Nha khoa hoc voi ly thuyet dao dong

8/14/2019 Nha khoa hoc voi ly thuyet dao dong

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NH KHOA HC NGUYN VN O VI L THUYT DAONG V CHUYN NG HN N

Tc gi: Cc cng s ca VS. Nguyn Vn o trong nhm nghin cu H ng lcPhi tuyn thuc Vin Khoa hc v Cng ngh Vit Nam:

GS.TS Nguyn Vn nh, PGS.TS Trn Kim Chi, PGS.TS Nguyn Dng

M u. Trong bi ny cc tc gi ch dm s lc im quanhng mc chnh ca con ng nghin cu khoa hc ca Vin sNguyn vn o. N th hin Anh lng say m, kht khao nghincu khoa hc; mt nh khoa hc y tm huyt, y nng lc vcn mn. Anh lun ng vai tr tin phong, m ng cho nhnghng nghin cu ln. Anh c nhiu cng hin cho s pht trinkhoa hc, cho s nghip o to v t chc nghin cu Khoa hc.

Anh ra i t ngt vo lc tr tu rt minh mn, nng lc thtsung mn; vo lc Anh c nhiu iu kin thun li nht sng

to, thc hin nhng c m ln ca mnh v s nghip khoahc v gio dc Anh ra i dang d bit bao tng, bao dnh to bo, bao hoi bo mong mun cng hin cho iAnh rai li cho chng ti nim tic thng v hn chng g c th bp ni.

A. M ng cho nhng hng nghin cu trong ngnh C hc

1. n vi hng nghin cu L thuyt dao ng phi tuyn

Con ng n vi hng nghin cu L thuyt Dao ng Phi tuyn - Lunn Ph tin s

Nm 1960, mt on cn b cao cp ca Vin Hn lm Khoa hc Lin X doPh ch tch - Vin s Cachennhicp lm trng on sang thm Vit Nam. Trongon c mt nh c hc ni ting - Vin s Knnhinc, chuyn gia hng u vL thuyt dao ng phi tuyn. Vin s Knnhinc c bui bo co rt c sccc kt qu nghin cu ca ng vi s c mt ca GS T Quang Bu (lc GS lPh ch nhim kim Tng th k U ban Khoa hc Nh nc Vit Nam).

L mt cn b tr (ging dy mn C hc l thuyt) ang chp chng, mym trn con ng nghin cu khoa hc, Anh hon ton b thu ht bi bithuyt trnh ca Vin s Knnhinc. Anh mnh dn trnh by vi GS TQuang Bu nh pht trin hng nghin cu L thuyt dao ng phi tuyn Vit Nam. GS T Quang Bu khuyn anh i theo hng nghin cu ca Vin sKnnhinc.

Ngy Anh o ln ng i nghin cu sinh, 3/12/1962, GS T Quang Bu vit mt l th tay bng ting Anh gii thiu Anh vi Vin s Knnhinc. Anh


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c tip nhn ngay vo Khoa Ton-C ca Trng i hc Tng hpLmnxp Mtxcva v lm vic di s hng dn trc tip ca Vin s.

Ma h nm 1965, Anh bo v Lun n Ph tin s vi ch Dao ng vn nh ca cc h ng lc vi cc b gim chn. Lun n cp n nhng bgim chn ng lc, trong vn cn c xem xt di quan im nng

lng.

Lun n Tin s

Sau khi v nc, Anh bt tay lun vo mt Chng trnh nghin cu quy mdi u : Kch ng thng s dao ng phi tuyn ca cc h ng lc. Victhc hin chng trnh ny v cng kh khn do phi s tn vo rng ni trongthi k chin tranh c lit nht. Cui nm 1976, khi c c i lm thc tp sinhcao cp nc ngoi, Anh mang theo mt tp hp cc cng trnh nghin cu honchnh dy trn 500 trang. Cng trnh ny tr thnh Lun n Tin s khoa hcm Anh bo v rt thnh cng vo thng 12/1976 ti Trng i hc VacsavaBa Lan sau hn ba thng hon tt cc th tc. Nh bo Hm Chu gi y l

bn Lun n Tin s Khoa hc gia rng su.

Trang ba bn tm tt Lun n Ph tin s vi nhan :

7/V 1965


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Mt phn lun n tin s ca Anh l pht trin mt hng nghin cu ca Vins Knnhinc v dao ng quan lin: Dao ng khng cng hng theo mtphng gy nn dao ng cng hng theo phng khc.

