . g . , i n a n o r m a t i v e a p p r o a c h t o s t a n d a r d s e t t i n g . The l a s t e x p r e s s i o n h a s been t a b u l a t e d f o r t h e model i n (31 and i s b e i n g u s e d a s i n p u t t o a n o p t i m i z a t i o n program.
E x o r e s s i o n s s u c h a s t h o s e i n ( i ) have been c a l l e d r e s i l i e n c e numbers, t h o s e i n ( i i ) r e s i l i e n c e f u n c t i o n s , s i n c e t h e y s t i l l a r e f u n c t i o n s o v e r s t a t e s p a c e . Another p o s s i b l e r e s i l i e n c e number c o u l d be
iii) Volume r e s i l i e n c e :
v being t h e volume on s t a t e s p a c e a s i n S e c t i o n 2.
T h i s number may be l e s s s i g n i f i c a n t due t o t h e f a c t t h a t i n h i g h e r - d i m e n s i o n a l models, t h e b a s i n s w i l l g e n e r a l l y have r a t h e r c o m p l i c a t e d s t r u c t u r e s , such t h a t t h e y c o u l d c o n t a i n a l a r g e volume w h i l e t h e boundary i s s t i l l c l o s e t o e a c h p o i n t i n t h e b a s i n . Of c o u r s e f o r t h e c o m p u t a t i o n of t h e s e numbers i n a
c o n c r e t e problem, t h e e x a c t l o c a t i o n s o f t h e s e p a r a t r i c e s have t o be d e t e r m i n e d . Two k i n d s of a p p r o a c h e s seem most s u i t a b l e f o r t h i s : e i t h e r a method p r o v i n g t h a t some r e g i o n l i e s wholly w i t h i n t h e domain o f a t t r a c t i o n (Lyapunov's method a s used i n [I 01, o r Zubov's method a s d e s c r i b e d i n [l 1 1 )
,
o r a d i r e c t d e t e r m i n a t i o n o f t h e s e p a r a t r i c e s a s s t a b l e m a n i f o l d s of co-dimension one. The l a t t e r method was used i n [7].6 . MATHEMATICAL DEFINITION: R2
R e s i l i e n c e of t h e second k i n d a s c o n c e i v e d i n t h i s paper w i l l o b v i o u s l y be a c o n c e p t r e l a t e d t o t h e s t r u c t u r a l s t a - b i l i t y i d e a of Smale [12]. I t i s w e l l known t h a t any n o t i o n of q u a l i t a t i v e e q u i v a l e n c e between s y s t e m s g i v e s r i s e t o a c o r r e s p o n d i n g n o t i o n o f s t r u c t u r a l s t a b i l i t y i n t h e g e n e r a l
s e n s e : a system w i l l have a c e r t a i n s t r u c t u r a l s t a b i l i t y p r o p e r t y i f i t s e q u i v a l e n c e c l a s s under t h e g i v e n e q u i v a l e n c e n o t i o n i s open i n some ~ ~ - t o ~ o l o ~ ~ . S t r u c t u r a l s t a b i l i t y in t h e t e c h n i c a l s e n s e i s c o n n e c t e d i n t h i s way t o t o p o l o g i c a l c o n j u g a c y , i - e . , t h e e x i s t e n c e o f a homeomorphism t r a n s f o r m i n g t h e s y s t e m s i n t o e a c h o t h e r . While s t r u c t u r a l s t a b i l i t y i s t o o s t r o n g a c o n c e p t , s i n c e it i s concerned w i t h t h e whole p h a s e p o r t r a i t and n o t w i t h p o s i t i v e t i m e a s y m p t o t i c s a l o n e , Q - s t a b i l i t y i s t o o weak s i n c e it i m p l i e s n o t h i n g a b o u t s t r u c - t u r a l change of t h e b a s i n s . I n [U]
