A method of chaos reduction for Hamiltonian systems is applied to control
chaotic advection. By adding a small and simple term to the stream function of
the system, the construction of invariant tori has a stabilization effect in
the sense that these tori act as barriers to diffusion in phase space and the
controlled Hamiltonian system exhibits a more regular behaviour

Aubry, Nadine

Benzekri, Tounsia

Chandre, Cristel

Goullet, Arnaud

Leoncini, Xavier

Lima, Ricardo

Vittot, Michel

English

arXiv

A method of chaos reduction for Hamiltonian systems is applied to control chaotic advection. By adding a small and simple term to the stream function of the system, the construction of invariant tori has a stabilization effect in the sense that these tori act as barriers to diffusion in phase space and the controlled Hamiltonian system exhibits a more regular behaviour

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Control of chaotic advectionTounsia Benzekri, Cristel Chandre, Xavier Leoncini, Ricardo Lima, MichelVittot, Arnaud Goullet, Nadine AubryTo cite this version:Tounsia Benzekri, Cristel Chandre, Xavier Leoncini, Ricardo Lima, Michel Vittot, et al.. Con-trol of chaotic advection. IFAC conference on Analysis and control systems , Ifac (Ed), 2006,pp.219-224. <hal-00102905>HAL Id: hal-00102905https://hal.archives-ouvertes.fr/hal-00102905Submitted on 2 Oct 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00102905, version 1 - 2 Oct 2006CONTROL OF CHAOTIC ADVECTIONT. Benzekri ∗,1 C. Chandre ∗ X. Leoncini ∗ R. Lima ∗M. Vittot ∗∗ Centre de Physique The´orique, CNRS Luminy, Case 907,F-13288 Marseille cedex 9, FranceA. Goullet ∗∗ N. Aubry ∗∗∗∗Department of Mechanical EngineeringNew Jersey Institute of Technology, Newark, New Jersey07102Abstract: A method of chaos reduction for Hamiltonian systems is applied tocontrol chaotic advection. By adding a small and simple term to the streamfunction of the system, the construction of invariant tori has a stabilization effectin the sense that these tori act as barriers to diffusion in phase space and thecontrolled Hamiltonian system exhibits a more regular behaviour.Keywords: Hamiltonian chaos, control.1. INTRODUCTIONChaotic advection (or Lagrangian chaos) was in-troduced by Aref (1988) to qualify a motion inwhich it is possible to generate chaotic trajecto-ries even if the flow is laminar. Such phenomenaare observed in a wide range of physical systems(Ottino (1989)) and has fundamental applicationsfor instance in geophysical flow (Behringer, etal. (1991); Brown and Smith (1991)). Chaoticadvection was mostly studied for two dimen-sional unsteady flow (Solomon and Gollub (1988);Solomon, et al. (2003); Camassa and Wiggins(1991)). The advantage of these studies is that ituses the theory of dynamical systems and in par-ticular the Hamiltonian theory for incompressibleflows; the phase space being the physical space inthis case. For these motions, experiments can beimplemented in a laboratory.Even though chaos is sometimes preferable in1 benzekri@cpt.univ-mrs.frmixing problems (Benzekri, et al.(2006)), we usu-ally want to suppress it for instance in chaoticadvection. Recently, a method to control chaoticdiffusion in Hamiltonian dynamical systems wasproposed by (Vittot, et al. (2005); Chandre, etal.(2005)). It was shown that it is possible toprevent chaotic diffusion by adding a small termto the Hamiltonian. This technique was used toa model describing anomalous transport in mag-netized plasmas, and applied experimentally on abeam of electrons produced by a long TravellingWave Tube (Chandre, et al.