By D. D. Joseph, K. Burns (auth.), David Rand, Lai-Sang Young (eds.)
Read Online or Download Dynamical Systems and Turbulence, Warwick 1980: Proceedings of a Symposium Held at the University of Warwick 1979/80 PDF
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Extra info for Dynamical Systems and Turbulence, Warwick 1980: Proceedings of a Symposium Held at the University of Warwick 1979/80
Example text
C h o o s e k so t h a t s o m e finite c o v e r by 8 9 c o n t a i n s one of t h e s e 8 9 V h a s f k v . ri Vj. # J3 Vj, j ' . Every c-ball 3 J so f k u N U' # 0 f o r a n y a - b a l l s U , U ' . For any x E F i x ( f m ) t a k e y s u c h t h a t p ( x , y ) < ~ and p ( f k y , x) < ~. f m - l x , Y, fy . . . x, fx ..... By e x p a n s i v e n e s s , *t F i x ( f m ) a Step 3. f k - l y , x is 89 ~[he c l o s e d a - o r b i t by a g e n u i n e p e r i o d i c o r b i t fm+kz = z. d i s t i n c t p o i n t s x give r i s e to d i s t i n c t p o i n t s z.
0 ~,<0 ! + T- ! - \ ] A2 4+ I + + m + 4F i g u r e 14. Remarks on r e g i o n s 2, 3; ~ < 0. In r e g i o n 2, k 1 > k 2 and t h e d o t t e d s o l u t i o n c u r v e h a s n e g a t i v e d e g r e e . 2 < ). i t h e r e a r e two c o m p e t i n g s t a b l e s o l u t i o n p o s s i b i l i t i e s in t h e two coordinate planes. In r e g i o n 3, X1 < )'2' and t h e d o t t e d s o l u t i o n s a r e s t a b l e ; t h e only s t a b l e s o l u t i o n s in t h e r a n g e )'1 < )' < X2" these are 31 In region 1, when ~ / 0 the evolution of the s y s t e m under q u a s i s t a t i c v a r i a t i o n s of X will be c o m p l e t e l y s m o o t h .
We c o n s i d e r a Lie subalgebra K;(IR n) of ~s One may think e . g . of the set of all r - p a r a m e t e r families of volume p r e s e r v i n g v e c t o r fields on ~Rm where m + r = n. k = 1,2 . . More examples will be given below. In the usual way for define the singular k - j e t space c~k(N n) and a~rk(lRn) r e s p e c t i v e l y , corresponding with the sets of truncated Xaylor s e r i e s of vector fields in K0(')(1Rn). F o r k = ~o we define the k - j e t spaces by taking the i n v e r s e limits : now the coefficient functions a r e f o r m a l power s e r i e s .