,
it was i l l u s t r a t e d t h a t b a s i n s c o u l d jump under a small p e r t u r b a t i o n of an Q - s t a b l e system ( h y p e r b o l i c f i x e d p o i n t s and c l o s e d o r b i t s ) , a non- r e s i l i e n t s i t u a t i o n a s e x p l a i n e d i n S e c t i o n 3 .A s a f u r t h e r d i s t i n c t i o n from t h e u s u a l m a t h e m a t i c a l con- c e p t s of s t r u c t u r a l s t a b i l i t y , R - s t a b i l i t y , e t c . , w e have t o n o t e t h a t w e w i l l n o t v a r y t h e s y s t e m o v e r a whole n e i g h b o r h o o d i n D i f f ( M ) o r X 1 ( M ) ( t h e s p a c e o f a l l d i s c r e t e o r c o n t i n u o u s s y s t e m s on M I . We t h e r e f o r e assume a s u b - m a n i f o l d P o f D i f f ( M ) o r X 1 (M) t o b e g i v e n s u c h t h a t t h e o r i g i n a l 41 € P . P c a n be t h o u g h t o f a s d e s c r i b e d by a f i n i t e s e t o f p a r a m e t e r s c o n t a i n e d i n 4 t h a t w e want t o v a r y ; t h e n P w i l l b e f i n i t e - d i m e n s i o n a l . But P c o u l d v e r y w e l l b e i n f i n i t e - d i m e n s i o n a l i f some e q u a t i o n s o f t h e model a r e a s s u m p t i o n s a n d i m p l i c i t l y i n e x a c t w h i l e o t h e r s a r e e x a c t i d e n t i t i e s . Then o n e m i g h t s t u d y r e s i l i e n c e o f t h e model t o a r b i t r a r y s m a l l v a r i a t i o n s o f t h e f i r s t s e t o f e q u a t i o n s ,
i n t h e s p i r i t o f Thom's i n s i s t e n c e on s t r u c t u r a l l y s t a b l e m o d e l s [ 1 3 ] . Many o f t h e m a t h e m a t i c a l d i s t i n c t i o n s i n what
f o l l o w s w i l l become t r i v i a l i n t h e c a s e o f s t a b l e e q u i l i b r i a a n d a r e i n c l u d e d p a r t l y f o r c o m p l e t e n e s s ; however, t h e r e a r e s t r o n g s u g g e s t i o n s t h a t n o n - t r i v i a l a t t r a c t o r s w i l l a p p e a r e v e n i n s i m p l e m o d e l s . A p a r t i c u l a r l y n a s t y example is g i v e n by t h e Lorenz a t t r a c t o r (see [ 1 8 ] ) .
W e a g a i n u s e t h e H a u s d o r f f d i s t a n c e t o f o r m u l a t e t h e con- d i t i o n t h a t b a s i n s a n d a t t r a c t o r s d o n o t v a r y much.
D e f i n i t i o n ( 6 . 1 . a ) (RZ, d i s c r e t e c a s e ) :
Given a s y s t e m 4 on M a n d a m a n i f o l d P a s a b o v e s u c h t h a t 4 € P
C
D i f f ( M ) , 4 i s c a l l e d r e s i l i e n t i n t h e s e c o n d s e n s e i f :i ) T h e r e e x i s t s a n e i g h b o r h o o d U o f 4 i n t h e ~ ' - t o p o l o g ~ on D i f f ( M ) s u c h t h a t a l l s y s t e m s @ ' U
n
P h a v e t h e same f i n i t e number o f a t t r a c t o r s ( a n d , t h e r e f o r e , t h e same number o f b a s i n s ! ) ;ii) The maps 4
'
b Ai (4'
) ( i - t h a t t r a c t o r o f 4'
) a n d 4'
I+B .
( 4 ' 1 ( c l o s u r e o f i - t h b a s i n o f 4 ' ) a r e c o n -1 1
t i n u o u s w i t h t h e C - t o p o l o g y on U
n
P a n d t h e Haus- d o r f f metric on t h e Ai a n dgi.