(2005)).Our goal in this paper is to use this methodof control of Hamiltonian chaos for the prob-lem of chaotic advection in a two dimensionaltime periodic flow. We apply this method withinthe framework of an experiment using magneto-hydrodynamic technique which shows that parti-cle trajectories in a time periodic flow are chaotic.More precisely, the experiment consists in an elec-tric current passing through a thin layer of an elec-trolytic solution with a free surface. The dynamicsof passive particles of a time periodic two di-mensional and incompressible flow is Hamiltonian.The Hamiltonian modeling the flow was derivedad hoc in (Solomon and Gollub (1988); Solomon,et al. (2003)). A comparison was made with theexperiment to validate the model. The equationof motion of these passive particles comes froma Hamiltonian of 1.5 degrees of freedom that westudy in the limit of weak amplitude oscillations.The model proposed to describe this phenomenais based on the following streamfunction:Ψǫ(x, y, t) = α sin(x+ ǫ sin t) sin y, (1)where α is the maximal vertical velocity in theflow, ǫ is the amplitude of the lateral oscillationsof the velocity field. The current interacts withan alternative magnetic field produced by mag-nets below the fluid. A chain of vortices are thenobserved. Time periodic dependence is imposedexternally with a plunger that oscillates slowly upand down, displacing the flow laterally, i.e., in thedirection perpendicular to the roll axes, giving riseto chaotic advection. The same phenomenon isobserved in Rayleigh-Be´nard convection. For thesteady state, rolls are formed periodically. It hasbeen shown that by imposing a sinusoidal flow,chaotic advection occurs.Under the perturbation, i.e. for ǫ 6= 0, the verticalheteroclinic connection breaks down and the sta-ble and unstable manifold intersect transverselythus generating chaotic advection of passive par-ticles.In this method, we seek for one of the simplestperturbations to create barriers around the bro-ken separatrix of the integrable case. We choosea perturbation depending only on the positionvariables.We will use the stream function (1) as the startingpoint to control or reduced the chaotic advection.2. LOCAL CONTROL METHODThe local control method has been extensivelydescribed in (Vittot, et al. (2005)) where thecorresponding rigorous results were proved. Wesummerize here the results of this paper. For aHamiltonian system with L degrees of freedom,the perturbed Hamiltonian isH(A,θ) = ω ·A+ V (A,θ),where (A,θ) ∈ RL × TL are action-angle likevariables and ω is a non-resonant vector of RL.Without loss of generality, let us consider a regionnear A = 0 (by translation), since the Hamilto-nian is nearly integrable, the perturbation V hasconstant and linear parts in actions of order ε, i.e.V (A,θ) = εv(θ) + εw(θ) ·A+Q(A,θ), (2)where Q is of order O(‖A‖2). Note that for ε = 0,the Hamiltonian H has an invariant torus withfrequency vector ω at A = 0 for any Q notnecessarily small. The controlled Hamiltonian isthen constructed as:Hc(A,θ) = ω ·A+ V (A,θ) + f(θ). (3)In this situation, the control term f only dependson the angle variables and is given byf(θ) = V (0,θ)− V (−Γ∂θV (0,θ),θ) , (4)where Γ is a linear operator defined as a pseudo-inverse of ω · ∂θ, i.e. acting on V =∑kVkeiθ·kasΓV =∑ω·k 6=0Vkiω · keiθ·k. (5)Note that f is of order ε2. This can be seen fromEq. (2) since f can be rewritten asf(θ) = ε2w(θ) · Γ∂θv −Q (−εΓ∂θv,θ) ,and Q is quadratic in the actions. For any pertur-bation V , Hamiltonian (3) has an invariant toruswith frequency vector close to ω. The equation ofthe torus which is constructed by the control isA = −Γ∂θV (0,θ), (6)which is of order ε since V (0,θ) is of order ε.In the next section, we will see that the amplitudeof the control term is small compared with theperturbation.2.1 Application to the suppression of chaoticadvectionIn this section, we apply the method previouslysummerized