T h e M a c k e y - G la s s sy s te m (2 ) is o n e o f th e b e s t k n o w n d e la y d iffe re n tia l e q u a tio n s.
T h e o rig in a l w o rk o f M a c k e y an d G la ss [ 17] s p a w n e d w id e a tte n tio n , b e in g c ite d b y m a n y p a p e rs w ith a b ro a d sp e c tru m o f to p ic s: fro m th e o re tic a l m a th e m a tic a l w o rk s to n e u ra l n e tw o rk s an d e le c tric a l e n g in e e rin g . N u m e ric a l e x p e rim e n ts sh o w th at, as e ith e r p a ra m e te r t [17] o r n [16] is in c re a se d , th e sy s te m u n d e rg o e s a series o f p e r io d - d o u b lin g b ifu rc a tio n s , a n d th e y le a d to th e c re a tio n o f an a p p a re n t c h a o tic attracto r.
In th is se c tio n , w e p re s e n t c o m p u te r-a s s is te d p ro o fs o f th e e x is te n c e o f a ttra c tin g p e rio d ic o rb its in M a c k e y - G la s s sy s te m (2 ) . W e u s e th e c la s sic a l v a lu e s o f p a ra m e te rs:
t = 2, j3 = 2 an d y = 1, a n d w e in v e s tig a te th e e x is te n c e o f p e rio d ic o rb its w ith n = 6 (b e fo re th e first p e rio d d o u b lin g ) an d n = 8 (a fte r th e first p e rio d d o u b lin g ) [ 16]. W e w o u ld lik e to stress, th a t w e a re n o t p ro v in g th a t th e s e o rb its are attractin g . T h is w o u ld re q u ire so m e C 1-e s tim a te s fo r th e P o in c a re m a p d e fin e d b y (2 ).
4 .1 O u tli n e o f th e M e t h o d f o r P r o v i n g P e r i o d ic O r b i t s
T h e s c h e m e o f a c o m p u te r-a s s is te d p r o o f o f a p e rio d ic o rb it c o n sists o f sev eral steps:
1. fin d a g o o d , fin ite re p re s e n ta tio n o f b o u n d e d sets in th e p h a s e s p a c e C k (o r in o th er su ita b le fu n c tio n sp ace),
2. c h o o se su ita b le se c tio n S an d so m e a p rio ri in itia l se t V on th e sectio n , 3. c o m p u te im a g e o f V b y P o in c a re m ap P>m on se c tio n S,
4 . p ro v e th a t th e m a p P>m , t h e s e t V , a n d t h e s e t W : = P>m( V ) all s a tisfy a ssu m p tio n s o f so m e fix ed p o in t th e o re m so th a t it im p lie s th e e x is te n c e o f a fix ed p o in t fo r P>m in V . T h is g iv es ris e to th e p e rio d ic o rb it in E q . ( 1).
To th is p o in t, w e h a v e p re s e n te d in g re d ie n ts n e e d e d in step s 1 an d 3. In S tep 4 , w e w ill u s e th e S c h a u d e r fix ed p o in t th e o re m [2 7 , 3 4 ]:
T h e o r e m 11 [ S c h a u d e r F i x e d P o in t T h e o r e m ] L e t X be a B a n a c h sp ace, let V c X b e n o n-em pty, convex, b o u n d e d se t a n d let P : V — X b e c o n ti n u o u s m a p p i n g such th a t P ( V ) c K c V a n d K is co m p a ct. T h e n th e m a p P h a s a f i x e d p o i n t in V.
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0
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Found Com put M ath (2018) 18:1299-1332 1323
on th e im a g e o f th e P o in c a re m ap . W e u s e an o b se rv a tio n fro m S ect. 3 .4 , a n d w e find th e le ft e ig e n v e c to r l o f th e m a trix dp ( T , x 0) c o rre sp o n d in g to e ig e n v a lu e 1, w h e re T is an a p p a re n t p e rio d o f th e a p p ro x im a te p e rio d ic o rb it fo r th e n o n -rig o ro u s sem iflo w p .