D e f i n i t i o n ( 6 . 1 . b ) ( R 2 , c o n t i n u o u s c a s e ) :
Assume t h e c o n t i n u o u s system { ( t ) g i v e n by a d i f f e r e n t i a l e q u a t i o n x = F ( x ) a s i n S e c t i o n 2, and P a sub-manifold of
x1
( M ) , F € P. Then F i s c a l l e d r e s i l i e n t i n t h e second s e n s e i f :i ) There e x i s t s a neighborhood U of F i n t h e
to to polo^^
such t h a t a l l systems { ( I t ) d e f i n e d by t h e F' U n P have t h e same f i n i t e number of a t t r a c t o r s ;
ii) The maps F P Ai ( F )
,
F I+ ( F ) a r e c o n t i n u o u s w i t h t h e1 1
C -topology on
-
U fl P and t h e Hausdorff m e t r i c on t h e Ai and B ~ .I f t h e manifold P i s f i n i t e - d i m e n s i o n a l , g i v e n by v a r i a t i o n s of some p a r a m e t e r s i n t h e f u n c t i o n a l form of ( ( o r F i n t h e
c o n t i n u o u s c a s e )
,
t h e c o n t i n u i t y c o n d i t i o n s ( 2 ) simply mean c o n t i n u i t y i n t h o s e parameters.Comparing t h i s d e f i n i t i o n w i t h t h e well-known s t a b i l i t y c o n c e p t s , we s e e t h a t we have p u t a v e r y weak c o n d i t i o n on t h e a t t r a c t o r s . Usual f o r m u l a t i o n s r e q u i r e t o p o l o g i c a l c o n j u g a t i o n on t h e a t t r a c t o r , 6 o r a t l e a s t t h a t n e a r b y systems g i v e homeo- morphic a t t r a c t o r s . Thus, t h e Lorenz a t t r a c t o r [ 18 ]
,
known t o be t o p o l o g i c a l l y u n s t a b l e , c o u l d v e r y w e l l be R 2 . T h i s weakening of t h e d e f i n i t i o n i s due t o t h e i n t e n d e d a p p l i c a t i o n s of t h e r e s i l i e n c e c o n c e p t ; one d o e s n o t want t o be b o t h e r e d w i t h " s u b - s h i f t s of f i n i t e t y p e " i n an e c o l o g i c a l model, f o r example. For t h e f i r s t s t e p s we a r e i n t e r e s t e d o n l y i n t h e l o c a t i o n of a t t r a c t o r s . However, a more r e f i n e d a n a l y s i s s h o u l d i n c l u d e t h e i n v a r i a n t - - u n d e r {Ot)--measures p i on t h e a t t r a c t o r s d e s c r i b i n g t i m e a v e r a g e s in t h e b a s i n s [ a ] . Some c l i m a t o l o g i s t s have e x p r e s s e d g r e a t i n t e r e s t in t h o s e a v e r a g e s , which c a n be used f o r d e f i n i n g mean r e s i l i e n c e i n S e c t i o n 5. Thus, f o r a6 ~ h i s means t h a t t h e o r b i t s of { ( I t ) on Ai (( I ) c a n be c a r r i e d i n t o t h e o r b i t s of { ( t ) on A i ( ( ) by a homeomorphism of t h e a t t r a c t o r s e t s .
s t a b l e f i x e d p o i n t , Rm would s i m p l y be Sii(? minimum d i s t a n c e t o t h e b a s i n boundary. Then t h e c o n d i t i o n s ( i i ) on t h e a t t r a c t o r s s h o u l d be r e p l a c e d by ( i i
'
) :The maps @ ' b
u .
( 4 ' ) a r e c o n t i n u o u s w i t h t h e weak t o p o l o g y1
on t h e s p a c e o f measures.' T h i s i m p l i e s ( i i ) s i n c e A = t h e i s u p p o r t o f
ui.
On t h e o t h e r hand, a p o s s i b l e g e n e r a l i z a t i o n m i g h t t a k e i n t o a c c o u n t t h a t " s p l i t t i n g " one a t t r a c t o r under v a r i a t i o n i n t o s e v e r a l - - s t i l l c l o s e t o g e t h e r - - d o e s n o t e s s e n t i a l l y c h a n g e t h e a s y m p t o t i c s t r u c t u r e of t h e s y s t e m . For e a c h a t t r a c t o r Ai of 4 , and e a c h p e r t u r b e d s y s t e m $ I , w e t h e n h a v e a f i n i t e s e t a i (4 ' 1 of a t t r a c t o r s of 4
'
( c l o s e t o g e t h e r ),
and w e r e p l a c e( i i ) by ( i i " ) :
t h e maps 4 ' t+
u
A ( @ ' ) A € a i ( 4 ' )and 4 t+ U q 4 l )
Bi
€ a
[where
9
i s t h e s e t o f a l l b a s i n s b e l o n g i n g t o a t t r a c t o r s i n a . ( 4 ' ) ] a r e c o n t i n u o u s .1
The s y s t e m a s d e f i n e d by
would s t i l l be r e s i l i e n t i n t h i s s e n s e t o v a r i a t i o n s o f
u
a r o u n d p = 0 Although t h e s t a b l e f i x e d p o i n t f o r p < 0 s p l i t s a tu
= 0 i n t o o n e u n s t a b l e and t w o s t a b l e p o i n t s , t h e a t t r a c t i n g p o i n t s a r e s t i l l c l o s e t o g e t h e r .I T h i s means t h a t t i m e a v e r a g e s o f c o n t i n u o u s f u n c t i o n s on t h e s t a t e s p a c e w i l l v a r y c o n t i n u o u s l y u n d e r v a r i a t i o n s o f 4.
