By H. S Hall
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Extra resources for A short introduction to graphical algebra
D. 121. Let the initial set Et 0 not coincide with the initial point to, and the be infinite dimensionaL manifold of solutions Ut 0‘ 0 Further, let there exist a point t*>tO such that t-T(t)Lt* for tLt*. ,n-l) (k=0,1,. x,(t) ~x~ (t) everywhere. then for t>t*, The case when the initial set Et* coincides with the point t*, from the mathematical point of view, is unusual, even for one retardation but, as simple examples show, the coalescence of solutions is possible and there are a l s o other sufficiently unusual cases.
1 4 ) t u r n s i n t o t h e f i n i t e equation x(t) = f(t,I$(t-T) (t-T)) (15) €or t < t < t+ T . Consequently, a t t h e p o i n t t=tO, 0 - 0 i n g e n e r a l , $ ( t o #) x ( t + 0 ) a n d , i n t h i s s e n s e , 4 0 t h e solution is discontinuous. I f the function f i s d i f f e r e n t i a b l e a s u f f i c i e n t number of t i m e s ( n o t less t h a n k-1 t i m e s ) and $ i s k t i m e s d i f f e r e n t i a b l e , t h e n t h e s o l u t i o n o f Eq. ( 1 3 ) i s d i f f e r e n t i a b l e , i n g e n e r a l , o n l y k-1 t i m e s ( s i n c e x(IC-1) ( t ) contains the t e r m 7 ak-lf (t-T) and f o r (k-1) (t-T t h e e x i s t e n c e o f t h e n e x t d e r i v a t i v e , i would be n e c e s s a r y t o r e q u i r e t h e e x i s t e n c e of + l k + l ) ( t - T ) , which w e d o n o t a s s u m e ) .
I f €=Of then a t t h e point t = O t h e r e occurs continual b i f u r c a t i o n of solution: x ( t ) = f ( t ) , where f ( t ) i s t h e d e r i v a t i v e o f a n8n-decreasing c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n , s a t i s f y i n g t h e c o n d i t i o n f (O)=O. 46 1. BASIC CONCEPTS Example 2. k(t) = -x(t-T(jc(t)))+t, tl_tol_O, @(t) = t, t5to. If %(t)