VS Knnhinc xt m hnh a trn dao ng cng bc khng cng hngquanh trc ca n, qua cc yu t qun tnh phi tuyn, gy dao ng lt quanh

ng knh. Anh o xt m hnh vt bi h n hi phi tuyn chu dao ngcng bc khng cng hng theo phng thng ng, gy dao ng cng hngtheo phng nm ngang.

Trn c s nhn xt c ch ca hin tng kch ng tham s m phng nykch ng phng kia nn Lun n c chn tn l: Kch ng thng s dao ngphi tuyn ca cc h ng lc.

Mt s kt qu ca Anh c Gio s Ali H. Nayfeh (M) dng lm ti liutham kho trong cng trnh ca ng.

Cng trnh Khoa hc u tin, cc thnh tu khoa hc

Cng trnh khoa hc u tin Anh cng b (trn Tp san Ton-L-Ho,UBKHNN, H Ni, N0 1, 1961) vi nhan p dng nguyn l cc i caPntriaghin vo mt vi bi ton C hc.

Tip theo l trn 100 bi bo v cc kt qu nghin cu xoay quanh cc vn ca Dao ng phi tuyn ca cc h ng lc v 13 cun sch chuyn kho, trong c nhiu cun Anh l ng tc gi vi Vin s Mitropolski. Mt phn ni dungca cc cc kt qu ny c th hin trong cc cun sch chuyn kho:1. Nguyen Van Dao. Nonlinear oscillations of high order systems, NCSR

Vietnam, Hanoi, 1979, 64p.2. Mitropolskii Yu. A., Nguyen Van Dao. Applied asymptotic methods in

nonlinear oscillations, Hanoi, 1994, 412p.3. Mitropolskii Yu. A., Nguyen Van Dao. Applied asymptotic methods in

nonlinear oscillations, Kluwer Academic Publishers, 1997, 342p.

Lun n Tin s, 12/1976


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4. Nguyen Van Dao, Nguyen Van Dinh. Interaction between nonlinearoscillating systems. Vietnam National University Publishing House, Hanoi,1999, 356p

Sau hn bn chc nm nghin cu v xy dng i ng, hng nghin cu

L thuyt Dao ng phi tuyn Vit nam pht trin, t c nhiu thnhtu v c cc nh khoa hc Quc t nhn nhn. Vin s Mitrplski nh girng: hnh thnh mt Trng phi H Ni trong hng nghin cu ny.

2. n vi hng Nghin cu Chuyn ng Hn n (Chaotic Motions)

Khai ph hng nghin cu Chaos trong cc H ng lc phi tuyn

Cho ti trc nm 2000, cc cng trnh ca Anh ch yu tp trung vo nghin

cu cc vn ca dao ng ca cc h ng lc phi tuyn theo hng truynthng.Bt u t nm 1999, Anh m ng i vo mt hng nghin cu rt mi

ca cc h phi tuyn: Chuyn ng hn n - Chuyn ng Chaos (ChaoticMotions).

Hn n - Chaos khng ch l hng nghin cu mi i vi chng ta ml hng nghin cu mi i vi th gii. N ch mi c pht hin v bt u

VS Mitropolski v GS Nayfeh (M) l khch mi ca Hingh C hc ton quc ln th VII. H ni. 12/2002


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c nghin cu khong hn bn chc nm gn y (vo u nhng nm 60 cath k XX) nhng n c nhng nh hng v ng gp rt ln n vic nghincu hu nh trong mi lnh vc: khoa hc, k thut, kinh t, k c khoa hc xhi. N thu ht s quan tm ca nhiu nh khoa hc v c bit c nghin cunhiu M, Nht Bn, hai ci ni sn sinh ra ngnh khoa hc ny.

S khai ph mi bao gi cng tht kh khn, rt t ngi cng tc (v trong lnhvc ng lc hc, Vit nam n thi im , cha c nh khoa hc no i sutm hiu hin tng Chaos), hn th na, bn thn Chaos l vn rt kh.