I f w e l o o k f o r a n u m e r i c a l e x p r e s s i o n c o n n e c t e d w i t h R 2 w e f i r s t m i g h t t r y :
D e f i n i t i o n ( 6 . 2 ) ( m i n i m a l r e s i l i e n c e ) :
Given a m e t r i c
d
on t h e " p a r a m e t e r m a n i f o l d ' P, andd e n o t i n g by S p ( p a r a m e t e r s e p a r a t r i x ) t h e s e t Sp = I $ € P I $ ' d o e s n o t s a t i s f y R 2 ) , we d e f i n e
E(@)
= Z ( 0 , s p ) mina n o r m a t i v e number s u c h a s R
min ( S e c t i o n 5 )
.
T h i s number, o f c o u r s e , t e l l s u s a r a n g e o f p a r a m e t e r v a r i a t i o n s w h i c h d o n o t i n d u c e q u a l i t a t i v e c h a n g e s i n t h e b e h a v i o r of t h e s y s t e m . By c h o o s i n g t h e p a r a m e t e r m a n i f o l d P i n d i f f e r e n t ways, o n e c o u l d t h e n s t u d y r e s i l i e n c e w i t h r e s p e c t t o v a r i o u s com- b i n a t i o n s o f p a r a m e t e r s .A n o t h e r p o s s i b l e d e f i n i t i o n o f a r e s i l i e n c e number more i n l i n e w i t h s t a n d a r d s e n s i t i v i t y a n a l y s i s i s s u g g e s t e d h e r e . Given t h e c o n t i n u o u s d e p e n d e n c e o f a t t r a c t o r s a n d b a s i n s r e q u i r e d by D e f i n i t i o n ( 6 . 1 1 , t h e i r " s p e e d " u n d e r p a r a m e t e r v a r i a t i o n may b e i n t e r e s t i n g . A l t h o u g h t h e p h a s e p o r t r a i t d o e s n o t c h a n g e - - t h e s y s t e m s m i g h t e v e n b e s t r u c t u r a l l y s t a b l e , i . e . , more t h a n j u s t R 2 - - a v e r y s e n s i t i v e d e p e n d e n c e o f b a s i n b o u n d a r i e s d o e s n o t c o r r e s p o n d t o t h e i n t u i t i v e c o n c e p t o f a r e s i l i e n t s y s t e m . A l a r g e r e d u c t i o n in s i z e o f a p a r t i c u l a r b a s i n is c o n s i d e r e d a l m o s t a s c a t a s t r o p h i c a s i t s c o m p l e t e d i s a p p e a r a n c e . W e t h e r e - f o r e p r o p o s e
D e f i n i t i o n ( 6 . 3 ) ( s p e e d r e s i l i e n c e ) :
Under t h e a s s u m p t i o n s o f D e f i n i t i o n ( 6 . 2 ) , a n d d e n o t i n g by Bh a b a l l o f r a d i u s h a r o u n d t h e s y s t e m @ i n t h e param- e t e r m a n i f o l d P,
'we d e n o t e r e s i l i e n c e v a l u e s c o n n e c t e d t o R 2 by a b a r .
w i t h d t h e Hausdorff m e t r i c o f c l o s e d s e t s i n s t a t e s p a c e and Ai,Bi
,...,
a s i n ( 6 . 1 ) :D e f i n i t i o n ( 6 . 4 ) (volume s e n s i t i v i t y r e s i l i e n c e ) :
gv
= l i m 1 s u p ( v ( B , )-
v ( B ; ) ( h-+OFi
6'
Bh Pw i t h B, t h e " d e s i r e d " b a s i n and n o t a t i o n s a s above. ~ u t a n y volume-type m e a s u r e s must be r e g a r d e d w i t h some r e s e r - v a t i o n , a s e x p l a i n e d i n S e c t i o n 5.
