Chaotic convection in a ferrofluid

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Chaotic convection in a ferrofluid

D. Laroze a,b,⇑, P.G. Siddheshwar c, H. Pleiner a

a Max Planck Institute for Polymer Research, D 55021 Mainz, Germanyb Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chilec Department of Mathematics, Bangalore University, Central College Campus, Bangalore 560 001, India

a r t i c l e i n f o

Article history:Received 22 September 2012Received in revised form 24 January 2013Accepted 25 January 2013Available online 4 February 2013

Keywords:Thermal convectionMagnetic fluidChaos

a b s t r a c t

We report theoretical and numerical results on thermally driven convection of a magneticsuspension. The magnetic properties can be modeled as those of electrically non-conduct-ing superparamagnets. We perform a truncated Galerkin expansion finding that the systemcan be described by a generalized Lorenz model. We characterize the dynamical systemusing different criteria such as Fourier power spectrum, bifurcation diagrams, and Lyapu-nov exponents. We find that the system exhibits multiple transitions between regular andchaotic behaviors in the parameter space. Transient chaotic behavior in time can be foundslightly below their linear instability threshold of the stationary state.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Chaotic behavior in thermal convection started with the work of Lorenz in the sixties [1]. The author found chaos in thesimplest Rayleigh–Benard convection using a prototype model to forecast the weather. The experimental observation of thechaotic behavior in this system was made by Libchaber and coworkers [2]. Other works in the chaotic convection are in sim-ple fluids [4,5], in binary fluids [3,6,7], in viscoelastic fluids [8–11], in porous media [12–14], in magnetohydrodynamics [15],in magnetic fluids [16] or in dielectrics [17], just to mention a few examples.

Ferrofluids are colloidal suspensions of magnetic nanoparticles dispersed in a carrier liquid. Typically, the particles’ diam-eter is of a few tenths of nanometers leading to gravitationally stable systems. Ferrofluids are superparamagnetic showing astrong response to external magnetic fields [18,19]. Convection in ferrofluids for different situations has been reported boththeoretically [20–45] and experimentally [46–57].

The purpose of the present paper is to analyze the chaotic convective behavior of an electrically nonconducting ferrofluid,in contrast to magnetohydrodynamic systems, where chaotic motion is accompanied by appropriate electric currents. Usinga truncated Galerkin expansion, similar to the Lorenz assumption [1], we derive a set of three nonlinear differential equa-tions for the amplitudes of flow, temperature and magnetic potential. We characterize the dynamical behavior of the systemthrough the calculation of Lyapunov exponents, bifurcation diagrams, and Fourier power spectra. Here we report thecomputation of complete phase diagrams for this generalized Lorenz system that include all physically stable phases, bothperiodic and chaotic. Generally, this is difficult to obtain, because computationally intensive calculations are needed, partic-ularly for models set up as a system of continuous ordinary differential equations. Recent work describing such methods anddifficulties can be found in Ref. [59].

The paper is organized as follows: In Section 2, the basic hydrodynamic equations for magnetic fluid convection arepresented. In Section 3 the generalized Lorenz equations are derived. In Section 4 the stability analysis of the stationary

1007-5704/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cnsns.2013.01.016

⇑ Corresponding author at: Max Planck Institute for Polymer Research, D 55021 Mainz, Germany. Tel: +49 (0) 6131 379165; fax: +49 (0) 6131 379340.E-mail address: [email protected] (D. Laroze).

Commun Nonlinear Sci Numer Simulat 18 (2013) 2436–2447

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solutions is calculated. In Section 5 and 6 the numerical simulations are performed and the results are explained. Finally, asummary is given in Section 7.

