A Note on Mathematical Aspects of Drive–Response Type Synchronization

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Chaos\ Solitons + Fractals Vol[ 09\ No[ 8\ pp[ 0346Ð0351\ 0888 Þ 0888 Elsevier Science Ltd[ All rights reserved 9859Ð9668:88:, ! see front matter

Pergamon

PII] S9859!9668"87#99012!3

A Note on Mathematical Aspects of DriveÐResponse Type Synchronization XIAO!SONG YANG\a\c C[ K[ DUAN b\c and X[ X[ LIAO d a

Department of Mathematics\ University of Science and Technology of China\ Hefei\ Anhui\ P[R[ China b Department of Physics\ University of Science and Technology of China\ Hefei\ Anhui\ 129915\ P[R[ China c Center for Nonlinear Dynamics\ Chongqing Institute of Post and Telecommunications\ Chongqing\ 399954\ P[R[ China d Department of Mathematics\ Huazhong University of Science and Technology\ Wuhan\ 329963\ P[R[ China "Accepted 11 April 0887#

Abstract*This note discusses some mathematical problems and presents several theorems\ which are fundamental in driveÐresponse type synchronization theory[ Þ 0888 Elsevier Science Ltd[ All rights reserved[

0[ INTRODUCTION

Synchronization in chaotic dynamical systems has been a theme in nonlinear sciences for several years ð0Ð04Ł\ and received considerable attention recently[ Some theorems have been established and some approaches have been proposed[ Nevertheless\ it is necessary to clarify the ideas behind these theorems and approaches " for example the theorem in ð04Ł is only true under more rigorous mathematical conditions#\ and put them on a rigorous mathematical foundation[ The objective of this note is to discuss some mathematical problems which are fundamental in synchronization theory[

1[ REVIEW OF SOME CONCEPTS

First we need to recall some concepts and terms from synchronization theory and dynamical system theory[ Consider systems x¾ f"x#

"0#

y¾ `"y\x#

"1#

where x$R n\ y$R n\ f\`]VWR n:R n are C0 maps[

 Author to whom correspondence should be addressed[ E!mail] rgxiaÝphys[ustc[edu[cn 0346

0347

XIAO!SONG YANG et al[

1[0[ Synchronization ð0Ł Let x"t\x9# and y"t\y9# be solutions to "0# and "1# respectively\ it is said that x"t\x9# and y"t\y9# are synchronized if lim =x"t\x9 #−y"t\y9 #=9

t:

"2#

1[1[ Generalized synchronization ð04Ł Suppose that there exists functional relation yH"x#] R n:R n\ denote by MH the manifold yH"x#\ x$R n[ let x"t\x9# and y"t\y9# be the solutions to "0# and "1# respectively[ If lim =y"t\y9 #−H"x"t\x9 ##=9

t:

"3#

holds for all the initial values x9 and y9 in the vicinity of MH\ then it is said that y"t\y9# and x"t\x9# are in generalized synchronization[ Remark 0[ For the convenience of the ensuing discussions\ the de_nitions quoted here are not general enough\ but the mathematical treatment is not essentially di}erent from the general situations[ Remark 1[ In the de_nitions above no condition is imposed upon initial states of x"t\x9# and y"t\y9#[ However in the current literature only the local synchronization "i[e[\ the additional condition is that the di}erence between the initial states of x"t\x9# and y"t\x9# can be as small as required# is widely discussed\ so that the linearization theory can be made use of[ The following discussions are also restricted to this case[ Similarly\ for discrete dynamical systems\ x"m¦0#F"x"m##\

"4#

y"m¦0#G"x"m#\y"m##\

"5#

synchronization and generalized synchronization can be de_ned as for continuous systems "taking m in place of t#[ According to convention in the literature\ "0# and "4# are called drive systems\ "1# and "5# response systems\ and the corresponding synchronization is called driveÐresponse "or masterÐ slave# type synchronization[