-
IRSP may w e l l be 0 e v e n i f t h e system i s R 2 , i f t h e l o c a t i o n o f a t t r a c t o r s o r b a s i n s d e p e n d s n o n - d i f f e r e n t i a b l y on t h e param- e t e r s ; s e e , f o r a n example, e q u a t i o n ( 6 . 3 ) . T h i s s y s t e m i s r e s i l i e n t o n l y i n t h e g e n e r a l i z e d s e n s e ( s t a b l e f i x e d p o i n t s a t
@)
mentioned t h e r e , however.For slow c h a n g e s o f t h e p a r a m e t e r s i n t h e s e n s e of S e c t i o n 3 , t h e c o n d i t i o n s a b o u t t h e v a r i a t i o n s of b a s i n s i n D e f i n i t i o n s
( 6 . 1 ) and ( 6 . 4 ) s h o u l d be l e f t o u t . Under t h e s e c o n d i t i o n s , a b a s i n boundary c a n n e v e r o v e r t a k e t h e s y s t e m on i t s c o u r s e t o - wards t h e a t t r a c t o r . I n D e f i n i t i o n ( 6 . 1 1 , o n l y t h e d i s a p p e a r a n c e o f t h e a t t r a c t o r s h o u l d be c o u n t e d a s n o n - r e s i l i e n t b e h a v i o r , a s i n c a t a s t r o p h e t h e o r y . Approaching such p a r a m e t e r v a l u e s w i l l i n v o l v e a s h r i n k i n g of t h e b a s i n a s o b s e r v e d i n s e v e r a l e c o l o g i c a l examples.
7 . CONCLUSION AND DIRECTION O F FURTHER RESEARCH
The t h e o r y o f d i f f e r e n t i a b l e dynamical s y s t e m s g i v e s a s a t i s f a c t o r y l a n g u a g e f o r d e s c r i b i n g t h e many d i f f e r e n t f a c e t s t h a t make up t h e r e s i l i e n c e c o n c e p t . A t t h e same t i m e , w e have been a b l e t o d i s t i l l from t h e a p p l i c a t i o n s a n i n t e r e s t i n g mathe- m a t i c a l c o n c e p t t h a t h a s n o t been s t u d i e d y e t i n o r d i n a r y s t a - b i l i t y t h e o r y . I n r e l a t i o n t o c a t a s t r o p h e t h e o r y , t h e r e s i l i e n c e c o n c e p t i s a two-fold e x t e n s i o n : it t a k e s more c o m p l i c a t e d
a t t r a c t o r s t h a n f i x e d p o i n t s i n t o a c c o u n t r i g h t from t h e s t a r t , and it also i n v o l v e s p r o p e r t i e s o f t h e b a s i n s .
The f o l l o w i n g d i r e c t i o n s o f f u r t h e r research--some o f them a l r e a d y u n d e r d i s c u s s i o n - - a r e s u g g e s t e d : 9
-
The m a t h e m a t i c a l t h e o r y o f r e s i l i e n c e s h o u l d bei n v e s t i g a t e d : g e n e r a l n e c e s s a r y o f s u f f i c i e n t c r i t e r i a f o r r e s i l i e n c e of t h e s t a t e s p a c e would be e x t r e m e l y i n t e r e s t i n g . Some s t a r t i n g p o i n t s f o r t h i s a r e con- t a i n e d i n S e c t i o n 6.
-
I n r e l a t i o n t o t h e " k i t c o n c e p t w - - c o n s t r u c t i n g a s y s t e m h a v i n g a p r e d e t e r m i n e d s t r u c t u r e of a t t r a c t o r s and basins--two t a s k s a r e i m p o r t a n t : s t a r t i n g a l i s t of a t t r a c t o r s r e l e v a n t f o r a p p l i c a t i o n s , and s t u d y i n g c o n s i s t e n c y q u e s t i o n s a l o n g t h e l i n e s o f [151.-
S t u d y i n g t h e c h a n g e o f t h e p h a s e p o r t r a i t a s we c r o s s a p a r a m e t e r s e p a r a t r i x : t h e problem of b i f u r c a t i o n . T h i s h a s been s u g g e s t e d a s a mechanism f o r g e n e r a t i n g t u r b u - l e n t and e r r a t i c b e h a v i o r o f s y s t e m s [ 171.