P le a s e n o te th a t l m ig h t b e c o n s id e re d a (p , n )- re p re s e n ta tio n w ith re m a in d e r p a rt se t to 0, an d th e re fo re , w e c a n d e fin e a (p , n )- s e c tio n b y
w h e re th e d o t p ro d u c t is c o m p u te d u sin g th e c o o rd in a te s o f (p , n )-re p re s e n ta tio n , i.e ., in th e v e c to r s p a c e R m , w h e re m is th e siz e o f a (p , n ) -re p re se n ta tio n , m = (n + 2) ■ p + 1.
S te p 3 . H a v in g a g o o d c a n d id a te fo r th e se c tio n S (d e fin e d b y (2 4 ) ), w e n e e d to in tro d u c e th e c o o rd in a te s on it. F o r th is, w e c re a te th e fo llo w in g m atrix :
N ow , le t C d e n o te th e m a trix o b ta in e d a fte r o rth o n o rm a liz a tio n o f c o lu m n s o f A.
P le a s e n o te th a t m a trix C acts on th e v a ria b le s c o rre s p o n d in g to th e re m a in d e r term s as an id e n tity . T h is fo llo w s fro m th e fa c t th a t li,[n+ 1] = 0. It is e a sy to see th a t all ( p , n )- re p re s e n ta tio n s th a t lie on th e se c tio n S are g iv en by:
fo r all y su ch th a t n 1 y = 0.
N ow , on se c tio n S , u s in g th e c o o rd in a te s d e fin e d b y th e m a trix C , w e d e fin e a c a n d id a te se t [V ], in a fo rm o f (p , n ) - f - s e t in a fo llo w in g m an n er. L e t [r] c R m— p (th e se c o rre s p o n d to v a ria b le s x 1' ^ fo r k < n ) a n d [B] c R p (th e se are b o u n d s for x 1’ ^ 1]— th e re m a in d e rs ) b e tw o in te rv a l b o x e s c e n te re d a t 0 su c h th a t d i a m ( n 1 [r ]) = 0. W e p u t [r0] : = [r] x [B ], a n d w e d e fin e (p , n ) - f - s e t [V ] by:
D ia m e te rs o f n [r0] fo r i > 2 are s e le c te d e x p e rim e n ta lly to fo llo w so m e e x p o n e n tia l law in p a ra m e te r k (i.e., d i a m & i q ’^ r ] ) & a k fo r 1 < i < n ), as th e p e rio d ic so lu tio n s to E q . (2 ) a re a t le a s t o f class C “ an d , i f x ( t ) > —1 fo r all t , th e n th e y s h o u ld b e a n a ly tic [ 18, 2 2 ]. T h e re m a in d e r [B ] is c h o se n in itia lly su c h th at d i a m ( B ) ^ d i a m ([ r0]). T h e re fo re , th e in itia l se le c tio n o f [r0] m a y n o t b e g o o d e n o u g h to sa tisfy a ss u m p tio n s o f T h e o re m 11 r ig h t aw ay. A s th e d y n a m ic s o f th e s y s te m is s tro n g ly c o n tra c tin g , w e h o p e to o b ta in a g o o d in itia l c o n d itio n b y th e fo llo w in g ite ra tio n . W e sta rt w ith [V ]0 = [V ] an d w e c o m p u te [V ] +1 = P>m( [ V ] ) n [V ] , u n til th e c o n d itio n P>m( [ V ] i ) c [V ]i is e v e n tu a lly m e t at so m e i stop. T h e n , th e in itial se t fo r th e c o m p u te r-a s s is te d p r o o f is [V ] = Vistop. B o th in itia l sets th a t are u s e d in c o m p u te r-a s s is te d p ro o fs in th is p a p e r w e re g e n e ra te d w ith su ch p ro c e d u re (s e e so u rc e c o d e s).
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l ■ x — l ■ x 0 = 0, (24)
y ^ x0 + C y ,
[V ] := x0 + C ■ [r0].