Hai nm u, Anh cng c nhm nghin cu tp trung vo c sch tm hiuvn . C ln Anh mi Gio s Nht Bn sang thuyt trnh. Mt vic cng rtkh khn l vic tnh li cc v d trong sch, bi v ng chm n cc nghimChaos th chng trnh tnh ton rt phc tp, mt nhiu thi gian (c nhiuchng trnh nh xy dng cc s phn nhnh, phi chy my tnh n hngchc ngy).

Sau thi gian tm hiu, Anh cng nhm nghin cu mnh dn ng k

hng nghin cu ny trong Chng trnh nghin cu c bn ca Nh nc vi ti H ng lc phi tuyn v Chaos. ti thc hin t nm 2001 n 2005v t c nhng kt qu ban u.

Anh c 8 cng trnh nghin cu v chuyn ng Chaos ca mt s h nglc. xut bn mt quyn sch chuyn kho vi nhan Nhp mn ng lchc phi tuyn v chuyn ng hn n (Nh xut bn HQG,2005).

Gii thiu s lc v khi nim chuyn ng hn n - Chuyn ng Chaos

Lu nay, nh lut II ca Newton mAF = c xem nh m u v cng l

kt thc cho vic nghin cu ng lc hc! Da vo , ngi ta c c s tinrng nu bit trc v tr, vn tc ban u v cc lc tng tc, th vi mt mytnh mnh, ngi ta c th d on c chuyn ng ca mt h trong tnglai lu di. Nhng tic rng, s xut hin ca cc my tnh nhanh v mnh nykhng phi lc no cng p ng c nim k vng vo kh nng d on trcmt cch v tn trong ng lc hc. Rt gn y, ngi ta pht hin ra rng,chuyn ng ca nhng h ng lc rt n gin khng phi lun lun c th d

bo c di trong tng lai. Nhng chuyn ng khng d bo c di hn nyc gn cho tn l chuyn ng Chaos (chuyn ng hn n) v vic nghincu chng gy ra cuc tranh ci v mt s tng ton hc mi si ng trongng lc hc. Nu ba trm nm trc, Newton (1687) a php tnh vi tch

phn, cng c ton hc hin i nht thi , vo cc nghin cu ng lc hc, thcng rt ph hp l ba trm nm sau, khi cc hin tng mi c pht hin trongcc h ng lc, th cc nh khoa hc cng phi x dng cc l thuyt ton hchin i nht v topo, hnh hc nghin cu nhng vn ny.

Tr li vi cc nghim ca h ng lc. Cng vi cc nh lut Newton, mtthi gian di, chng ta ch ngh rng nu nghim ca h ng lc cn b giam hm


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trong mt min gii ni trong khng gian trng thi th chng ch c th l mttrong hai trng thi sau:

1- Trng thi dng, hay cn bng c hnh thnh do s hao tn nnglng bi ma st, trng thi ny c Newton m t bi hnh nh qu to nmtrn mt t .

2- Trng thi dao ng, c th tun hon hoc tun hon, c Newtonm t nh l chuyn ng u n ca Tri t, Mt trng v cc hnh tinh khcxung quanh Mt tri.

Vi chc nm gn y, cc nh khoa hc mi pht hin ra cn c mt trngthi th ba ca chuyn ng, l chuyn ng Chaos. Nhng chuyn ng nycng b gii ni, b giam hm nh nhng chuyn ng tun hon v tun hon,nhng khng h lp li v khng th c c mt d on di hn.

Hin nay cha c mt nh ngha chnh xc cho mt nghim Chaos, bi v nkhng th biu din c qua cc hm ton hc chun. Mc d vy, ngi tathng c nhn nh chung rng: nghim Chaos l mt nghim khng tun hon

vi mt s c im nhn dng c bit.Nghim Chaos c xc nh nh mt trng thi yn nh gii ni, min htca nghim Chaos trong khng gian trng thi khng phi l mt i tng hnhhc n gin nh mt s hu hn im, mt ng cong kn hay mt xuyn,... mn c cu trc hnh hc phc tp, c gi l tp ht l (strange attractor) hay tpht Chaos (chaotic attractor), c th nguyn phn hnh (fractal dimension). Phca cc tn hiu Chaos c c tnh ca mt di rng lin tc, ngha l c tnh chtnh ph ca mt hin tng ngu nhin, ch khng ch ri rc nh ph ca tp httun hon hay tun hon.