For t h e r e s i l i e n c e c o n c e p t , i t s s t u d y g i v e s u s a n u n d e r s t a n d i n g o f "what g o e s wrong. 'I-
Numerical methods f o r t h e c a l c u l a t i o n o f boundary b a s i n s s h o u l d be d e v e l o p e d and t r i e d i n s i m p l e models; a l i s t o f c a n d i d a t e s i s g i v e n a t t h e e n d o f S e c t i o n 5.-
The c o n n e c t i o n s between t h e c h o i c e o f r e s i l i e n t measure ( d i s t a n c e , volume i n s t a t e s p a c e , e t c . ) a n d o u r know- l e d g e o r o u r i m p l i c i t a s s u m p t i o n s a b o u t t h e s t r u c t u r e of p e r t u r b a t i o n s o f t h e s y s t e m s h o u l d be d e s c r i b e d i n d e t a i l .ACKNOWLEDGEMENT
T h i s p a p e r c o u l d n o t have been w r i t t e n w i t h o u t t h e con- t i n u o u s i n t e l l e c t u a l i n s p i r a t i o n d u e t o d i s c u s s i o n s w i t h i n t h e IIASA r e s i l i e n c e g r o u p , n o t a b l y w i t h W. H a f e l e and
C.S. H o l l i n g . The r o l e of t h i s g r o u p - - l o o s e and u n f o r m a l i z e d t h o u g h it may be--is h e r e b y g r a t e f u l l y acknowledged.
y ~ o s t o f them, o f c o u r s e , a r e a l s o r e l e v a n t f o r t h e g e n e r a l s t r u c t u r a l s t u d y o f d y n a m i c a l s y s t e m s , a p a r t from t h e q u e s t i o n o f r e s i l i e n c e . A s a f o r t h c o m i n g p a p e r [ I 61 h o p e s t o show, t h i s t h e o r y c a n e x t e n d o u r u n d e r s t a n d i n g o f t h e dynamics o f a s y s t e m
"beyond n u m e r i c a l i n t e g r a t i o n . "
APPENDIX
A s h o r t i n t r o d u c t i o n t o t h e math'ematical f o u n d a t i o n s o f t h e t h e o r y c a n be found i n P. W a l t e r s . 1 ° Only a b r i e f l i s t o f i n f o r m a l d e f i n i t i o n s w i l l t h e r e f o r e be g i v e n h e r e . W e s h a l l d e a l w i t h t h e d i s c r e t e c a s e
(an
= f n ; D) and t h e c o n t i n u o u s c a s e (C) s i m u l t a n e o u s l y .A f i r e d ~ o i n t o f $ i s a n x o € M w i t h bt ( x 0 ) = x 0 f o r a l l t.
A p e r i o d i c p o i n t
lax closed
o r b i t (C) is a n x o € M w i t h bt(xO) = x O f o r some t .The non-wandering s e t R i s d e f i n e d a s t h e s e t o f x € M which d o n o t wander in t h e f o l l o w i n g s e n s e : x wanders i f t h e r e i s a neighborhood U ( x ) w i t h Q t U ( x )
n
U(x) =0
f o r a l l t l a r g e enough. Non-wandering is t h e w e a k e s t r e c u r r e n t - l i k e p r o p e r t y ; o f c o u r s e , a l l p e r i o d i c p o i n t s / c l o s e d o r b i t s , b u t i n g e n e r a l a l s o more c o m p l i c a t e d t h i n g s , l i e i n R.A f i x e d p o i n t x o i s h y p e r b o l i c i f t h e J a c o b i a n o f f ( D l / aF. 1
t h e matrix
-
ax, ( C ) / a t x o h a s no e i g e n v a l u e s on t h e u n i t c i r c l e (D)/on t h e i m a g i n a r y a x i s ( C ) . I n t h i s c a s e , t h e s t a b l e and u n s t a b l e m a n i f o l d s wS(x0) and w U ( x O ) c a n be d e f i n e d a s p o i n t s t e n d i n g e x p o n e n t i a l l y t o x o a s x + +m, x + - a ,r e s p e c t i v e l y . They a r e smooth s u b - m a n i f o l d s o f M, i n v a r i a n t u n d e r bt. A h y p e r b o l i c f i x e d p o i n t c a n n o t d i s a p p e a r u n d e r a small p e r t u r b a t i o n o f $.