to reduce chaotic transport of passivetracers. With this method, we create isolated bar-riers of transport. In particular, a barrier createdbetween two convection rolls allows one to reducethe diffusion across these rolls. Let us first givesome results in the integrable case.For ǫ = 0, the Hamiltonian is integrable, thetrajectories of advected particles coincide withstreamlines. The phase space, which is here thephysical space, is characterized by a chain of rollswith separatrices localized at x = mπ, with m ∈Z. The fluid is limited by two invariant surfacesy = π and y = 0 corresponding to the top andbottom roll boundaries.In the integrable case, as mentionned in (Camassaand Wiggins (1991)), the flow is characterized byhyperbolic fixed points along the two invariantsurfaces y = π and y = 0 and localized at x = mπ,m ∈ Z. These points are joined by a verticalheteroclinic connection for which the stable andunstable manifolds coincide.In order to built a barrier, we select a surfacewhich is located around x = 0. However we couldalso create a barrier at x = π (mod 2π)We map the time-dependent stream function Ψǫgiven by Eq. (1) into an autonomous Hamilto-nian written as H(x, y, E, τ) = E + Ψǫ(x, y, τ).This corresponds to an extension of phase space(x, y, E, τ), where A = (x,E) and θ = (y, τ) aremomenta and positions. We assume that at t = 0,τ(0) = 0 for H . Therefore the equation of motionof H are the same as the ones for Ψǫ since τ = t.We rewrite the autonomous Hamiltonian in theform:H(x, y, E, τ) =E + α sin(x+ ǫ sin τ) sin y,=H0(x,E) + Ψǫ(x, y, τ), (7)and develop the stream function in Taylor seriesaround x = 0H(x, y, E, τ) =E + ǫv(y, τ) + ǫw(x, y, τ)x+Q(x,E, y, τ), (8)where H0(x,E) = E is the integrable part,ǫv(y, τ) = Ψǫ(0, y, τ) andǫw(x, y, τ) = ∂xΨǫ(0, y, τ),Q(x,E, y, τ) =∞∑l=11(l + 1)![∂l+1x Ψǫ(0, y, τ)]xl+1.The frequency vector ω is given byω = (∂H0/∂x, ∂H0/∂E) = (0, 1) and is resonant.In order to compute the operator Γ given byEq.(5), we rewrite Ψǫ by expanding sin(ǫ sin t)and cos(ǫ sin t) as series of Bessel functions of firstkind:Ψǫ(x, y, t) = α sin y sinx(Jǫ0 + 2∑n≥1J ǫ2n cos 2nt)+ 2α sin y cosx(∑n≥1J ǫ2n+1 sin(2n+ 1)t),where J ǫl = Jl(ǫ) for l ∈ N, and Jl are Besselfunctions of the first kind.The control term is given by Eq.(4):f(y, t) = Ψǫ(0, y, t)−Ψǫ(−Γ∂yΨǫ, y, t).(9)Since Ψǫ does not depend on E, we only have tocompute Γ∂yΨǫ(0, y, t) and thenf(y, t) = α sin(ǫ sin t) sin y−α sin [−α cos yCǫ(t) + ǫ sin t] sin y,(10)0 1 2 3 4 5 600.511.522.53 xy(a) 0 1 2 3 4 5 600.511.522.53 xy(b) −2 0 2 4 6 800.511.522.53 xy(c) Fig. 1. Streamlines at (a) t = 0 and (b) t =3π/4, and (c) Poincare´ section of the streamfunction (1). The parameters are α = 0.6 andǫ = 0.63.whereCǫ(t) = Γ sin(ǫ sin t),=∑n≥0−22n+ 1J ǫ2n+1 cos(2n+ 1)t. (11)The equation of the invariant torus is given byEq. (6)x = α cos yCǫ(t). (12)The controlled stream function is given by:Ψc(x, y, t) = α sin y (sin(x+ ǫ sin t) + sin(ǫ sin t)− sin [−α cos yCǫ(t) + ǫ sin t]) . (13)We notice that with a small modification of thestream function, provide that α and ǫ are suchthat | αǫ |< 1 , there is an exact formula givingthe equation of the torus. This invariant torussuppress the chaotic transport from roll to rolland then along the channel.To illustrate numerically these results, we considerfixed values of the parameter α and ǫ. Similarresults hold for other values of the parameters αand ǫ.Streamlines of the streamfunction (1) are depictedon Fig. 1(a) and (b) at two different times t = 0and t = 3π/4 respectively and for ǫ = 0.63 andα = 0.6. We observe closed curves correspondingto lateral oscillations in the x-direction with a pe-riodic displacement of−ǫ sin t. For the same valuesof ǫ and α, a Poincare´ section of the dynamics ofthe