Tnh cht in hnh quan trng nht ca chuyn ng Chaos l n c bit

nhy cm vi s thay i ca iu kin ban u, c ngha l: nhng khc nhaurt nh u vo, b khuch i v to nn s khc nhau rt ln u ra . Scc k nhy cm vi iu kin ban u ca nghim Chaos c ngi ta gn chomt ci tn rt p, rt sinh ng l Hiu ng cnh bm(Butterfly effect) vc din t bi mt cu ni y n tng v vn hoa: Mt ci vy cnh ca mtcon bm bm Bc Kinh, Trung Quc hm nay, c th gy bo t cho bangCalifornia (M) vo thng sau.

Nh vy, vic tm nghim Chaos dn n mt lot vn : tp ht l, ph phnng, s ph thuc nhy cm vo iu kin ban u m tiu chun in hnh l cs m Liapunov dng , xy dng hm mt .... Mt trong nhng vn c ngha nht, c bit trong thc tin, l xc nh c min cc tham s tng ngvi nghim Chaos.

Sau mt thi gian di nghin cu, Gio s Yoshisuke Ueda (ngi u tin phthin ra hin tng Chaos trong cc dao ng ca dng in vo nm 1961) nhn xt:Ngi ta gi Chaos l mt hin tng mi, nhng n lun lun tn tixung quanh ta. Tht ra chng c g mi v n, ch c iu ngi ta khng ch ti n. (People call Chaos a new phenomenon, but it has always been around.Theres nothing new about it , only people did not notice it).


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Mt iu ng ni l vic pht hin ra nhng nghim Chaos trong cc h nglc thot tin khng phi l cc nh ton hc, m l cc chuyn gia v k thut khih nghin cu cc bi ton thc t v kh tng thu vn, v dao ng ca dngin v sau ngi ta mi dng cc cng c ton hc nghin cu bn chtca hin tng ny mt cch su sc.

3. Bc u n vi C hc Nano

Vic ch to ra cc vt liu Nano v pht minh ra cng ngh Nano c xeml mt trong nhng thnh tu khoa hc k thut c sc v kch thc cui thk XX. Trong th k XXI ny, nhng nghin cu v cc vt liu v cng ngh

Nano c th to ra mt cuc cch mng mi trong sn xut.

Anh o tm hiu, nm bt vn khoa hc v cng ngh c vai tr cchmng ny v nhn nhn n trn quan im C hc.

Trong hi ngh C hc ton quc k nim 25 nm thnh lp Vin C hc(10/4/2004) ti H Ni, Anh c bo co khoa hc vi nhan C hc Nano.Trong bo co ny Anh trnh by nhng thng tin tng qut v hng nghincu C hc Nano, nhng thnh tu t c trn th gii v xut phnghng nghin cu C hc Nano nc ta.

Trong C hc Nano, vt liu c xem xt kch thc c mt phn t camt ( 910 m ), gn vi kch thc ca phn t. kch thc ny ca vt liu, s

Bo co trong hi tho KH Mng Sinh nht ln th 80 VS Frolov K. V.:The Study of Chaotic Phenomena in a strong nonlinear Mathieu Oscillator

Matxcva 7/2002


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xut hin hiu ng lng t , v th cc nh lut Newton, cc quy lut kt cu, mast, tng tc trong c hc kinh in khng cn chnh xc na m phi kthm cc hiu ng in t, lc phn t,..

Nhn trc vai tr quan trng ca C hc Nano trong tng lai ca khoa hcc bn v cng ngh, Anh c mt s kin ngh :

1. T chc mt nhm nghin cu thm d, tm kim thm thng tin v lnhvc ny: ni dung khoa hc, cc t chc nghin cu ti cc nc Ngnhc hc thuc Hi ng Khoa Hc T Nhin ti tr cho nhm nghin cu.Cc ti la chn cn c nh hng v vic nghin cu c hc phcv cho pht trin cng ngh Nano Vit Nam, cho vic pht trin cngngh sn xut cc sn phm cu trc Nano. c bit lu tm n lnh vcC hc vt liu.

2. Tm hiu chng trnh o to cn b C hc Nano cc nc. Chuynmt s cn b C hc tr v gii cc trng i hc v Vin nghin cu

sang nghin cu v C hc Nano3. T chc mt s seminar v C hc Nano

4. Tham gia cc hat ng hp tc quc t trong lnh vc C hc Nano, ccn b i trao i khoa hc cc nc v vn ny.

Trong Hi ngh, Anh cng ng vin khuyn khch cc cn b khoa hc trmnh dn i vo hng nghin cu mi ny. Mong rng chng ta c th thc hinc cc nguyn tm huyt ca Anh.

