On the structure of the space of linear systems of differential equations with periodic coeffiсients

We examine linear systems of differential equations 1 : ( ) ( ) n i i j j j j x a t x b t  l &  , i  1,..., n with continuous -periodic coefficients. The system 1 induces the autonomous system 1 : ( ) ( ), 1 n p i i i j j j x a s x b s s  l &   & on Rn S1, where S1 = R/ Z. The system lp has the unique extension lp on RPn  S1. By trajectories of system l in Rn  S1 ( RPn  S1 ) we will mean trajectories of system lp ( lp ). Let us consider linear systems l as elements of Banach space LSn of continuous -periodic functions 2 11 1 (,...,,..., ) : n n nn n a a b b R  R  with norm, : max max max{ ( ), ( )} i j t i j i a t b t  l. The system LSn l is said to be structurally stable in 1 n  R S (in 1 n  RP S ) if l has a neighborhood V in LSn such that for any system l%V we may find a homeomorphism h : Rn  S1  Rn  S1 ( h : RPn S1  RPn  S1, h(Rn  S1)  Rn  S1 ) which maps oriented trajectories of system %l onto oriented trajectories of system l. Let 0LSn  be the set of systems LSn l whose multiplicators do not belong to the unit circle. Theorem 1. The set 0LSn  is open and everywhere dense in LSn. A system LSn l is structurally stable in Rn  S1 if and only if it belongs to the set 0LSn . Let LS2   be the set of systems LS2  l whose multiplicators are real, distinct and different from -1 and 1. Let s , s  , ns , ns , nu  and nu  be subsets of LS2   consisting of systems l with multiplicators 1 , 2  for which 1 2   1   ( 2 1   1   ) if s l  ( s l  ), 1 2 0     1 ( 1 2 1      0 ) if ns l  ( ns l  ), 1 2 1     ( 1 2     1) if nu l  ( nu l  ). Theorem 2. 1) A system LS2 l is structurally stable in RP2  S1 if and only if it belongs to the set LS2  . 2) For any system LS2  l the corresponding system lp in RP2  S1 is a Morse-Smale system. 3) The sets s , s , ns , ns , nu  and nu  are classes of topological equivalence in LS2  . The paper also describes bifurcation manifolds of codimension one in the space LS2 .