streamfunction (1) for initial conditions with-out control are represented in Fig. 1(c). It showsthat the transport along the channel is greatlyenhanced. The phase space is characterized byregular (quasiperiodic) trajectories and a chaoticregion around and between the rolls.When we add the control term, the streamlines0 1 2 3 4 5 600.511.522.53 xy(a) 0 1 2 3 4 5 600.511.522.53 xy(b) −2 0 2 4 6 800.511.522.53 xy(c) Fig. 2. Streamlines at (a) t = 0 and (b) t =3π/4, and (c) Poincare´ section of the streamfunction (13). The parameters are α = 0.6and ǫ = 0.63.of the controlled stream function (13) are thenslightly modified (non-uniformly in y) as it canbe seen on Fig. 2(a) and (b) for ǫ = 0.63 andα = 0.6 at two different times t = 0 and t = 3π/4respectively. Moreover, the displacement of theserolls remains parallel to the x-axis as it is forthe stream function given by Eq. (1). A Poincare´section of the dynamics of passive tracers in aflow described by the stream function Ψc givenby Eq. (13) is represented in Fig. 2(c). We noticefirst that there are invariant surfaces which havebeen created around x = 0 (mod 2π) (bold curves)which prevent the diffusion of passive particlesalong the channel. Moreover we observe a regular-isation of the dynamics whithin the cell boundedby the two barriers x = 0 and π (mod π).In Fig. 3, we depict a numerical simulation of thedynamics of a dye in the fluid. The left columnshows the evolution of the tracers for the streamfunction Ψǫ given by Eq. (1). The right columnshows that the control term regularizes the dy-namics and prevents any spread of a dye withina cell which is limited by two barriers created bythe stream function Ψc given by Eq. (13).For | αǫ |<< 1, the control term (13) can besimplified such as:fs(y, t) = −α22sin 2y cos(ǫ sin t)Cǫ(t). (14)We remark that the control term fs is of order α2ǫand does not depend on x as it is expected fromthe method.In order to test the robustness of the method andto try an experimentally more tractable pertur-bation, we truncate the series given by Eq. (11)by considering only the first term of the series−2 0 2 4 6 80123xy−2 0 2 4 6 80123 xy−2 0 2 4 6 80123xy−2 0 2 4 6 80123 xy−2 0 2 4 6 80123xy−2 0 2 4 6 80123 xy−2 0 2 4 6 80123xy−2 0 2 4 6 80123 xyFig. 3. Numerical simulation of the dynamicsof a dye at t = 30, t = 50, t = 70 andt = 140 (from top to bottom) : left columnfor the stream function (1) and right columnfor the stream function (13). The parametersare α = 0.6 and ǫ = 0.63.Cǫ(t) in Eq. (11) and cos(ǫ sin t) which leads tothe following stream functionΨs(x, y, t) = α sin(x+ ǫ sin t) sin y−α2J ǫ1 sin 2y cos t. (15)The perturbation has a norm supx,y,t | V (x, y, t) |=ǫ and the norm of the the simplified control termfs issupy,t| fs(y, t) |= α2J ǫ1 . (16)Therefore the ratio r between the norm of thecontrol term and the perturbation is given byr ≈ αǫ.The dynamics of the stream function Ψs is de-picted in Fig. 4 for ǫ = 0.63 and α = 0.6. Wesee that invariant surface has been created aroundx = 0 (mod 2π) and that the control term still re-duces significantly the chaotic advection but thereare some advected particles along the channel. Weobserve that an other invariant surface has beencreated around x = π. This is due to the factthat the flow given by the stream function Ψs isinvariant under the symmetryt→ t, x→ x+ π, y → y + π.It results from this symmetry, a regularisation ofthe dynamics within the cell bounded by the twobarriers x = 0 and π (mod 2π). This symmetry isan approximate one ( up to order | α2ǫ |) for thestream function Ψc. Thus the control term is ableto regularize its dynamics also in this region.−2 0 2 4 6 800.511.522.53 xyFig. 