4. Kt lun

Cch y t hm, ngy 21/7/2007, Nguyn Th tng V Vn Kit n thmgia nh VS Nguyn Vn o. Trong li cm tng ng vit: ...Ti tht xcng bi hi v thng tic Anh v cng - Mt nh tr thc ln y tm huyt,nhiu tng, lun sng to v vn ti ci mi .

Trong khoa hc, Anh th hin rt r l con ngi nh vy. T nhng ngy u la tui 20, chp chng i vo nghin cu khoa hc v cho n nhng nmthng cui Anh lun mit mi lm vic, khai ph m ng v n dt c mt tpth nghin cu theo hng ca Anh. Anh cng lun gip , bi dng, toiu kin cho cc cn b khoa hc tr vn ln. Mt hot ng khc trong lnh ny

l Anh rt chu kh vit sch chuyn kho. Anh Bt u vit sch t nm 1969, tinay Anh vit 13 cun sch chuyn kho.

anh mt nt ni bt l kh nng t chc, on kt, hi t cc nh khoa hc pht huy sc mnh ca h phc v t nc.

B. Cng trnh khoa hc cui cng


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Mt tip cn s ca cc Chuyn ng Hn n trong Chn t Duffing Van Der Pol

AN NUMERICAL APPROACH OF CHAOTIC MOTIONS IN

A DUFFING-VAN DER POL OSCILLATOR

Nguyen Van Dao, Nguyen Van Dinh,Vietnam National University, Hanoi

Tran Kim Chi, Nguyen DungVietnamese Academy of Science and Technology

Abstract. In the present paper the influence of the excitation

frequency ( ) and the forcing amplitude (e) on the chaoticbehaviour of the system governed by equation

2 2 3(1 ) sinx x k x x x e t && & (1)

will be examined. This equation is a Van der Pol one with a forcing term esinvt, where , , , ,e and are constants,overdot denotes the derivative with respect to time t .When e=0 ,

0, 0 we have the classical Van der Pol equation whichrepresents a selfexcited oscillator with the amplitude * 2 /a

and frequency . Our discussion was focused upon variation ofthe excitation frequency and the forcing amplitude e . Thebifurcation diagrams for acquiring the overview of equation (1)and the Liapunov exponent method will be used [3,4,5,6,9]. For a concrete case, the parameter regions in which eitherperiodic or chaotic motions exist were shown. In two precedingcases, the first case, when is control parameter, it changessuddenly from periodic motion to chaotic motion, correspondingto Hopf bifurcation. In the second case, it is the double periodprocess and leads to chaotic motion.Chaotic attractors illustrate

the complexity of the motion of the system under consideration.1. Summary of the case of small parameters

First, we recall briefly some known results of deterministic motions in (1) forthe case of smallness of the coefficients. It is supposed that is close to thenatural frequency , namely :

2 2 , (2)


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here is a detuning parameter and is a small positive one .Applying to (1) theasymptotic method [2] and using the amplitude and phase variables ( ,a ) given

by

cos( ),

sin( ),

cos( ) sin( ) 0,

x a t

x a t

a t a t

&

&&

(3)we have

).sin(]sin)([

),sin(]sin)([

++++=

++++=

ttexxxkxa

ttexxxkxa

32

32

1

1

(4)

Since a and are slowly varying functions of time, the change in their valuesduring a time period T = 2 / is very small. Hence, in the first approximation onemay replace equations (4) by their time averages over (t, t+T )assuming a and to be constant :

).sin(

],cos)([

eaaa

eaaka

+=

=

3

2

4

3

2

4

11

2

(5)

The steady-state equations are

.sin

,cos)(

0300

0200

43

41

eaa

eaka

=+

=

(6)

By eliminating the phase 0 from these equations we obtain,E]A)-[(1A 222 =+ (7)

where

,,][,,

=+

====

202

22

222

2

202

04

31

44a

ke

kE

a

aaA (8)

/2=

a is the amplitude of the purely self-excited van Der Pol oscillator.Below only the behaviour of forced oscillations with the frequency which isclose to will be considered.

