Behavior near periodic orbit of functional differential equation

Hale, J. K.

English

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Behavior Kear a Periodic Orbi t  of Functional D i f f e ren t i a l  Equations 
* 
T h i s  research has  been supported i n  -part by t he  United S ta tes  A i r  Force under Grant 
No. AFO$R-693-67, by t h e  Nat ioml  Science Fmndation under Grant No. GP-7471, by 
the United S ta tes  Arrny under Grant No. DA--31-124-_4RO-D-270 and i n  pa r t  by the  
National Aeromautics and Space Administration under Grant No, NGR-40-002-013. 
https://ntrs.nasa.gov/search.jsp?R=19680022527 2020-03-12T10:07:40+00:00Z
Behavior near a periodic o rb i t  of funct ional  d i f f e r e n t i a l  equations 
by 
Jack K. Hale 
n I. Introduction. Let C = C([- r ,O] ,E  ) be the  space of continuous 
functions mapping the  i n t e r v a l  [ - r , O ]  i n t o  an n-dimensional r e a l  or 
complex vector space En. For any cp i n  Cy def ine 11y11 = s ~ p - ~ ~ ~ . . ~  
I c p ( f 3 ) l  where 1x1 i s  the  norm of a vector x i n  En, If x i s  a 
continums function mapping any i n t e r v a l  [a-r ,  a+A)  i n t o  E ~ ,  A > o 
and t i s  a given element of [a ,  a t A ) ,  define x i n  C by xt(0) = 
x( t t8 ) ,  -r 5 8 5 0. If F i s  a function taking R x C i n t o  E , R = 
t 
n 
(-w,oo), then a funct ional  d i f f e r e n t i a l  equation i s  defined by the  re la -  
t i o n  
%(t) = F(t,xt) 
where %(t) denotes t h e  r i g h t  hand der ivat ive with respect t o  t. A 
solut ion of (1) with i n i t i a l  value cp a t  Q i s  a continuous function 
x = x(a,y) defined on [b-r,  a+A), A > 0 such t h a t  x = Q and x 
s a t i s f i e s  (1) f o r  t i n  [b, 0 4 ) .  It i s  wel l  known t h a t  F(t,cp) can- 
b 
tinuous i n  t,cp and loca l ly  Lipschitzian i n  cp implies t h e  existence and 
uniqueness of a solution of (1) through (a,cp) f o r  any a,cp. Further- 
more, t h e  solut ion x(b,cp)(t) i s  continuous i n  (b,cp,t) i n  i t s  domain of 
def in i t ion .  It w i l l  be assumed i n  the  sequel t h a t  F i s  smooth enough 
t o  ensure t h i s  l a t t e r  pmperty and, i n  fact, F(t,cp) i s  continuous i n  
t,cp and has  a continuous f i r s t  der iva t ive  w i t h  respect t o  cp. 
If F(t,cp) = f (y )  i s  independent of t, system (1) i s  referred 
t a  as an autmamms funct ional  d i f f e r e n t i a l  equation 
2 
( 2 )  %(t) = f(xt). 
I n  t h i s  case, it i s  no r e s t r i c t i o n  t o  assume t h a t  CT = 0 and write t h e  
solution as ~(9). An o rb i t  O ' ( 9 )  of (2) through cp i s  defined i n  C 
and not i n  En as 0 ( c p )  = UtZOxt(Cp). The qua l i ta t ive  theory of auto- 
nomous functional d i f f e r e n t i a l  equations i s  concerned with the  study of 
t he  geometrical properties of the  o rb i t s  of (2) .  
+ 
We b r i e f l y  out l ine 
below some of t he  r e su l t s  t h a t  have been obtained alolng t h i s  l i n e  for  
equilibrium points and periodic orbi ts .  
