Quasi-periodic Flows in Time-Delay Systems
In this chapter, from Luo (2014 ), period-m flows to quasi-periodic flows in time-delayed nonlinear dynamical systems will be presented. The analytical solutions of quasi-periodic flows in autonomous time-delayed systems will be discussed, and the analyti
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Quasi-periodic Flows in Time-Delay Systems
In this chapter, from Luo (2014), period-m flows to quasi-periodic flows in time-delayed nonlinear dynamical systems will be presented. The analytical solutions of quasi-periodic flows in autonomous time-delayed systems will be discussed, and the analytical solutions of quasi-periodic flows in periodically forced, time-delayed nonlinear dynamical systems will be presented. The analytical solutions of quasi-periodic solutions in free and periodically forced, time-delayed vibration systems will be presented.
4.1
Time-Delay Nonlinear Systems
Consider quasi-periodic flows in autonomous, time-delayed nonlinear systems, and the analytical solution of quasi-periodic flows relative to period-m flow is given as follows. Theorem 4.1 Consider a nonlinear, time-delayed, dynamical system as x_ ¼ Fðx; xs ; pÞ 2 Rn
ð4:1Þ
where Fðx; xs ; pÞ is a C r -continuous nonlinear vector function (r 1). (A) If such a time-delayed dynamical system has a period-m flow xðmÞ ðtÞ with finite norm jjxðmÞ jj and period T ¼ 2p=X, there is a generalized coordinate transformation for the period-m flow of Eq. (4.1) in the form of ðmÞ
xðmÞ ðtÞ ¼ a0 ðtÞ þ sðmÞ
xsðmÞ ðtÞ ¼ a0
ðtÞ þ
1 X
k k bk=m ðtÞ cosð hÞ þ ck=m ðtÞ sinð hÞ; m m k¼1 1 X
k k bsk=m ðtÞ cos½ ðh hs Þ þ csk=m ðtÞ sin½ ðh hs Þ m m k¼1
© Springer International Publishing Switzerland 2017 A.C.J. Luo, Periodic Flows to Chaos in Time-delay Systems, Nonlinear Systems and Complexity 16, DOI 10.1007/978-3-319-42664-8_4
ð4:2Þ
115
116
4 Quasi-periodic Flows in Time-Delay Systems sðmÞ
with a0
ðmÞ
sðmÞ
ðmÞ
¼ a0 ðt sÞ; bk ð0Þ
ðmÞ
sðmÞ
¼ bk ðt sÞ; ck ðmÞ
ðmÞ
ðmÞ
¼ ck ðt sÞ; hs ¼ Xs and
ðmÞ
¼ ða01 ; a02 ; ; a0n ÞT ;
a1 a0 ðkÞ
a2 bk=m ¼ ðbk=m1 ; bk=m2 ; ; bk=mn ÞT ; ðkÞ
a3 ck=m ¼ ðck=m1 ; ck=m2 ; ; ck=mn ÞT sð0Þ
a0
sðmÞ
sðkÞ
bsk=m ¼ ðbsk=m1 ; bsk=m2 ; ; bsk=mn ÞT ;
sðkÞ
csk=m ¼ ðcsk=m1 ; csk=m2 ; ; csk=mn ÞT
a1 a2 a3
sðmÞ
sðmÞ
sðmÞ
¼ ða01 ; a02 ; ; a0n ÞT ;
ð4:3Þ
which, under jjxðmÞ ðtÞ xðmÞ ðtÞjj\e and jjxsðmÞ ðtÞ xsðmÞ ðtÞjj\es with prescribed small e [ 0 and es [ 0, can be approximated by a finite term transformation xðmÞ ðtÞ as ðmÞ
xðmÞ ðtÞ ¼ a0 ðtÞ þ sðmÞ
xsðmÞ ðtÞ ¼ a0
ðtÞ þ
N0 X
k k bk=m ðtÞ cosð hÞ þ ck=m ðtÞ sinð hÞ; m m k¼1 N0 X
ð4:4Þ
k k bsk=m ðtÞ cos½ ðh hs Þ þ csk=m ðtÞ sin½ ðh hs Þ m m k¼1
and the generalized coordinates are determined by a_ ¼ f s0 ða; as ; pÞ
ð4:5Þ
where k0 ¼ diagðInn ; 2Inn ; ; NInn Þ; ð0Þ
ðmÞ
ðkÞ
ðkÞ
a1 a0 ; a2 bk=m ; a3 ck=m ; sð0Þ
a1
sðmÞ
a0
sðkÞ
; a2
ðkÞ
bsk=m ; a3 csk=m
ð0Þ
a1 ¼ a1 ; ð1Þ
ð2Þ
ðNÞ
ð1Þ
ð2Þ
ðNÞ
a2 ¼ ða2 ; a2 ; ; a2 ÞT bðmÞ ; a3 ¼ ða3 ; a3 ; ; a3 ÞT cðmÞ ; sð0Þ
as1 ¼ a1 ; sð1Þ
sð2Þ
sðNÞ T
sð1Þ
sð2Þ
sðNÞ T
ðmÞ
ðmÞ
as2 ¼ ða2 ; a2 ; ; a2 as3 ¼ ða3 ; a3 ; ; a3
Þ bsðmÞ ; Þ csðmÞ ;
ðmÞ
F1 ¼ F0
ðmÞ
F2 ¼ ðF11 ; F12 ; ; F1N ÞT ;
4.1 Time-Delay Nonlinear Systems
117 ðmÞ
ðmÞ
ðmÞ
F3 ¼ ðF21 ; F22 ;
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