The oscillation described by the equation (1) with stationary amplitude 0a andphase 0 : 0 0cos( )x a t has the frequency of the external force only. Theself-excited oscillation is entrained by the external excitation .The synchronizedoscillation (3) is characterized by the entrainment of the auto-periodic frequency


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by the external one.The synchronization effect is observed only when the excitingfrequency is close enough to the natural frequency .

The amplitude curves with various values of external excitation (E) are given inthe Figure 1 for the case 0= [1]. For E=0, i.e. for the zero external excitation,we find the results for the classical Van der Pol oscillator:

1) A=0 with arbitrary ,2) =0 , A=1.

Therefore ,the resonance curves degenerate into the line A=0 ( -axis) and thepoint =0,A=1 .

If E is small but different from zero ,we expect A to be nearly 1 or nearlyzero so that one of the response curves would be oval which is approximately thecircle

2 2 2( 1)A E

with centre at =0 , A=1 . In addition, the other branch runs near the -axis.The oval expands with increasing E. When E increases, the resonance curves firstconsist of two branches, up to the critical value E= 2 / 27 for which the two

branches join at =0, A=1/3, then with further increase of E the resonancecurves have only a single branch .

From the Figure 1 one can see that under certain conditions the frequency ofthe free oscillation is canceled out and is replaced by a synchronized oscillation,i.e. by an oscillation whose frequency is that of the external force ,namely:

1) For a given amplitude of the exciting force (E), the synchronization effect isobserved when the exciting frequency is close enough to the naturalfrequency of the oscillator . The larger the amplitude of the excitingforce ,the greater the frequency interval over which the synchronizationoccurs.

2) For a given exciting frequency ,the oscillator is synchronized when theexciting amplitude is large enough . The closer the exciting frequency is to, the lower its threshold amplitude is.


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2. The case of arbitrary parameters

2.1 Some concepts connected to bifurcation

Bifurcation is a concept used to indicate a qualitative change in the features ofa dynamical system, such as the number and type of solutions, under the variationof one or more parameters on which the considered system depends. These

parameters are called the control parameters, and parameter values at which

bifurcations occur are called bifurcation values. A bifurcation diagram is a graphof the state variables versus the parameters [3,6,9].The bifurcation diagram provides a summary of the essential dynamics and is

therefore an important tool for examining the prechaotic or postchaotic changes ina dynamical system under parameter variations. The Poincare map can be used toconstruct the bifurcation diagrams for continuous differential equations. When thedata are sampled using a Poincare map, it is very easy to observe period doublingand Hopf bifurcations. It is useful because one characteristic precursor to chaoticmotion is the appearance of subharmonic periodic vibrations.

Well examine two following concrete cases: a) The frequency is the controlparameter b) The forcing amplitude e is the control parameter.

2.2 The frequency is the control parameter.

We go back to the system (1) with 7.02 , 1=k , =0.6, = -1, e = 5 anduse the frequency as a control parameter. Poincare sections for orbits of thissystem are constructed by using the excitation frequency . For each orbit of the

E2=1

A

Fig.1 Amplitude curves for the Duffing Van der Pol oscillator ()with various values of external excitation


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system the discrete points ( )(),( nTxnTx ) are collected at time intervals of /2=T (the period of the external excitation force). The bifurcation diagram

shown in Figure 2 was generated by incrementing the control parameter in stepsof = 0.0001. The graph consists of the points )),(( nTx , where the values

)(nTx correspond to the attractor realized at each value of .From Figure 2 it is clear that as increased through 780895.0h , there is an

abrupt transition from the point attractor to an aperiodic one, so a Hopf bifurcationof a periodic solution (the Poincare section consists of only one point) occurs. For

h < , the state of the system is periodic, when the control parameter exceeds the

threshold value h , the system evolution is attracted to chaotic attractor, then thesystem undergoes a subcritical Hopf bifurcation. The attractors, both before andafter the bifurcation, are shown in Figure 3(a, b). Figure 3(a) describes the periodicattractor with its Poincar section consisting of one point (*) connected to =0.78.With = 0.782 a chaotic motion occurs, Figure3(b) describes its attractor. Well

consider this case more detail below.With the values of the frequencies h < , the Poincar sections consist of onepoint, the motions are periodic with the period equating the one of the externalforce. Beyond the periodic region occupying much of the interval h


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every parameter , x takes a finite number of values (more than one), so that thecorresponding motions are subharmonic oscillations.