2. Basic equations

We consider a layer of thickness d, parallel to the xy-plane, with very large horizontal extension, of a ferrofluid subject toa vertical temperature gradient and gravitational field g ¼ �gz. The magnetic fluid properties are modeled as those of anelectrically nonconducting superparamagnet. An external vertical magnetic field H ¼ H0z is assumed to be present. The statictemperature difference across the layer is imposed by fixing the temperatures at the layer boundaries, Tðz ¼ 0Þ ¼ T0 þ MTand Tðz ¼ dÞ ¼ T0. Within the Boussinesq approximation, the dimensionless equations for the perturbations from the quies-cent, heat conducting state, can be written as [45]

r � v ¼ 0; ð1ÞP�1dtv ¼ �rpeff þr2v þ RaRðh;/Þ; ð2Þdth ¼ r2hþ vz; ð3Þð@zz þM3ð@xx þ @yyÞÞ/� @zh ¼ 0; ð4Þr2/ext ¼ 0; ð5Þ

where fv; h;/g are the dimensionless perturbations of the velocity, the temperature, and the magnetic potential, respec-tively. Here dtf ¼ @t f þ v � rf is the material derivative, peff is the effective pressure which contains the static hydrodynamicpressure and the gradient term of the magnetic force, and R ¼ P1ðh;/ÞzþM1ðrhÞð@z/Þ with P1 ¼ ð1þM1Þh�M1@z/.

We have kept four dimensionless numbers in (1)–(5): The Rayleigh number, Ra ¼ aT gbd4=jm, accounting for buoyancy

effects, the Prandtl number, P ¼ m=j, relating viscous and thermal diffusion time scales, M1 ¼ bv2T H2

0=ðq0gaTð1þ vÞÞ describ-

ing the strength of the magnetic force relative to buoyancy, and M3 ¼ ð1þ vÞ=ð1þ vþ vHH20Þ, a measure for the deviation of

the magnetization curve from the linear behavior M0 ¼ vH0. In these dimensionless numbers different physical quantitiesappear such as q0 the reference mass density, cH the specific heat capacity at constant volume and magnetic field, vT thepyromagnetic coefficient, j the thermal diffusivity, vH the longitudinal magnetic susceptibility, aT the thermal expansioncoefficients, aH the magnetic expansion coefficients, m the static viscosity, and b ¼ MT=d the applied temperature gradient.

The Rayleigh number Ra is the main control parameter and can be varied by several orders of magnitude, relevant valuesin the present case are Ra � 102 � 103. Typical values for P in ferrofluids are P � 100 � 103 with P � 10 for aqueous systems.The magnetic numbers are field dependent with M1 � 10�4 � 102 and M3 J 1 for typical magnetic field strengths [27]. Notethat M1 is directly proportional to H2

0, while M3 is only a weak function of the external magnetic field. Other magnetic num-bers have been suppressed, since their values are of the order 10�5 in ferrofluids, with negligible effects on the balance equa-tions [27].

3. Generalized Lorenz equations

In this section we derive a set of ordinary differential equations using a truncated Galerkin method in the same spirit asLorenz [1]. For the sake of simplicity, the analysis is limited to two-dimensional flows. In particular, we assume a two-dimen-sional pattern, which is laterally in the x-direction infinite and periodic with wave number k, describing parallel convectionrolls along the y-axis. In this case, we express the velocity field in terms of the stream function, w, defined byv ¼ f�@zw;0; @xwg. Therefore, the set of equations can be written as

P�1dtr2þw ¼ Ra ½1þM1�@xh�M1@xz/ð Þ þ RaM1 ½@xh�½@zz/� � ½@zh�½@xz/�ð Þ þ r4

þw; ð6Þdth ¼ @xwþr2

þh; ð7Þ0 ¼ ð@zz þM3@xxÞ/� @zh; ð8Þ

where dtf ¼ @t f þ ½@xw�½@zf � � ½@zw�½@xf �;r2þf ¼ @xxf þ @zzf and r4

þf ¼ @xxxxf þ 2@xxzzf þ @zzzzf . We impose idealized boundaryconditions at z ¼ ð0;1Þ for the temperature, the scalar magnetic potential and the stream function, respectively

