Neural, Parallel, and Scientific Computations 24 (2016) 65-82

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1 Neual, Paallel, and Scientific Computations 24 (216) 6-82 DYNAMICAL BEHAVIORS ANALYSIS AND NUMERICAL SIMULATION OF THE LORENZ-TYPE SYSTEM FOR COUETTE-TAYLOR FLOW HEYUAN WANG College of Sciences, Liaoning Univesit of Technolog, Jinhou 1211, China ABSTRACT. In this pape, we investigate the poblem of dnamical behavios and numeical simulation of the Loen-tpe sstems fo the incompessible flow between two concentic otating clindes. The estimation of Hausdoff dimension of its attacto is discussed, and the globall eponentiall attactive set and positive invaiant set of the chaotic sstem ae studied via Lapunov function. We pesent a detailed numeical esult of the whole pocess fom bifucation to chaos, and anale the evolutiona mechanism of the dnamical behavio of the sstem. Moeove, b using numeical simulation esults of attactos, bifucation diagam, Lapunov eponent spectum and Poincae map, etun map of the sstem we show abundant and comple dnamical behavios of the sstem, and eplain successive tansitions of Couette-Talo flow fom Lamina flow to tubulence in the epeiment. Ke Wods Couette-Talo flow, the Loen sstem, attacto, Poincae map. 1. Intoduction Thee have been a lot of investigations which concen with otating flow between two concentic clindes (abbeviate fequentl as Couette-Talo Flow), and Couette-Talo flow is the tpical otation flow poblems, it povides a paadigm fom lamina to tubulent tansition, and we efe the eades to [1 12]. The oute clinde is fied while the inne clinde otates at a constant angula velocit ω, when ω is small, the basic flow is the Couette flow consisting of cuent lines which ae coaial cicles. When ω eceeded a citical value ω 1c, this basic flow becomes unstable and a new complicated flow is obseved, and it is aismmetic and stationa, called the Talo votices (the Talo vote flow). In fact, this is a bifucation phenomenon, mathematicall, the instabilit epesents a supecitical stead bifucation fom Couette flow to Talo vote flow. This egime is still stationa, and the otational invaiance of the flow is not boken. The second tansition beaks both the time and otational smmet, the egime is now peiodic and the cells assume a wav Suppoted b the Natual Science Foundation of China ( ), scientific eseach foundation of education depatment of Liaoning povince (L213248) and science and technolog funds of Jinhou cit (13A1D32) Received Decembe 31, $1. c Dnamic Publishes, Inc.

2 66 H. WANG fom. These waves move unifoml about the Z-ais, so that in a suitable otating fame, the flow appeas stationa. Such peiodic motions ae called otating waves. The thid tansition leads to a quasi-peiodic flow, which ae called modulated wav Talo votices. Subsequent bifucations follow seveal possible outes but geneall lead to tubulence afte a few steps. When both clindes ae otated in opposite diections, even iche outes to tubulence ae obseved. Pevious eseach wok mostl focused on the stabilit and bifucation theo of the flow, It is mainl used to eplain and anale the vaious kinds of vote and the evolution pocess of the vaious votices in the epeiment, as well as the wa of the tansition fom lamina flow to tubulent flow, and the eistence and simulation of chaotic attactos of the tubulent state ae ael involved in the liteatue. As the global attacto of the Couette-Talo flow between two concentic clindes is ve comple and difficult of simulation, and the occuence of tubulence is usuall deived fom instabilit of a small numbe of modes, we use the simplified mode analsis method fo numeical simulation. Low-dimensional analsis of fluid sstems ae of inteest in ode to captue the essential of thei behavio, the theoetical basis is theo of the inetial manifold and appoimate inetial manifold (the ae consideed to be a low dimensional smooth manifold containing the global attacto and the eponential attacto of all obits), namel, the complicated dnamical behavio of the infinite dimensional dnamical sstems can aise fom a few simple coupled odina, and be distinguished b the simple equations. This simplified mode analsis method not onl can ovecome the toublesome difficult of the Couette-Talo flow poblem, but also involve some essential featues of the flow, which is ve meaningful to discuss nonlinea phenomena of Navie-Stokes equation, fo eample, the bifucation, tubulence etc. Although the dnamic behavio of the Loen-tpe equation is not eactl the same as the actual flow of the Couette-Talo flow, it can not onl achieve the minimum degee of feedom of simulation but also eflect some essential featues of the flow, this is a valuable attempt to use the simple model to eflect some of the featues of comple poblems. Of couse, the fascinating and confusion vote wave obseved in the epeiment befoe the occuence of tubulence ma be beond the epessed scope of finite model Loen-tpe equations, so we can t epect to obtain all the details of this comple poblem, the focus of ou stud is intepetation of thee tpical flow of the Couette-Talo flow befoe evolution into tubulence and some featues of the tubulence behavio afte flow tansition to chaos and its simulation. Chen and Hsien [1987] investigated a model sstem of evolutiona equations fo aismmetic Couette-Talo flow, but the model of tuncation seemed too simple, thee wee no Hopf bifucation and chaos behavios in thei sstem. Heuan Wang [212] deive a thee-model sstem b using spectal Galekin method, and discove chaos phenomena. The eistence of its attacto was given, some numeical simulation esults