A concrete case of Chaotic motion. We consider a concrete case for theparameters 2 0.7, k 1, 0.6, 1, =e 5 and h >= 782.0 . The aperiodicappearance of x (see Fig.4) suggests that the system under consideration ischaotic.

To verify that motion realized at 782.0= is chaotic, we need to show thesensitivity to initial conditions on its attractor. We choose two points separated by

70 10

=d close to the attractor and examine initiated evalutions from them. Figure

5 illustrates the variation of the separation dwith time t. The exponential growth

of separation dfor 12020 120. Therefore, there is a positive Liapunov exponentassociated with the chaotic orbit at 782.0= and its approximate value is 0.0495. Much more insight can be gained from a Poincar section (Figure6)consisting of stroboscopic points at instants /2( nt= 0.782), =n 0,1,2 ... of theorbit of the system (9) in the space ),( xx . Figure 6 shows the next 10000 points

-8

0

8

-3.5 0 3.5

x

x

-8

0

8

-3.5 0 3.5

x

x

Figure 3. (a) Periodic attractor (),(b) Chaotic attracror ()

(a) (b)

-3

0

3

40170 40270 40370 40470

x

x

Figure 4. Aperiodic appearance of x(t) ().


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after the transition decays about the first 1000 periods. The corresponding attractorof the chaotic solution is presented in Figure 3(b).

2.3 The forcing amplitude e is the control parameterWe examine a graph of x versus the forcing amplitude e at 702 .= , k =1 ,

=0.6 , = -1, = 0.837 in order to detect bifurcations. The bifurcation diagramis shown in Figure 7. In this numerically constructed bifurcation diagram, thediscrete points on the Poincare section of the attractor realized at each value e aredisplayed.

Obviously, from Figure 7, we can observe the sequence of period-doublingbifurcations. First, with one of values e in the interval (4.7 , 4.7645125), thePoincare section consists of five points (five dark points in Fig.8), so there existthe subharmonic motions with the period equaling five times of the period of the

-2.5

0

2.5

-2 -0.5 1

x

x

Figure 6. Poincaresection

1.00E-07

1.00E-05

1.00E-03

1.00E-01

0 50 100 150 20010-

7

10-5

10-3

10-1

t

d

Figure 5. Sensitivity to initialconditions

4.7208 4.7417 4.7625 4.7833

e

4.8042 4.825

x

-1.82

0

1.54

Figure 7. . Illustration of bifurcations on the Poincaresection


16/20

external force (Figure 8). At the value e 4.7645125, the first period-doublingbifurcation occurs. After the bifurcation, with the values e which is in the rightvicinity of the value e 4.7645125, the subharmonic motions with the periodequaling twice the period of the previous motions appear, the Poincare sectionsconsist of ten points (Fig. 9), and so on. The chaotic attractor realized at e =4.8042 appearing after a sequence of period-doubling bifurcations is shown inFigure 10, and Figure 11 is corresponding attractor. The largest Liapunovexponent evaluated is positive ( )0553.0 defines sensitivity to initial conditionson the chaotic attractor.

-2

0

2

-1.9 -0.95 0 0.95

x

x

Figure 9. Ten points in the Poincare section for e =

-3

0

3

-2 0 2

x

x

Figure 11. Poincare sectionfor e=4.8042

-7.5

0

7.5

-3 0 3

x

x

Figure 10. Chaotic attractorfor e=4.8042


17/20

The sequence of period-doubling bifurcations is one route to chaos and it is

commonin many dynamical systems. It is particularly interesting because it may becharacterized by a certain universal constant (called the Feigenbaums constant)that do not depend on nature of the concrete systems. This constant is consideredas a specify criterion to determine if a system becomes chaotic by observing the

prechaotic periodic behavior.If the first bifurcation occurs at parameter value e1, the second at e2, and so on,

then this constant is defined as

....6692016.4lim1

1==

+

kk

kk

k ee

ee

Table 12 shows a list of the parameters at which period-doublings occur in thePoincare map corresponding to the system (1) for 702 .= , k =1 , =0.6 , =-1, = 0.837 and e is used as the control parameter.

Period Parameter e Ratio510204080

160.

4.76451254.7717130

4.773274994.773611

4.77368297

..