O P . W a l t e r s , "An O u t l i n e o f S t r u c t u r a l Stability Theory, "
i n H.-R. G r k , e d . , A n a l y s i s and Computation o f E q u i l i b r i a and Regions o f S t a b i l i t y , CP-75-8, I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d Systems A n a l y s i s , Laxenburg, A u s t r i a , 1 9 7 5 .
A c o n t i n u o u s h y p e r b o l i c s p l i t t i n g o v e r a c l o s e d s u b s e t N
C
M i s g i v e n by d e f i n i n g i n a c o n t i n u o u s way a s p l i t t i n g o f t h e t a n g e n t space Tx ( M ) = :E 8 E: ( f o r t h e s e c o n c e p t s s e e ~ a l t e r s ' O ) i n t o d i r e c t i o n s where @ i s e x p o n e n t i a l l y c o n t r a c t i n g o r expanding, r e s p e c t i v e l y (D). I n t h e con- t i n u o u s c a s e , Tx(M) = E ~ ( M ) = :E $ :E $ Fx, where F i s a one-dimensional subspace a l o n g t h e d i r e c t i o n of t h e v e c t o r f i e l d a t x.@ s a t i s f i e s Axiom A i f
1 ) The p e r i o d i c p o i n t s ( D ) / c l o s e d o r b i t s ( C ) a r e d e n s e i n Q ;
2 ) There e x i s t s a c o n t i n u o u s h y p e r b o l i c s p l i t t i n g o v e r Q. (Axiom A t h u s d e a l s o n l y w i t h t h e b e h a v i o r of @ on t h e non-wandering s e t . ) I n t h i s c a s e , Q c a n be p a r t i t i o n e d i n t o b a s i c s e t s hi;
t h e y a r e t h e p r o p e r g e n e r a l i z a t i o n s of f i x e d p o i n t s . Every Ai h a s i t s s t a b l e and u n s t a b l e m a n i f o l d w s ( A i ) and w u ( A i ) w i t h p r o p e r t i e s a n a l o g o u s t o t h o s e of f i x e d p o i n t s . An a t t r a c t o r i s a c l o s e d minimal i n v a r i a n t s u b s e t A C N w i t h an open neighborhood U c o n t r a c t i n g t o A i n t h e f u t u r e
( A =
n .
T h i s i s t h e p r o p e r g e n e r a l i z a t i o n of a s t a b l e t > Of i x e d p o i n t . The b a s i n B of an a t t r a c t o r A i s t h e set of x M t e n d i n g t o A a s t + m; i t i s open. I n t h e c a s e of Axiom A , a b a s i c set hi i s an a t t r a c t o r i f .
ws
( A i ) i s open;t h e n wS(bi) i s t h e b a s i n of h i .
An a t t r a c t o r i s c a l l e d s t r a n g e i f i t i s n o t a smooth sub- m a n i f o l d of M , e . g . , t h e Lorenz a t t r a c t o r . Warning: most a t t r a c t o r s a r e s t r a n g e !
A s e p a r a t r i x ( i n t h e t e r m i n o l o g y of [ 4 ] ) i s a s t a b l e m a n i f o l d of co-dimension one. I n t h e c a s e of Axiom A ,
t h e basin b o u n d a r i e s a r e t o be found among t h e s e p a r a t r i c e s . For a n Axiom A a t t r a c t o r A , t h e r e e x i s t s a n i n v a r i a n t
measure
u
on A f o r which t h e f o l l o w i n g is t r u e :f o r a l m o s t a l l x i n t h e b a s i n of A and a l l c o n t i n u o u s f u n c t i o n s f on M (Bowen-Ruelle theorem)
.
Sou
a l l o w s u s t o c a l c u l a t e t i m e a v e r a g e s .The ~ ~ - t e ~ o t o ~ ~ on t h e s p a c e of a l l d y n a m i c a l s y s t e m s on M" i s d e f i n e d in t h e f o l l o w i n g way: two s y s t e m s 4 and 9 ' a r e C - c l o s e i f , t o g e t h e r w i t h t h e i r d e r i v a t i v e s u p t o o r d e r r r , t h e y are u n i f o r m l y c l o s e a s maps 4; and @ t from M t o M f o r a f i x e d t.