3 DIFFERENTIAL INCLUSIONS 67 wee pesented [Heuan Wang 212]. In this pape, we futhe stud dnamical behavios of the thee-model Loen-tpe sstem, discuss the estimation of Hausdoff dimension of its attacto, and pesent numeical simulation esults of attactos, bifucation diagam, Lapunov eponent spectum, Poincae map and etun map of dnamical behavios. The outline of the pape is as follows. In section 2 we discuss the estimation of Hausdoff dimension of its attacto. In section 3 we stud the globall eponentiall attactive set and positive invaiant set of the thee-mode sstem. In section 4 we pesent numeical simulation esults of attactos, bifucation diagam, Lapunov eponent spectum, Poincae map and etun map of the sstem, and anale the evolutiona mechanism of the dnamical behavio of the sstem. 2. The thee-mode Loen-tpe equation and the estimation of Hausdoff dimension of its attacto Heuan Wang intoduce eigenfunctions of Stokes opeato in the clindical gap egions as basis function of Fouie epansions, and deive the following thee-model Loen-tpe sstem b a suitable model tuncation of the Navie-Stokes equations fo the incompessible flow between two concentic otating clindes in peiodic bounda conditions in the Z-ais [Heuan Wang 212] Ẋ = C 1 X + C 2 ReZ C 3 Y Z, (2.1) Ẏ = C 4 Y + C XZ, Ż = C 6 RX C 7 Z C 8 XY. whee Re is Renolds numbe, X, Y, Z is Fouie coefficients, the ae function of t, and C 1,...,C 8 ae positive paamete. The model equations (2.1) ae the same as the model sstem of Chen and Hsien [1987], which is a Loen-tpe sstem. Because the coefficients satisf C 1 + C 4 > C 7 in the model sstem of Chen and Hsien [1987], thee is no Hopf bifucation even fo lage Renolds numbe Re. On the conta, due to disappeaance of the esticted condition C 1 + C 4 > C 7 in the model sstem (2.1), thee eists Hopf bifucation and chaos [Heuan Wang 212]. Fo the sstem (2.1) thee eists tivial stationa solution O = (,, ) and two nontivial stationa solutions P +, P [Heuan Wang 212]. In ode to discuss the stabilit of the nontivial stationa solutions P ± Heuan Wang [212] intoduces the following tansfomation: t = sτ, X = s 2, (2.2) Y = s 3, Z = s 1.