Found Com put M ath (2018) 18:1299-1332 1325
O b s e rv e th a t w e are n o t v e ry c a re fu l in th e c h o ic e o f c o o r d i n a te s o n the s e c t io n — w e sim p ly c h o o s e so m e b a sis o rth o n o rm a l to th e n o rm a l v e c to r l o f th e se c tio n h y p e rp la n e . D e fin ite ly b e tte r c h o ic e w o u ld b e to u s e a p p ro x im a te e ig e n v e c to rs o f th e P o in c a re m a p , b u t in th e c a s e o f stro n g ly a ttra c tin g p e rio d ic o rb its, it is e n o u g h to c h o o s e a g o o d sectio n . O b s e rv e also , th a t th e o rth o n o rm a l m a trix is e a sy to in v e rt rig o ro u sly , w h ic h is an im p o rta n t step in th e c o m p a ris o n o f th e in itia l set an d its im a g e b y th e P o in c a re m ap.
4 .3 A t t r a c t i n g P e r i o d ic O r b i t s in M a c k e y - G l a s s E q u a t i o n f o r n = 6 a n d n = 8
In th is se c tio n w e p re s e n t tw o th e o re m s a b o u t th e e x is te n c e o f p e rio d ic o rb its in M a c k e y - G la s s e q u a tio n . A s th e y d e p e n d h e a v ily on th e e s tim a te s o b ta in e d fro m th e r ig o ro u s n u m e ric a l c o m p u ta tio n s, w e w o u ld lik e to d is c u ss first th e te x tu a l p re s e n ta tio n o f n u m b e rs u s e d in th is se c tio n a n d h o w th e y a re re la te d to th e in p u t/o u tp u t v alu es u s e d in rig o ro u s c o m p u ta tio n s.
In th e rig o ro u s n u m e ric s, w e u s e in te rv a ls w ith en d s b e in g re p re s e n ta b le c o m p u te r n u m b e rs. T h e re p re s e n ta b le n u m b e rs are im p le m e n te d as b i n a r y 3 2 o r b i n a r y 6 4 d a ta ty p e s d e fin e d in IE E E S ta n d a rd fo r F lo a tin g P o in t A rith m e tic (IE E E 7 5 4 ) [2 8 ], so th a t th e y are sto red (ro u g h ly s p e a k in g ) as s ■ m ■ 2e , w h e re s is th e sig n b it, m is th e m a n tis s a an d e th e e x p o n e n t. S u c h a re p re s e n ta tio n m e a n s th a t m o s t n u m b e rs w ith a fin ite re p re s e n ta tio n in th e d e c im a l b a s e a re n o t re p re s e n ta b le (e.g ., n u m b e r 0 .3 3 3 3 ).
In th is p a p e r, fo r b e tte r re a d a b ility , w e are g o in g to u s e th e d e c im a l re p re s e n ta tio n o f n u m b e rs w ith th e fixed p re c is io n (u su a lly 4 d e c im a l p la c e s), so w e h a v e re w ritte n c o m p u te r p ro g ra m s to h a n d le th o s e v a lu e s rig o ro u sly . F o r e x a m p le , if w e w rite in th e te x t th a t a = 0 .3 3 3 3 , th e n w e p u t th e fo llo w in g rig o ro u s o p e ra tio n in th e code:
a = [3 3 3 3 , 333 3 ] y [1 0 ,0 0 0 , 1 0 ,0 0 0 ].
T h a t is, all n u m b e rs p re s e n te d h e re in th e o re m s a n d /o r p ro o fs sh o u ld b e re g a rd e d b y th e re a d e r as th e re a l, rig o ro u s v a lu e s, ev en i f th e y are n o t re p re s e n ta b le in th e se n se o f IE E E 7 5 4 sta n d a rd .