4. Poincare´ section of the stream function(15) . The parameters are α = 0.6 and ǫ =0.63.In order to see more clearly the effect of thecontrol term, we study the diffusion propertiesof the system. The mean square displacement< r2(t) > of a distribution of M particles (oforder 3000) is computed as a function of time:< r2(t) >=1MM∑i=1‖xi(t)− xi(0)‖2, (17)where xi(t), i = 1, ...,M is the position of the i-thparticle at time t obtained by integrating Hamil-ton’s equations with initial conditions xi(0). Wefind that < r2(t) > grows linearly for the consid-ered time interval (see Fig. 5(a)). The transport isthen assumed to be described as normal diffusionand the corresponding diffusion coefficient can bedetermined from the slope of < r2(t) > versus t:D = limt→∞< r2(t) >t. (18)Figure 5(b) shows the values of D as a function ofǫ with and without control term (15) determinedfrom the mean square displacement for t > 1000.We remark that the diffusion coefficient with thecontrol term (15) is significantly smaller than theuncontrolled case.3. CONCLUSIONWe presented in this paper an application of arecently developed control technique to Rayleigh-Be´nard convection. We derived analytically theexpansion of the control term required to reducechaotic advection present in the original prob-lem. Using Poincare´ sections, and tracking thedynamics of a dye, we showed the efficiency ofthe method. The slightly deformed streamlinesdue to the control terms prevent any large spreadthrough the channel but confine the material linesinside some deformed rolls. The chaotic behaviorof the flow was significantly reduced by takingonly the first order of control term.H. Aref, Stirring by chaotic advection, J. FluidMech. 143, 1 (1984).S. Balasuriya, Optimal perturbation for enhancedchaotic transport, Physica D 202, 155 (2005).0 500 1000 1500 2000 2500 3000 3500050100150200250300t <r2(t)> ε=0.2 ε=0.18 ε=0.12 ε(a) 1 2 3 4 5 610−610−510−410−310−210−1100 εD(b) Fig. 5. (a) Mean square dispacement < r2(t) >versus time t obtained for Hamiltonian(1) with three different values of ǫ =0.12, 0.18, 0.2 and (b) Diffusion coefficient Dversus ǫ obtained for stream function (1) (cir-cle) and stream function (15) (square)R.P. Behringer, S. Meyers and H. Swinney, Chaosand mixing in geostrophic flow, Phys. Fluids A 3,1243 (1991).T. Benzekri, C. Chandre, X. Leoncini, R. Lima,M. Vittot, Chaotic advection and targeted mixing,Phys. Rev. Lett. 96, 124503 (2006).M.G. Brown, K.B. Smith, Ocean stirring andchaotic low-order dynamics, Phys. Fluids 3, 1186(1991).R. Camassa, S. Wiggins Chaotic advection inRayleigh-Be´nard flow, Phys. Rev. A 43, 774(1991).C. Chandre, G. Ciraolo, F. Doveil, R. Lima, A.Macor, M. Vittot, Channelling chaos by buildingbarriers, Phys. Rev. Lett. 94, 074101 (2005).J.M. Ottino, The Kinematics of mixing: streching,chaos, and transport, Cambridge U.P., Cambridge(1989).T.H. Solomon, N.S. Miller, C.J.L Spohn, J.P.Moeur, Lagrangian chaos: transport, coupling andphase separation, AIP Conf. Proc. 676, 195(2003).T.H. Solomon and I. Mezic Uniform, resonantchaotic mixing in fluid flows, Nature, 425, 376(2003).T.H. Solomon, J.P. Gollub, Chaotic particle trans-port in Rayleigh-Be´nard convection, Phys. Rev. A38,6280 (1988).A.D. Stroock, S.K.W. Dertinger, A. Ajdari, I.Mezic, H.A. Stone, G.M. Whitesides,Chaotic mixerfor microchannels, Science 295, 647 (2002).M. Vittot, C. Chandre, G. Ciraolo and R. Lima,Localized control for nonresonant Hamiltoniansystems, Nonlinearity 18,423 (2005).H. Willaime, O.Cardoso, P. Tabeling, Spatiotem-porel intermittency in lines of vortices, Phys. Rev.E, 48, 288 (1993).S. Wiggins, Chaotic transport in dynamical sys-tems, Springer-Verlag (1992).

Control of chaotic advection

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Control of chaotic advection

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