4 68 H. WANG Due to C 3 C >, C 6, we set s = 1 C 1, s 1 = C 1 C3 C, s 2 = then (2.1) can be ewitten as a Loen-tpe sstem: ẋ = σ( ) + c, (2.3) ẏ = +, ż = b, C 1 C 7 C 6 C3 C Re, s 3 = C 1C 7 C 3 C 6 Re, whee = C 2C 6 C 1 C 7 Re 2, c = C2 7 C 8 C 2 6 C 3Re 2, b = C 4 C 1, σ = C 7 C 1. In contast to Loen sstem, the sstem (2.3) has an additional nonlinea tem c. In the following we discuss the estimation of Hausdoff dimension of its attacto. Fiing,, eplacing with + σ +, and setting p = [ σ + c( + σ) ], we ewite sstem (2.3) as (2.4) ẋ = σ + p + c (1) ẏ = σ (2) ż = b b( + σ) (3). Let E be Hilbet space, and Y E be a subset of E, I be inde set. Given d R +, ǫ >, we denote b µ H (Y, d, ǫ) the quantit inf i I d i, i be adius of a famil ball coveing Y of E, and i ǫ (fo all i I). Set µ H (Y, d) = lim µ H (Y, d, ǫ) = sup µ H (Y, d, ǫ) ǫ µ H (Y, d) is called d-dimensional Hausdoff measue of Y. Thee eists d [, ) such that µ H (Y, d) = fo d > d, µ H (Y, d) = + fo d < d, the numbe d is called the Hausdoff dimension of Y, and is denoted d H (Y ) (R. Temam [12]), and the following conclusion is quoted fom R. Temam [12]. Theoem 2.1. Let H be a Hilbet space, and X H be a compact set,and S be a continuous mapping fom X into H such that SX X. We assume that L is unifoml diffeentiable on X, i.e., Fo eve u X, thee eists a linea opeato L(u) L(H) and Su Sv L(u)(v u) sup, as ǫ u,v X,< u v X <ǫ u v whee, X ae nom of H and X espectivel, and we assume the following: ǫ> sup L(u) L(H) < +, u X sup ω d (L(u)) < 1 fo some d >, u X then the Hausdoff dimensional of X is less than o equal to d.

5 DIFFERENTIAL INCLUSIONS 69 We pesent estimation of Hausdoff dimension of attacto in the following conclusion, in the poof of the following theoem 1 we efe to the method shown in liteatue [12], but we have discoveed a few mistakes in the pocess of poof in liteatue [12], we have evised these eos. Since ou sstem contains the c, the following discussion become moe complicated. Definition of ω d, ω m (L), α m (L), m H, (, ) Vm H is quoted fom R. Temam [12], Theoem 2.2. Fo thee-mode equations (2.3) Hausdoff dimension of its attacto satisfies d H 2 + s, and s (, 1). Poof. Sstem (2.4) can be witten in the fom σ p c du = F(u) = F(,, ) = σ + +. dt b + b( + σ) Let U = ξ(t) H = R 3, and we conside the following initial-value poblem (2.) whee { du dt = F (u) U U() = ξ, F (u) U = A 1 U + A 2 U + B(u)U, σ σ A 1 = 1, A 2 = σ b c( + σ) c c B(u) =,, u = (,, ) R 3. Fo initial-value ξ 1, ξ 2, ξ 3 R 3, solution U = (U 1, U 2, U 3 ) of the initial-value poblem (2.) satisfies U i (t) = L(t, u )ξ i, t >. u is stationa solution of the sstem (2.4). L(t, u ) is linea opeato fom R 3 into R 3 In m H(m = 2, 3) we conside L(t, u ) : U() = ξ( R 3 ) U(t)( R 3 ). d dt U 1 U 2 U 3 = U 1 U 2 U 3 TF (u) d dt U 1 U 2 = U 1 U 2 T(F (u) Q), whee Q = Q 2 (t, u ; ξ 1, ξ 2 ) is othogonal pojection fom R 3 into span{u 1, U 2 }, notation T is quoted fom R. Temam [12]. U 1 U 2 U 3 (t) = ξ 1 ξ 2 ξ 3 ep[ (σ + b + 1)t]