An equilibrium or c r i t i c a l  point of (2)  i s  a constant function 
Without any loss i n  generali ty,  we may a s sme  t h i s  
x = 0; t h a t  i s ,  f(0) = 0. 
which s a t i s f i e s  (2) .  
equilibrium point i s  
equation associated w i t h  x = 0 i s  
The l inear  var ia t iona l  
where L(cp) i s  the  der ivat ive of f a t  0 evaluated a t  cp.  The map- 
ping L: C +En i s  a continuous l inear  operator and therefore has  t h e  
form 
where 7 i s  an n X n matrix with elements of bounded variation. The 
charac te r i s t ic  equation fo r  ( 3 )  i s  
( 5 )  
The solution 
3 
x = o  of (2) i s  called nondegenerate i f  h , = o  i s  not a 
solution of ( 5 )  and elementary i f  no solution of ( 5 )  has  zero r e a l  part .  
UI 
If no solution of ( 5 )  has zero r e a l  par t ,  then the space C 
can be decomposed as the  d i r e c t  sum of  two l inear  subspaces, C = S 8 U, 
which ars invariant under the  mapping induced by the  solutions of ( 3 )  
such t h a t  t he  solutions of ( 3 )  are  decaying exponentials on 
ing.exponentials on U. The s e t  U i s  f i n i t e  dimensional. The s e t s  
S and grc~w-  
S and U are  re€erred t o  as %he s table  and unstable manifolds, re- 
spectively, of ( 3 ) .  The s e t  S 'may a l so  be characterized as  the  s e t  
of i n i t i a l  val.ues of those solutions of ( 3 )  which remain i n  a neighbor- 
hood of zero f o r  a l l  t 2 0. The se t  U i s  characterized i n  the  same 
manner except for  t 5 0. 
The l a t t e r  characterization of S and U may a l so  be used 
t o  define the  s table  and unstable manifolds and u fo r  t h e  solution 
x = 0 
[ 11 t h a t  an elementary c r i t i c a l  point p of (2 )  exhibi ts  i n  C a saddle 
point s t ruc ture  i n  the  sense t h a t  the  s t ab le  and unstable manifolds of 
p are  loca l ly  homeomorphic t o  the  s table  and unstable manifolds of 
t he  l inear  var ia t iona l  equation. 
of (2)  With t h i s  def in i t ion ,  it was shovn by Hale and Perello 
Recently, Oliva [ 21 has formulated the  concept of fwictional 
d i f f e r e n t i a l  equations on a compact manifold. 
the  family of a l l  f which have a continuous bounded first der ivat ive 
such t h a t  system (2) has  only elementary (or nondegenerate) c r i t i c a l  
He a l so  has s h m n  t h a t  
4 
1 
points i s  open and dense i n  t h e  C -topology of a l l  f having a con- 
tinuous bounded first derivative.  
If system (2) has a nonconstant co-periodic solution p, then 
the  l inear  var ia t iona l  equation r e l a t ive  t o  p i s  
i(t) = L ( t , z t )  
whire L(t+cu,cp) = L(t,cp) i s  the  der ivat ive of  f a t  p evaluated 
at cp. 
tinuous l inear  3perators T( t ,c r ) ,  t 2 cr,  taking C i n t o  C by defining 
To the  l inear  equation (6), one can associate a family of con- 
where z(cr,cp) i s  the  solution of (6) w i t h  i n i t i a l  value cp a t  6. 
Stokes [ 3 ]  has  shown t h a t  the  spectrum of t he  operator 
s is ts  only of paint spectrum (except f o r  
T(accu,a) con- 
(O]), i s  independent of Q 
and he j u s t i f i a b l y  c a l l s  t h e  s p e e t r m  of t h i s  operator the ‘character- 
i s t i c  mult ipl iers  of (7).  -
Since p i s  a nonconstarit w-periodic solution o f  ( 2 ) ,  
t h e  der ivat ive 5 i s  a non t r iv i a l  w-periodic solution of (6) and 
therefore  m e  of the  charac te r i s t ic  a u l t i p l i e r s  of (6) i s  one. If 
i s  a generator f o r  t he  generalized eigenspace of the  mult ipl ier  
one of T(ow,a),  then 5 i s  a generator f o r  t he  generalized eigen- 
Q: 
5 
space of the  mult ipl ier  one of T(a+co,a) f o r  any a. I n  such a case, 
we say the  o rb i t  I' generated by the  u-periodic solution p i s  non- 
degenerate. If I' i s  nondegenerate and a l so  no other mult ipl iers  of 
(6) have modulus one, we say I' i s  elerfientary. The theory of non- 
-
degenerate and elementary periodic o rb i t s  i s  not complete a t  the  pres- 
ent time although a few r e s u l t s  a r e  available.  