In th e p ro o fs, w e re fe r to c o m p u te r p ro g ra m s m g _ s t a b l e _ n6 an d m g _ s t a b l e _ n8. T h e ir s o u rc e c o d e s, to g e th e r w ith in s tru c tio n s on th e c o m p ila tio n p ro c e s s , c a n b e d o w n lo a d e d fro m [2 3 ]. T h e c o d e s w e re te s te d on a la p to p w ith Intel® Core™
I7 -2 8 6 0 Q M C P U (2 .5 0 G H z ), 16 G B R A M u n d e r 6 4 -b it L in u x o p e ra tin g sy ste m (U b u n tu 12.04 L T S ) an d C /C + + c o m p ile r g c c v e rs io n 4 .6 .3 .
4 .3 .1 C a s e n = 6
O u r first re s u lt is fo r th e p e rio d ic o rb it fo r th e p a ra m e te r v a lu e b e fo r e th e first p e rio d - d o u b lin g b ifu rc a tio n .
W ith n = 6, n u m e ric a l e x p e rim e n ts c le a rly sh o w th a t th e m in im a l p e rio d o f th e p e rio d ic o rb it is a ro u n d 5 .5 8 . In o u r p ro o f, h o w ev er, d u e to th e p ro b le m w ith th e loss o f th e re g u la rity a t th e g rid p o in ts , th u s th e n e e d to u s e th e “ lo n g e n o u g h ” tra n sitio n tim e , w e c o n s id e r th e se c o n d re tu rn to th e sectio n .
F o C 'n
1
Springer uN u m e ric a l e x p e rim e n ts in d ic a te th a t th e o rb it is a ttra c tin g w ith th e m o s t sig n ific a n t
Found Com put M ath (2018) 18:1299-1332 1327
w h ic h g u a ra n te e s th e C n+1-re g u la rity o f th e so lu tio n s a n d th e c o m p a c tn e ss o f the m a p P>m (in C k n o rm fo r k < n ). T h e in c lu sio n c o n d itio n P>m ([x0]) c [x0] o f the S c h a u d e r fix ed p o in t th e o re m is c h e c k e d rig o ro u s ly ; se e o u tp u t o f th e p ro g ra m for d e ta ils . T o g eth er, th e s e tw o facts g u a ra n te e th e n o n -e m p tin e s s o f S u p p (n+1) ([x0]).
T h e tra n s v e rs a lity is g u a ra n te e d w ith l ( x ) > 0 .2 8 2 8 fo r all x e C n+1 n P ( x 0). T h e d is ta n c e in C0 n o rm is rig o ro u s ly e s tim a te d to
lix — x ||C0 < 0 .0 1 9 0 2 8 6 7 6 8 1 .
S im ilarly , w e h a v e v erifie d th e o th e r n o rm s; se e o u tp u t o f th e p ro g ra m . □ T h e e x e c u tio n o f th e p ro g ra m re a liz in g th is p r o o f to o k a ro u n d 12 se c o n d s on 2 .5 0 G H z m a c h in e .
T h e d ia m e te r o f th e e s tim a tio n fo r p e rio d T (also fo r th e la s t step [e1, e2]) o b ta in e d fro m th e c o m p u te r-a s s is te d p r o o f is c lo s e to 1.15 ■ 1 0 —4.
A g ra p h ic a l re p re s e n ta tio n o f th e e stim a te s o b ta in e d in th e p r o o f is fo u n d in F ig . 4 .
4.3 .2 C a s e n = 8
F o r n = 8 w e c o n s id e r th e p e rio d ic o rb it a fte r th e first p e rio d d o u b lin g . T h is tim e th e p e rio d o f th e o rb it is lo n g e n o u g h to o v e rc o m e th e in itia l lo ss o f re g u la rity , so w e c o n s id e r th e first re tu rn P o in c a re m ap.