6 7 H. WANG ω 3 (L(t, u )) = theefoe, ω 3 (L(t, u )) ep[ (σ + b + 1)t]. sup U 1 U 2 U 3 (t) ξ i H, ξ =1,i=1,2,3 Let Λ i, µ i (i = 1, 2, 3) be unifom Lapunov numbes and unifom Lapunov eponents (R. Temam [12]), then B simila manne we have Λ 1 Λ 2 Λ 3 = lim t ω 3 (t) 1/t = ep[ (σ + b + 1)], µ 1 + µ 2 + µ 3 = (σ + b + 1). U 1 U 2 = ξ 1 ξ 2 ep t Assume that ξ 1 ξ 2, thus U 1 U 2, t >. whee m = ma(1, b, σ). T(F (u(c)) Q(c))dc. T(A 1 + A 2 ) Q = T(A 1 ) Q 1 + b + σ m Set ϕ i = ( i, i, i ), i = 1, 2, 3 be an othogonal basis of R 3, theeafte = 3 3. We find T(B(u) Q) = = = 2 (B(u)ϕ i )ϕ i i=1 2 [( c i i c i i + i i + i i i i i i ) c( + σ) i i ] i=1 2 [(1 c) i i + ( 1 c) i i c( + σ) i i ] i=1 = (1 c) (1 + c) c( + σ) 3 3. When c, we have 1 c 1 + c, then T(B(u) Q) = (1 c) (1 + c) c( + σ) c c( + σ) c c( + σ) 1 1 c u(t) + c( + σ), 2 whee u(t) = , ϕ i = 2 i + 2 i + 2 i. Theefoe T(B(u) Q) 1 2 (1 c)ρ δ, whee ρ is absobing adius of the sstem when c < ( < δ 1).

7 DIFFERENTIAL INCLUSIONS 71 Theeafte U 1 U 2 ξ 1 ξ 2 ep((k 2 + δ)t), whee k 2 = (σ + b + 1) + m + 1 ρ 2 (1 c) + c( + σ), theefoe ω 2 (L(t, u )) ep((k 2 + δ)t) ω 2 (t) ep((k 2 + δ)t), Λ 1 Λ 2 ep(k 2 ), µ 1 + µ 2 < k 2. Assume that d H = 2 + s, < s < 1, t t 1 (δ), we get d H = 2 + s 2 + When c, estimate is diffeent slightl, namel, t t 1 (δ), k 2 + δ σ + b k 2 + δ. T(B(u) Q) (1 c) (1 + c) c( + σ) 1 (1 + c) u(t) + c( + σ). 2 Accodingl, T(B(u) Q) 1 2 (1+c)ρ δ, theefoe, k 2 = (σ+b+1)+m+1 2 ρ (1+c), then d H = 2 + s 2 + k 2 +δ k 2 +δ+σ+b+1. We complete estimation of Hausdoff dimension of attacto. 3. The globall eponentiall attactive set and positive invaiant set In ode to stud the globall eponentiall attactive set and positive invaiant set of the thee-mode Loen-tpe sstem we ewite sstem (2.3) as ẋ = σ( ) + c, (3.1) ẏ = + a, ż = b, whee a = C 2C 6 C 1 C 7, c = C2 7 C 8 C 2 6 C 3, b = C 4 C 1, σ = C 7 C 1, = Re 2. Fist, we pesent a basic definition hee, let X = (,, ) and suppose X(t) = X(t, t, X ) is a solution of sstem (3.1), and we denote the solution of sstem (3.1) satisfing the initial value X(t, t, X ) = X. Definition 3.1. If thee eists a constant numbe L > such that fo V (X ) > L, V (X(t)) > L, impl lim t + V (X) L, then Ω = {X V (X(t)) L} is said to be a globall attactive set of sstem (3.1). Definition 3.2. If fo an X Ω and an t > t, impl X(t, t, X ) Ω, then Ω = {X V (X(t)) L} is said to be positive invaiant set.