A s y s t e m 4 i s c r - s t r u c t u r a t t y - s t a b l e i f f o r a l l 4 '
, cr-
c l o s e enough t o 4 , t h e r e e x i s t s a homeomorphism12 h of M t r a n s f o r m i n g o r b i t s of 4 i n t o o r b i t s of 4 ' . Thus, u p t o a t o p o l o g i c a l d e f o r m a t i o n , 4 ' l o o k s e x a c t l y l i k e 4.
Here, M h a s t o be compact, o r awkward t e c h n i c a l problems w i l l o c c u r .
I 2 ~ h e r e a r e good r e a s o n s f o r i n s i s t i n g o n l y on b i c o n t i n u i t y o f h, i n s t e a d of on d i f f e r e n t i a b i l i t y .
A s y s t e m 4 i s ~ ~ - Q - s t a b l e i f t h e a b o v e h o l d s a t l e a s t on t h e non-wandering sets R o f 4 and Q o f 4 ' ( s o t h a t h t r a n s f o r m s R i n t o Q ' ) . Here, w e a r e i n t e r e s t e d o n l y i n n o n - t r a n s i e n t b e h a v i o r : on R . I f 4 is Q - s t a b l e , i t must s a t i s f y Axiom A.
A s y s t e m 4 i s t o p o l o g i c a l l y s t a b l e i f a l l c r - c l o s e 4 ' h a v e non-wandering s e t s homeomorphic t o t h e o n e o f 4 . The L o r e n z a t t r a c t o r is n o t e v e n t o p o l o g i c a l l y s t a b l e .
H o l l i n g , C.S., " R e s i l i e n c e a n d S t a b i l i t y o f E c o l o g i c a l S y s t e m s , " IIASA RR-73-3, I n t e r n a t i o n a l I n s t i t u t e
f o r A p p l i e d S y s t e m s A n a l y s i s , L a x e n b u r g , A u s t r i a , 1973.
C l a r k , W.C. ( 1 9 7 5 ) , " N o t e s on R e s i l i e n c e M e a s u r e s , " and C l a r k , W.C., C.S. H o l l i n g , a n d D.D. J o n e s ( 1 9 7 5 ) ,
"Towards S t r u c t u r a l View o f R e s i l i e n c e " ; i n t e r n a l p a p e r s , I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s , L a x e n b u r g , A u s t r i a .
Avenhaus, R . , e t a l . , " N e w S o c i e t a l E q u a t i o n s , " i n t e r n a l p a p e r , I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s , L a x e n b u r g , A u s t r i a , 1975.
G r i . , H. -R.
,
" S t a b l e M a n i f o l d s a n d S e p a r a t r i c e s , " i n t e r n a l p a p e r , I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s , Laxenburg, A u s t r i a , 1975.S v e t l o s a n o v , V . , " Q u a n t i t a t i v e D e f i n i t i o n o f S t a b i l i t y , R e s i l i e n c e a n d E l a s t i c i t y , " i n t e r n a l c o m u n i c a t i o n , I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s , L a x e n b u r g , A u s t r i a , 1975.
J o n e s , D.D., " A n a l y s i s o f a Compact P r e d a t o r P r e y Model
-
I.The B a s i c E q u a t i o n s and B e h a v i o r , " i n t e r n a l p a p e r , I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s , Laxenburg, A u s t r i a , 1974.
~ r i i m m , H.-R., Appendix I11 i n [ 3 ] .
R u e l l e , D . , a n d R. Bowen, " E r g o d i c T h e o r y f o r Axiom A A t t r a c t o r s , " p r e p r i n t , I n s t i t u t d e s H a u t e s E t u d e s S c i e n t i f i q u e s , B u r e s - S u r - Y v e t t e , 1974.
H a f e l e , W . , " Z i e l f u n k t i o n e n , " i n B e i t r a g e z u r K e r n t e c h n i k
-
Karl W i r t z gewidmet zum 65. G e b u r t s t a g , KFK-2200,
~ L - I 1 7 8 , K e r n f o r s c h u n g s z e n t r u m K a r l s r u h e , 1975.
G a t t o , M., S. R i n a l d i , a n d C. W a l t e r s , "A P r e d a t o r - P r e y Model f o r D i s c r e t e - T i m e Commercial F i s h e r i e s , "
RR-75-5, I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s , L a x e n b u r g , A u s t r i a , 1975.
C a s t i , J . , " Z u b o v ' s P r o c e d u r e s f o r E s t i m a t i n g t h e Domain o f A t t r a c t i o n ,
"
i n G r k , H. -R. ( e d . ),
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