ez s i ly  prctve t h a t  a nondegenerate w-periodic o rb i t  I' of (2)  i s  iso- 
lated i n  the  sense t h a t  there  i s  a neighborhoDd V of I' such t h a t  V'\Y 
contains no w-periodic o rb i t s .  
For example, one can 
Halanay [ 41 has proved the following more complicated s ta te -  
ment: If the d i f f e r e n t i a l  difference equation 
k ( t )  = g ( x ( t ) , x ( t - r ) )  
has  a nondegenerate periodic o rb i t  l', then there  i s  an c0 > O  such 
t h a t  the  equation 
(3) H( t) = h( x( t) ,x( t-r), E) 
has a nondegenerate periodic o rb i t  for 5 E and To = I? 
provided t h a t  h( x,y, E) 
di f fe ren t iab le  i n  x and y, and h(x,y,O) = g(x,y). 
0 
i s  continuous i n  i t s  arguments, continuously 
We have'obtained an extension of the  r e su l t  of Halanay 
t o  a ra ther  general c l a s s  of functional d i f f e r e n t i a l  equations. Suppose 
. 
6 
n x: R -+E 
y ( ~ ) .  Then 
i s  a continuous function, t = ( l + f 3 ) ~ ,  f3 > -1 and x ( t )  = 
ranges over t he  in t e rva l  [ -r , O ] ,  r = r/( S+f3). 1 1 where 5 = e/(l-cf3) 
We say a function f: c + 
di f fe ren t iab le  function x: R + E  , the  function f ( x  ) = f ( y  ) 
considered as a function of (3 i s  continuously different iable .  Any f 
i s  of c lass  (SI i s  f o r  any con t inuxs ly  
n 
t T J P  
of the  form 
where F and the g a re  czxd~inu3usly d i f fe ren t iab le  vector functions 
and the  7 
j 
a re  matrices D f  bounded var ia t ion i s  of c l a s s  (S)  . 
j 
The following r e s u l t  i s  true: Suppose G ( c p , ~ )  i s  of c l a s s  
(S) ,  i s  continuous i n  c p , ~ ,  continuously d i f fe ren t iab le  i n  cp and 
G(cp,O) = f(cp) If system (2)  has a nondegenerate periodic o rb i t  I?, 
then there  i s  an E~ > 0 such t h a t  the  system 
2( t )  = G ( x ~ , E )  
has a nondegenerate periodic o rb i t  rE f o r  IcI s eo and P = r. 
0 
The ideas i n  the  proof of the above theorem a r e  as f3llows. 