8 72 H. WANG Definition 3.3. If thee eists constant numbes L >, M > and an X R 3 such that fo V (X ) > L, V (X(t)) > L, impl the following eponentiall estimate inequalit: V (X(t)) L (V (X ) L)e M(t t ), then Ω = {X V (X(t)) L} is said to be a globall eponentiall attactive set of sstem (3.1). Theoem 3.4. Let (3.2) V = 2 + ( 1 + c ) [ ( 2 + σ 1 + c ) ] 2 a (3.3) L = b2 [σ + (1 + c )a]2 2b 1 when V (X ) L, and V (X(t)) L, then sstem (3.1) has the following estimate inequalit fo globall eponentiall V (X(t)) L (V (X ) L)e (t t ), and lim t + V (X(t)) L. i.e. Ω = {X V (X(t)) L} is the globall eponentiall attactive set and positive invaiant set of the sstem (3.1). Poof. We constuct a famil of genealied adicall infinite and positive definite Lapunov functions as ( V = c ) ( ( 2 + σ 1 + c ) ) 2 a Computing the time deivative along the positive half-tajecto of sstem (3.1), we have ( (3.4) V = 2ẋ c ) [ ( ẏ + 2 σ 1 + c ) ] a ż = V + F(X) whee ( F(X) = F(,, ) = (1 2σ) c ) 2 (1 2b) 2 [ ( 2 σ c ) ] [ ( a (1 b) + σ c ) 2 a]. Because F(,, ) is quadatic function, its local maimum is the global maimum. Let F = 2(1 2σ) =, F (1 = 2 + c ) =, ( 1 + c F [ = 2(1 2b) 2 σ + we have =, =, = [σ+(1+ c )a](1 b) ( deivative of F(X) at the point 1 2b,, [σ+(1+ c )a](1 b) 1 2b ) ] a (1 b) =, 2 F = 2(1 2σ) <, when σ > 1 2 2, ) <, when 2 F 2 = 2 (1 + c, then we obtain the following two-ode ), ( 1 + c ) >,

9 So DIFFERENTIAL INCLUSIONS 73 2 F = 2(1 2b) <, when b > 1 2 2, 2 F = 2 F = 2 F =. sup F(X) = F(X) ( =, =, = X R 3 [ = b2 σ + ( ) ] 1 + c 2 a = L 2b 1 [ σ + ( 1 + c ] ) a (1 b) ) 1 2b Fom (3.4), we have dv dt V + L, and we get the following globall eponentiall estimate V (X(t)) L (V (X ) L)e (t t ), when V (X ) L, V (X(t)) L, and we take the uppe limit of the both sides fo V (X(t)) L (V (X ) L)e (t t ), then lim t + V (X(t)) L. Theefoe, Ω = {X V (X(t)) L} is the globall eponentiall attactive set and positive invaiant set of the sstem (3.1). 212] (4.1) 4. The Numeical Simulation and Analsis of Dnamical Behavios Heuan Wang obtains the following concete thee-mode equations [Heuan Wang ẋ = 4.2( ) , ẏ = , ż = The numeical computation esults indicate that sstem (4.1) has complicated dnamical behavios with inceasing the paamete = Re 2. In this section we pesent the numeical epeiment esults. 1) Fo < < 1 = , the stationa solutions O of sstem (4.1) is stable, in this case O is the onl attacto of the sstem. As passes though 1, equilibium solutions O becomes unstable, simultaneous with the loss of stabilit of O, the two fied points P ± ae bon, fo 1 < , the stationa solutions P ± of sstem (4.1) is stable, Numeical evidences indicate that an andoml chosen initial data is attacted b one of them, so the ae global attactos (see Fig. 1, 2). 2) At = the two stable stationa solutions P ± become unstable because a pai of comple conjugate eigenvalues of Jacobian mati at stationa point P ± cosses the imagina ais, each of the two stable stationa solutions P ± loses stabilit undegoing a Hopf bifucation, and each geneates a unstable peiodic obit (Fig. 3). 3) At = , the unstable peiodic obits gives ise to a new obit, the new obits wind up aound two of the stationa solutions P ±, instead of onl one like the pevious ones (Fig. 4), the motions become moe and moe complicate as gows,