1[2[ Ome`a limit set ð05Ł Consider a dynamical system x¾ f"x# "or x"m¦0#f"x"m###\ x$R n\ the Omega limit set v"x¹ # of a point x¹ $R n is the collection of all accumulation points for the set "x"t\x¹ #=t$"t9 \##"or"x"m\x¹ #=n$N##\ where x"t\x¹ #"x"m\x¹ ## is the orbit with initial value x¹ [

2[ MAIN RESULTS

To develop the main theorems\ the following lemmas are needed[ Lemma 0[ Suppose that the following system

A note on mathematical aspects of driveÐresponse type synchronization

0348

x¾ A"t#x\

"6#

n

where A"t# is a matrix and x$R \ is asymptotically stable\ and b"t#:9 as t:[ Then any solution to the system "7#

x¾ A"t#x¦b"t# approaches 9 as t:\ where b"t#"b0 "t#\===\bn "t##[ Proof[ An exercise in virtue of variation of constants formula[ Lemma 1[ Suppose that the following discrete system x"m¦0#A"m#x"m#\

"8#

where A"m# is a matrix and xeR n\ satis_es that the norm >A"m#>9¾r³0\ m0\1\===\ rconst[\ and b"m#:9 as m:[ Then any solution x"m# to x"m¦0#A"m#x"m#¦b"m#

"09#

>A>9 limsupðAx\AxŁ0:1 :ðx\xŁ0:1 limsup>Ax>:>x>

"00#

approaches 9 as m:[ Where

and b"m#"b0 "m#\===\ bn "m##[ Proof[ Clearly\ the solution to "8# with initial value x9 can be expressed as m

m−0m−k−0

x"m¦0#t A"i#x9 ¦ s i9

k9

t A"i#b"k#[

"01#

i9

Hence m

m−0 m−k−0

>x"m¦0#>¾t >A"i#x9 >¦ s i9

k9

t >A"i#>>b"k#> i9

m−0

¾rm¦0 >x9 >¦ s rm−k b"k# k9 m

rm¦0 >x9 >¦ s exp""m−k#lnr#b"k#[ k9

From the last expression it follows that x"m¦0#:9 as m:[

"02#

Theorem 0[ Let x"t\x9# and y"t\y9# be the solutions to "0# and "1# respectively[ Then y"t\x9# can synchronize with x"t\x9# if and only if f"x#`"x\x#

"03#

holds for x$v"x9#\ and the variational equation of "1# 1` "x\x#Dy 1y

Dy¾ 

"04#

is asymptotically stable for xx"t\x9#[ Proof[ For {only if| part\ suppose that f"z#`"z\z# for a point z$v"x9# and meanwhile y"t\y9# can synchronize with x"t\x9#[ Then y"t\y9# can be expressed as y"t\y9#x"t\x9#¦Dy"t#[ Therefore

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XIAO!SONG YANG et al[

1` Dy¾  "x"t\x9 #\x"t\x9 ##Dy"t#¦`"x"t\x9 #\x"t\x9 ##−f"x"t\x9 ##¦o"Dy#[ 1y

"05#

By the assumption at the begining\ Dy"t#:9 as t:[ Meanwhile\ f"z#`"z\z# implies that there exists a constant c×9\ a sequence ti:\ and N×9\ such that =f"x"ti \x9 ##−`"x"ti \x9 #\x"ti \x9 ##=−c\[i×N

"06#

for i×N[ Since v"x9# is compact set\ then =1`:1y= is bounded for ti and there exists an M×9 such that

b

b

1` "x"ti \x9 #\x"ti \x9 ##Dy ³c:3\=o"Dy#=³c:3\[i×M[ 1y

"07#

=Dy¾ "ti #=−c:1\[i×max"M\N#[

"08#

Consequently

On the other hand\ Dy¾ "t# is continuous\ then Dy"t#:9 leads to Dy¾ "t#:9 as t:\ a contradiction[ Hence f"z#`"z\z#\ and the de_nition of synchronization implies that "04# is asymptotical stable[ For {if| part\ by the conditions given in this theorem\ the following holds lim "`"x"t\x9 #\x"t\x9 ##−f"x"t\x9 ##9\

t:

"19#

which together with the asymptotical stability of nonautonomous system Dy¾ 

1` "x"t\x9 #\x"t\x9 ##Dy 1y

"10#

and Lemma 0 show that the solution to "05# approaches 9 as t goes to in_nity[ The proof is completed[ Theorem 1[ Let x"m\x9# and y"m\y9# be the solutions to "2# and "3# respectively[ If y"m\y9# can synchronize with x"m\x9# then F"x#G"x\x#\x$v"x9 #

"11#

On the other hand\ if FG on omega limit set v"x9# and >1G:1y>9 ¾c on v"x9#\ then y"m\y9# can synchronize with x"m\x9#[ Proof[ This theorem can be proved in the same way as the proof of Theorem 0[ Lemma 1 is used in proving Theorem 1[ For generalized synchronization\ two theorems are given as follows[ Theorem 2[ If there exists functional relation H]R n:R n\ such that lim =y"t\y9 #−H"x"t\x9 ##=9

"12#

`"H"x#\x#DH"x#f"x#

"13#

t:

Then

holds for x$v"x9#[ Where x"t\x9# and y"t\y9# are solutions to "0# and "1# respectively[ DH is the Jacobian of H[ On the other hand\ if "13# holds on v"x9# and Dy¾ Dy `"H"x"t\x9 ##\x"t\x9 ##Dy is asymptotically stable\ then for y9 near H "x9#\ we have

"14#

A note on mathematical aspects of driveÐresponse type synchronization

0350

lim y"t\y9 #H"x"t\x9 ##[

"15#

t:

Proof[ Let y"t\y9#H"x"t\x9##¦Dy\ then ¾ "x"t\x9 ## Dy¾ y¾ "t\y9 #−H `"y"t\y9 #\x"t\x9 ##−DH"x"t\x9 ##f"x"t\x9 ## `"H"x"t\x9 ##¦Dy\x"t\x9 ##−DH"x"t\x9 ##f"x"t\x9 ## Dy `"H"x"t\x9 ##\x"t\x9 ##Dy¦`"H"x"t\x9 ##\x"t\x9 ## −DH"x"t\x9 ##f"x"t\x9 ##f"x"t\x9 ##¦o">Dy>#

"16#

Now omit higher order terms\ we have Dy¾ Dy `"H"x"t\x9 ##\x"t\x9 ##Dy¦`"H"x"t\x9 ##\x"t\x9 #−DH"x"t\x9 ##f"x"t\x9 ##

"17#

Keep in mind the compactness of v"x9# and Lemma 0\ we can complete the rest of the proof in the similar way as in proof of Theorem 0[ From the arguments above\ it is easy to prove the same assertion for discrete system[ Theorem 3[ Let x"m\x9# and y"m\y9# be the solutions to "4# and "5# respectively\ if there exist a functional relation H] R n:R n\ such that lim =y"m\y9 #−Hðx"m\x9 #Ł=9\

"18#

G"x\H"x##DH"x#f"x#

"29#

m:

then

holds for x$v"x9#[ On the other hand\ if "00# holds for x$v"x9# and >Dy G"x"m\x9 #\H"x"m\x9 ##>9 ³c³0\[m$N

"20#

then for y9 in the vicinity of H"x9#\ the following holds lim >y"m\y9 #−H"x"m\x9 ##>9[

m:

"21#

3[ FURTHER DISCUSSION

The results of previous section indicate that if synchronization is expected to occur with respect to a trajectory x"t\x9# of drive system\ then the identity of drive and response systems on omega set v"x9# is to be met\ not necessarily in the whole space or in a domain of phase space[ In the case that one hopes the synchronization to occur for almost every trajectory of drive system\ the identity of drive and response systems should be hold on the so called likely limit set VV"f# " for the de_nition see ð05Ł#[

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