h ¼ w ¼ @2z w ¼ @z/ ¼ 0: ð9Þ

For the numerical simulations in the lateral direction we will restrict ourselves to the fundamental mode, neglecting higherharmonics in the x-direction. This assumption can be made since we consider a large container. In the z-direction across thelayer a multimode description will be used where necessary. Higher harmonics describe deviations of the variables from thelinear regime. According to the boundary conditions we can expand the functions in the following way [1]

kwðt; z; xÞ ¼ �a1ðtÞ sinðpzÞ sinðkxÞ; ð10Þhðt; z; xÞ ¼ a2ðtÞ sinðpzÞ cosðkxÞ þ a3ðtÞ sinð2pzÞ; ð11Þ/ðt; z; xÞ ¼ a4ðtÞ cosðpzÞ cosðkxÞ þ a5ðtÞ cosð2pzÞ: ð12Þ

D. Laroze et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 2436–2447 2437


Note that similar to the Lorenz model we consider the effect of second harmonics only in the temperature (and conse-quently in the scalar magnetic potential). The second harmonic of the stream function has been neglected under the assump-tion of small convective motions [1]. This term could be important for the study of large-scale convection.

Substituting these trial functions into Eqs. (6)–(8), multiplying the equations by the corresponding orthogonal eigenfunc-tions, and integrating in space over the wavelength of a convection cell,

R pk�p

k

R 10 dxdz, yields a set of ordinary differential equa-

tions for the time evolution of the amplitudes

1P

_a1ðtÞ ¼ �q2a1ðtÞ � q4ra2ðtÞ½1�M13a3ðtÞ�; ð13Þ

_a2ðtÞ ¼ �q2a2ðtÞ � a1ðtÞ � pa1ðtÞa3ðtÞ; ð14Þ

_a3ðtÞ ¼ �4p2a3ðtÞ þp2

a1ðtÞa2ðtÞ; ð15Þ

where q2 ¼ p2 þ k2; r ¼ Ra=Ras, and M13 ¼ pk2M1M3=ðp2 þ k2½1þM1�M3Þ. Here Ras is the stationary Rayleigh number ob-

tained from linear stability analysis [20]

Ras ¼q6ðk2M3 þ p2Þ

k2 k2½1þM1�M3 þ p2� �

We remark that the equation for the scalar magnetic potential is independent of time and the magnetic amplitudes areslaved and determined by a4ðtÞ ¼ �pa2ðtÞ=ðk2M3 þ p2Þ and a5ðtÞ ¼ �a3ðtÞ=ð2pÞ. Note that this set of three differential equa-tions can be viewed as a generalized Lorenz system for ferrofluid convection. The magnetic effects in Eq. (13) appear in thenonlinear term proportional to a2a3. Note that similar systems were presented in Ref. [16] and in Ref. [17] for magneto- andelectro-convection, respectively. In the following section Eqs. (13)–(15) are analyzed in detail.

Apart from the two dimensional roll pattern considered here, the system could exhibit three dimensional patterns likesquare or hexagonal ones. To compare their stability range with the roll patterns requires a complete three dimensional anal-ysis, which is well beyond the scope of the present work.

4. Stability analysis

For the analysis of our generalized Lorenz system it is convenient to use an equivalent normalization defining a new timescale s ¼ q2t and new variables AðsÞ ¼ pa1ðsÞ=ðq2

ffiffiffi2pÞ;BðsÞ ¼ �pra2ðsÞ=

ffiffiffi2p

and CðsÞ ¼ �pra3ðsÞ. Hence, Eqs. (13)–(15) readas

A0ðsÞ ¼ PðBðsÞ � AðsÞÞ þ Q 13BðsÞCðsÞ; ð16ÞB0ðsÞ ¼ rAðsÞ � BðsÞ � AðsÞCðsÞ; ð17Þ

C0ðsÞ ¼ AðsÞBðsÞ � 4p2

q2 CðsÞ; ð18Þ

where f 0 ¼ df =ds and Q13 ¼ PM13=ðprÞ. Note that when M1 ! 0 or M3 ! 0 (Q1;3 ! 0) and q2 ! q2RB ¼ 3p2=2, the Lorenz sys-

tem is exactly recovered. We remark that the generalized system still has the reflection symmetry fA;B;Cg ! f�A;�B;Cg ofthe original Lorenz system [1], which implies that if fA;Bg are solutions so are f�A;�Bg. The latter degeneracy will not beshown explicitly in the following. We will now analyze the stability of the fixed points and perform full numericalsimulations.