4.6001444.6566774.668751

..

Table 12. Feigenbaums constant in the Poincare mapLiapunov exponent. To determine the chaotic motion in the system described by(1) it is necessary to calculate the largest Lyapunov exponent. Ifd0is a measure of

-7

0

7

-3 0 3

x

x

Figure 8. Periodic orbit and itsPoincare section (five points) for


18/20

the initial distance between the two starting points, at a small but later time thedistance is

0 2td t d

The divergence of chaotic orbits can only be locally exponential, since if the

system is bounded, as most physical experiments are, d(t) cannot go to infinity.Thus, to define a measure of this divergence of orbits, we must average theexponential growth at many points along a trajectory. One begins with a referencetrajectory and a point on a nearby trajectory and measures 0/d t d . When d(t)

becomes too large (i.e, the growth departs from exponential behavior), one looksfor a new nearby trajectory and defines a new 0d t . One can define the firstLyapunov exponent by the expression

2

10 0 1

1log

Nk

kN k

d t

t t d t

the motion is chaotic if the corresponding largest Lyapunov exponent is positive.For this calculation ([7]), in the case of concrete case of Chaotic motion in (2.2),we represent the equation (1) in the form :

1 2

2 3

2 1 1 2 10.7 (1 0.6 ) 5sin 0.782

1.

x x

x x x x x z

z

(9)

Let ),,( 21 zxxu = is a three dimension vector and ),( 0** utuu = is a reference

trajectory of the system (3), where 0u is the initial condition. The variationalequation corresponding to this reference trajectory is

A= ,

where *uu = and the matrix A depends on *u

** * *2 *21 2 1 1

00 1

3.91cos 0.7820.7 1.2 3 1 0.6

0 0 0

A z x x x x

. (10)

If this initial condition is chosen at random, then it is likely to have a componentthat lies in the direction of the largest positive eigenvalue ofA. It is the change inlength in this direction that the largest Lyapunov exponent measures.

After a given time interval 1k kt t , take

1

;

0;kk

k k

td t

d t t


19/20

The time evolution of the Lyapunov exponent is presented in Fig.14. The largestLyapunov exponent value is a positive number 0.0698>0, which shows thechaotic character of the motion of the system (10). This means that two trajectoriesstarting closely one to another in the phase space will move exponentially awayfrom each other for small times on the average

( ) tdtd 20= ,

where 0d is the initial distance between two adjacent starting points at 0tt= and d is the distance between two these points at the moment t . In Fig. 15, weshow how the distance d between evolutions initiated from two points separated

by =0d 10-7 varies with time. Both of the initial points are located close to theattractor. The separation clearly grows exponentially in the range 10 t 125.

3. Conclusion

The bifurcation diagrams and Liapunov exponent have been used for detectingchaotic regimes in the system described by equation (1). Our discussion wasfocused upon variation of the excitation frequency and the forcing amplitude e .For a concrete case, the parameter regions in which either periodic or chaotic

0

0.05

0.1

0.15

0.2

0 500 1000 1500 2000 2500

cycle

Fig. 14. Time evolution of the largest Liapunovexponent

1.00E-07

1.00E-05

1.00E-03

1.00E-01

1.00E+01

0 100 200 300

t

d

Fig. 15. Sensitivity to initial conditions on theattractor


20/20

motions exist were shown. Chaotic attractors (Fig. 5, 10) illustrate the complexityof the motion of the system under consideration. The bifurcation diagram showsso clearly the motions of system (1) with respect to the observed parameters. Intwo preceding cases, the first case, when is control parameter, it changessuddenly from periodic motion to chaotic motion, corresponding to Hopf

bifurcation. In the second case, it is the double period process and leads tochaotic motion. Lyapunov exponent value and some other criterions have used todetermine chaotic motion of a solution.

*

* *

Cng trnh khoa hc ny ca Anh l bo co mi, s bo co vo ngy th hai,11/12/2006 ti Hi ngh Quc t Gii tch phi tuyn v C hc ngy nay tngy 11 n 14/12/2006 ti thnh ph H Ch Minh. Nhng Anh ra i mi mivo chnh bui sng hm . S nghip khoa hc thc s i theo Anh n hi

th cui cng. Khi chun b bo co ny chng ai ng rng y l bo co cuicng ca Anh.

H Ni, 29/7/2007

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