10 74 H. WANG eventuall geneating a chaotic attacting set, namel, the stange attacto appeas (Fig. 1). 4) Fo < < , the stange attacto shinks into a limit ccles gaduall (Fig. 11 1), if followed backwads with deceasing, this bifucation is an invese bifucation. ) Fo < < , inceasing we found futhe bifucation points, since the obits become inceasingl inticate equiing highe pecision, we did not look fo futhe bifucations. So we cannot definitel state whethe we have just a finite numbe of bifucations and the onl obstacle to obseve moe seems to be the numeical pecision needed, then the stange attacto appeas again, Fig pesents thee kinds of quasi-peiodic state, Fig. 19,2 descibes two kinds of attacto in diffeent. With the inceasing of the paamete, a stong hsteesis phenomenon(i.e., coeistence of stable attactos) appeas, in some intevals hsteesis takes place between closed obits and toi (Fig. 21 3). 6) Fig. 36 shows bifucation diagams of the sstem (4.1), Fig. 37 ae the coesponding lagest Lapunov eponents, we can see that egion of the positive maimum Lapunov eponents in Fig. 37 and chaotic egion of the bifucation Fig. 36 is consistent. 7) Fig. 38 shows Poincae section of the sstem (4.1) ( = ); Fig. 39 shows etun map of the sstem (4.1) ( = ); Fig. 4 pesent the powe spectum of the sstem (4.1), the indicate that chaos featue of the sstem (4.1). 8) Fom bifucation diagams Fig. 36 we find that the chaotic egion contains peiodicobit windows of vaing width, the stange attactos and limit ccles appea altenatel at some paamete. Though moe delicate and difficult calculation we obtain the following details: at = and = the stange attacto shinks into a limit ccles(fig. 14, 1, 24), at = and = the limit ccles become unstable and the peiodic obits geneated b the peiod doubling bifucation continue to double in eve faste succession eventuall geneating a chaotic attacting set (Fig. 16 2) at = the limit ccles gaduall evolved into a tous (Fig. 29 3), if followed backwads with deceasing, this bifucation is an invese bifucation (Fig. 36)

11 DIFFERENTIAL INCLUSIONS 7 Fig. 1 ( = 3.342) Fig. 2 ( = 3.837) Fig. 3 ( = 4.1) Fig. 4 ( = 4.617) Fig. ( = 4.714) Fig. 6 ( = 4.928) Fig. 7 ( = 8.914) Fig. 8 ( = 1.921) Fig. 9 ( = ) Fig. 1 ( = )

12 76 H. WANG Fig. 11 ( = ) Fig. 12 ( = 2.811) Fig. 13 ( = ) Fig. 14 ( = 3.713) Fig. 1 ( = 3.713) Fig. 16 ( = 3.721) Fig. 17 ( = ) Fig. 18 ( = 32.21)

13 DIFFERENTIAL INCLUSIONS Fig. 19 ( = 3.796) Fig. 2 ( = ) Fig. 21 ( = ) Fig. 22 ( = ) Fig. 23 ( = 4.896) Fig. 24 ( = 42.8) Fig. 2 ( = ) Fig. 26 ( = 48.89)

14 78 H. WANG Fig. 27 ( = 2.896) Fig. 28 ( = ) Fig. 29 ( = ) Fig. 3 ( = ) Fig. 31 ( = ) Fig. 32 ( = ) Fig. 33 ( = ) Fig. 34 ( = )

15 DIFFERENTIAL INCLUSIONS Fig. 3 ( = 1.843) Fig Poincae section = 1 4 lapunov eponents paamete Fig. 37 Fig. 38 ( = ) n+1 34 powe n f Fig. 39 ( = ) Fig. 4 The numeical esults 1) 3) can be used to eplain successive tansitions phenomena of Couette-Talo flow fom Lamina flow to tubulence in the epeiment, fo eample, in the egime < < 1 the onl attacto is O, this epesents a situation in which the fluid is at est, this stable equilibium egime coespond to Couette flow. The stead states P ± both epesent time-independent convective flow pattens, this coespond to the Talo votices and the otating waves. The quasi-peiodic obits in the Figs. 9 ae with espect to modulated wav Talo votices, stange attacto in the Fig. 1 coespond to tubulence (chaos), and so on. These simulation esults indicate that the tansition fom stable equilibium point unstable equilibium point peiodic quasi-peiodic tansient chaos chaos, simila to one of the epeimental esults of Gollub and Benson [198]. Although ou numeical esult is not completel compatible with the epeimental esult, it is simila to the epeimental esult qualitativel. In ode to check the behavios of this low-dimensional