To obtain a periadic solut io3 of (10) which f a r  E = 0 coincides with the  
7 
t h e  equation 
For a fixed E and f3 su f f i c i en t ly  small, one shows there  a re  a scalar  
function B ( E , ~ )  and an w-perioStic function P*(T,E,B), P*(T,o,o) = 
p( 7) , sa t i s fy ing  
The hypotheses of the  theorem imply t h a t  B(0,O) = 0 and aR(E,B)/aP f 0 
f o r  E = 0,  f3 = 0. The impl ic i t  function theorem implies t he  existence 
of a ( 3 ( c ) ,  g ( 0 )  = 0, such t h a t  B(E ,@(c) )  = 0 and, therefore,  t he  
existence of a periodic solution of (10) i 
The discussion of an elementary periodic o rb i t  i s  much more 
d i f f i c u l t .  TZI the  author's knowledge, the  f irst  r e s u l t  dealing w i t h  
t he  s t a b i l i t y  pmper t ies  of a periodic x b i t  of (2) i s  due t o  Stokes 
[TI. He proved the  following thearem: If I' i s  nondegenerate and a l l  
charac te r i s t ic  mult ipl iers  of (6) except one have modulus l e s s  than one, 
then there  i s  a neighborhood V of I' such t h a t  any solution of (2) 
wlth i n i t i a l  value i n  V must approach I' exponentially as t + m  and 
with asymptotic phase; t h a t  is, there  a re  pos i t ive  constants K,a such 
t h a t  for  any 9 i n  V, there  i s  a constant y w i t h  
For the  case when some mul t ip l ie rs  have modulus 
one, some fur ther  information i s  a l so  avai lable ,  Suppose 
constant and consider the  l i nea r  va r i a t iona l  equation f o r  
of (2) r e l a t i v e  t o  p( t+y) .  There ex i s t  two subspaces of 
U( y), such t h a t  any solution of the  l i nea r  equation which 
8 
greater  than 
y i s  a given 
the  solutions 
c y  s(r) and 
+ O  as  t+.o 
a r e  s e t s  
which a r e  loca l ly  homeomorphic t o  S(y) and U(y) such t h a t  any solu- 
t i o n  which remains i n  a ce r t a in  &neighborhood of zero f o r  t 2 0 
Ss( y), Us( y) associated w i t h  t h e  camplete va r i a t iona l  equation 
( t  5 0) must have i t s  
s e t  Y i s  the  s tab le  
neighborhood V of I' 
i n i t i a l  value 
manifold of a 
i f  t h e  o r b i t  
a t  t = 0 on Ss(r)  (Us(y) ) .  A 
periodic o r b i t  T' r e l a t i v e  t o  a 
I' i s  pos i t ive ly  s tab le  r e l a t ive  
t o  i n i t i a l  values on 9 and any solut ion which remains i n  V fo r  t Z 0 
must have i t s  i n i t i a l  value on Y arid approach r as t -+a, The un- 
s tab le  manifold i s  defined analogously for t 5 0. We have been able  t o  
prove t h a t  
a r e  s t ab le  
s e t  2.( i s  
the  s e t s  9 = U 0s- ( P  r +ss(r)) and u="o~m ( P  r +U6 (r)) 
and unstable manifolds, respectively,  of I'. Finally,  t h e  
e i the r  homeomorphic t o  Uo( 6) X (a  c i r c l e )  or a generalized 
Mobius band w i t h  cross section Uo(6) .  
time t h a t  
generalized MDbius band w i t h  cross section 
It has not been shown a t  t h i s  
i s  e i the r  homeomorphic t o  So(&)  X (a  c i r c l e )  or  a 
So( 6). 
9 
R”f ere’n’c’e’s 
[lJ Hale, J. K. and C. Perello,  The neighborho3d of a singular p d n t  
of functions1 d i f f e r e n t i a l  equations. Contrib, D i f f .  Eqs. 3( 1.9641, 
351-375 
[ 2 ]  Oliva, W. M., Functional d i f f e r e n t i a l  equations on compact mani- 
folds  and an approximation theorem. To appear i n  J. D i f f .  Eqs. 
[3] Stokes, A., A Floquet theory fo r  funct ional  d i f f e r e n t i a l  equations. 
Proc. Nat. Acad. Sei. USA, 48( 1962), 1330-1334. 
[ 41 
[5]’  
Halanay, A., D i f f e ren t i a l  Equations, Academic Press, 1966. 
Stokes, A., On the  s t a b i l i t y  of a l i m i t  cycle of an autonomous 
functional d i f f e r e n t i a l  equation. Contrib. D i f f .  Eqs. 3( 1964), 
121-140. 


Behavior near a periodic orbit of functional differential equations

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