N u m e ric a l c o m p u ta tio n sh o w s th a t th e o rb it is a ttra c tin g w ith th e 10 m o s t sig n ifican t e ig e n v a lu e s o f th e m a p P>m e s tim a te d to be:
ReX 0.3090 — 0.1359 — 0.0067 — 7.58 ■ 10—4 6.58 ■ 10—4 —1.23 ■ 10—4 2.184 ■ 10—5
ImX 6.265 ■ 10—6
|XI 0.3090 0.1359 0.0067 7.58 ■ 10—4 6.58 ■ 10—4 1.23 ■ 10—4 2.272 ■ 10—5
T h e o r e m 13 There exis ts a T - p e r i o d i c s o l u ti o n x w ith p e r i o d T e [1 1 .1 3 5 0 , 11.1353]
to Eq. (2 ) f o r p a r a m e t e r s y = 2, a = 1, t = 2 a n d n = 8. M oreover,
x — x C 0 < 0 . 0 1 2
x — x C1 < 0 .0 6 x — x C 2 < 0 .2 0
x — x C 3 < 0 .5 2 x — x C 4 < 1.25
f o r x d e fi n e d by
x ( t ) = 0 .9 4 8 0
, ' 2 n \ ( 2 n
+ 0 .0 4 7 7 ■ co s ( — ■ 1 ■ t 1 — 0 .0 6 8 9 ■ sin I — ■ 1 ■ t
1 n
FoCTI
H h
Ll=]°J
X
1 .4
1.2
1
(N
§
0.80.6
0 .4
0 .4 0.6 0.8
X ( t )
1.2 1 .4
Fig. 4 Top: approximate function x (blue) and estimates on the value of the true solution obtained from computer-assisted proof (red). Bottom: solution plotted as parametric curve r(t) = (x(t), x(t — t)) (Color figure online)
+ 0 .2 5 1 6 • co s ( r~ n • 2 • t ^ + 0 .1 1 2 0 • sin . 2 • t
, ' 2 n \ ( 2 n
+ 0 .0 2 4 2 • co s ( — • 3 • t
I
+ 0 .0 6 0 4 • sinI
— • 3 • t, ' 2 n \ ( 2 n
— 0 .0 3 8 6 • co s ( — • 4 • M — 0 .0 1 9 1 • sin I — • 4 • t FoCJI
1
uFound Com put M ath (2018) 18:1299-1332 1329
X
1 .4
1.2
0.8
0.6
0 .4
0 .4 0.6 0.8 1.2 1 .4
X ( t )
Fig. 5 Top: approximate function x (blue) and estimates on the value of the true solution obtained from computer-assisted proof (red). Bottom: solution plotted as parametric curve r(t) = (x(t), x(t — t)) (Color figure online)
+ 0 .0 1 3 2 • co s ( . 5 . t ^ — 0 .0 0 6 8 • sin • 5 • t
, ' 2 n \ ( 2 n
— 0 .0 1 9 7 • co s ( — • 6 • t 1 + 0 .0 1 9 8 • sin I — • 6 • t
, ' 2 n \ ( 2 n
+ 0 .0 0 7 7 • co s ( — • 7 • t j — 0 .0 1 3 4 • sin I — • 7 • t
FoCTI
1
Springer u0 .0 0 1 8 • cos 0 .0 0 5 3 • cos
0 .0 0 0 5 • cos
Pr o o f T h e p r o o f fo llo w s th e sa m e lin e s as in th e c a s e o f T h e o re m 12 (e x c e p t th is tim e w e c o n s id e r th e first re tu rn to th e se c tio n ). T h e re fo re , w e ju s t list th e p a ra m e te rs fro m th e p ro o f.
T h e d ia m e te r o f th e e s tim a tio n fo r p e rio d T (also fo r th e la s t step [e 1, e 2]) o b ta in e d fro m th e c o m p u te r-a s s is te d p r o o f is c lo s e to 3 .8 9 9 • 1 0 —6 . A g ra p h ic a l re p re se n ta tio n o f th e e s tim a te s o b ta in e d in th e p r o o f is fo u n d in F ig . 5 .
T h e e x e c u tio n tim e w as a ro u n d 1 2 m in . T h is in c re a s e w h e n c o m p a re d to n = 6 is d u e to m u c h la rg e r re p re s e n ta tio n siz e in th is c a s e w h ic h affects th e c o m p le x ity o f m a trix an d a u to m a tic d iffe re n tia tio n a lg o rith m s w h ic h w e are u sin g .