16 8 H. WANG model, we have obtained a five-dimensional sstem fo the same poblem, the obtained esults ae ve simila. The five-dimensional sstem also ehibits a peiod doubling scenaio. The bifucations occu at diffeent Renolds numbes. In both cases, the qualitative phenomena ae essentiall the same. Accoding to the numeical computation esults and analsis, we pesent stabilit of the thee-mode and coesponds to the actual flow of Couette-Talo flow b the following table 4.1. Table 4.1 Equilibium points popet of thee-mode sstem and coesponds to the actual flow of Couette-Talo flow -value < < 1 As > 1 inceases futhe equilibium Stable node Saddle node O One diection is unstable, the othe two diections ae stabilit P +, No Stable Stable Saddle P equilibium node focus point Unstable Phase Tends to Tends to spial line limit ccles tajecto of stable stable Tends to subcitical sstem (2.1) equilibium equilibium P + o P Hopf O P + o P bifucation the actual Couette flow the Talo votices iegula flow of and otating taveling tubulent C-T flow waves chaotic. Conclusions This pape has studied a thee-mode Loen-tpe sstem of the incompessible flow between two concentic otating clindes. Dnamical behavios of the chaotic sstem, including the stange attacto, bifucations, and some univesal featues of chaos behavios have been investigated both theoeticall and numeicall b changing paamete. Compaed with the classical Loen sstem, ou thee-mode sstem diffes onl b the ight-hand side fo the -deivative whee is, as an addition, a tem c, ou thee-mode sstem does not epoduce the qualitative featues of Loen sstem fom which it has been obtained as an etension, no stable attacting peiodic obit being pesent at high values of the paamete, in some intevals hsteesis takes place between closed obits and toi. Even if the thee-mode model studied dose not epoduce the inteesting phenomena of the Loen sstem, we think it is

17 DIFFERENTIAL INCLUSIONS 81 also inteesting b itself. In fact its phenomenolog, studied quite in detail, appeas so ich and vaied to ampl justif the pesent wok. REFERENCES [1] F. Chen, D. Y. Hsien, A Model Stud of Stabilit of Couette Flow, Comm. on. Applied Math. and Comp., 12:22 33, [2] H. Y. Wang, Dnamical Behavios and Numeical Simulation of Loen Sstems fo The Incompessible Flow Between Two Concentic Rotating Clindes, Intenational Jounal of Bifucation and Chaos, 22():22-27, 212. [3] H. Y. Wang, The chaos behavio and simulation of thee model sstems of Couette-Talo flow, Acta Mathematica Scientia, 3(2): , 21. [4] C. M. Gassa Feugainga, O. Cumeollea, K. S. Yangb, I. Mutabaia, Destabiliation of the CouetteCTalo flow b modulation of the inne clinde otation, Euopean Jounal of Mechanics - B/Fluids, 719:14 46, 213. [] Richad J. A. M. Stevens Siegfied Gossmann, Robeto Veicco, D. Lohse, Optimal Talo- Couette flow: Diect numeical simulations, Jounal of Fluid Mechanics, 34: , [6] Olivie Can, Eic See, Identification of comple flows in Talo-Couette counte-otating cavities, C. R. Acad. Sci. Pais t. 329, Seie II, , 21. [7] Pascal Chossat, Gead Tooss, The Couette-Talo poblem, Spinge Velag, New Yok, [8] H. L. Swinne, J. P. Gollub, Hdodnamic, Instablilities and the Tansition to Tubulence, Spinge Velag, New Yok, [9] G. I. Talo, Stabilit of a Viscons Liquid Contained Between Two Rotating Clindes, Phil. Tans. Ro. Soc., London, A223: , [1] D. G. Thomas, B. Khomami, R. Sueshkuma, Nonlinea dnamics of viscoelastic Talo- CCouette flow, effect of elasticit on patten selection, molecula confomation and dag, J. Fluid Mech., 62:33 C382, 29. [11] C. D. Andeeck, S. S. Liu, and H. L. Swinne, Flow egimes in a cicula Couette sstem with independent otating clindes, J. Fluid Mech., 64:1 183,1986. [12] R. Temam, Infinite dimensional dnamic sstem in mechanics and phsics, Appl. Math. Sci., Spinge Velag, New Yok, 2. [13] J. P. Gollub S. V. Benson, Man outes to tubulent convection, J. Fluid Mech., 1:449, 198.