The system of Eqs. (16)–(18) has the general form Y0 ¼ FðYÞ with Y ¼ fA;B;Cg. There are five stationary solutions (two ofwhich are degenerate), which are calculated from Y0 ¼ 0. The first one is the trivial, motionless solution Y1 ¼ f0;0;0g, whichexists for any value of the control parameter r. Generally, there are other solutions

A2;3 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiPgW�

p�

ffiffiffiffiUpþ P þ rQ 13

� �2ffiffiffi2p

PQ13

; ð19Þ

B2;3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiPgW�

pffiffiffi2p

Q13

; ð20Þ

C2;3 ¼rQ13 � P �

ffiffiffiffiUp

2Q 13; ð21Þ

where g ¼ 4p2=q2;W� ¼ �ffiffiffiffiUp� P � Q 13ð�2þ rÞ, and U ¼ P2 þ 2PQ13ð�2þ rÞ þ Q 2

13r2. These solutions describe stationaryconvection, only when they are real. Indeed, in the case r > 1 there is U > 0 and W�?0 meaning that one solutionY2 ¼ fA2; B2; C2g is real and describes a stationary convective state, while Y3 ¼ fA3;B3;C3g does not (since A3 and B3 are imag-inary). For r < 1 both solutions are complex, since W� < 0, and do not describe a physical realizable state. In Fig. 1 the typicalbehavior of A2 and A3 is shown as a function of r. The former state converges to the trivial one at r ¼ 1, equivalently to the

2438 D. Laroze et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 2436–2447


Fig. 1. The flow amplitudes A2 and A3 of the stationary solutions as a function of the reduced Rayleigh number r. Solid (blue) lines show the real parts, whiledashed (red) ones depict imaginary parts. Only A2 is real for r > 1 and describes the stationary, homogeneous convection state. The fixed parameters arek ¼ p=

ffiffiffi2p

; P ¼ 10;M1 ¼ 10, and M3 ¼ 1:1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of thisarticle.)

Fig. 2. Critical reduced Rayleigh number rc , where the stationary convection state becomes linearly unstable, as a function of M1 � H20. The fixed parameters

are k ¼ p=ffiffiffi2p

; P ¼ 10;M3 ¼ 1:1.

Fig. 3. (Color online) Stream function amplitude, AðsÞ, as a function of time s for three different values of r in the stationary regime forM1 ¼ 1;M3 ¼ 1:1; k ¼ p=

ffiffiffi2p

and P ¼ 10. The purple (continuous), red (dashed), and blue (dashed-dotted) curves are for r ¼ 1;3;5, respectively. (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)



Lorenz system [1], while the latter is non-zero and imaginary, due to a non-zero Q 13. We note that for r > 1 there are two realsolutions, A2 and the trivial one, A1 ¼ 0, whose stability we will consider next.

To analyze the stability of each homogeneous solution YH let us suppose that YðsÞ ¼ YH þ dYðsÞ where dYðsÞ is a fluctu-ation such that jdYj � 1. The linearized equation around YH is dY0 ¼ J � dY where J is the Jacobian matrix

J ¼�P P þ CHQ 13 BHQ 13

r � CH �1 �AH

BH AH �g

0B@

1CA: ð22Þ

The associated eigenvalue problem of J produces the secular equation (with dY0 ¼ fdY)

f3 þ p2f2 þ p1fþ p0 ¼ 0; ð23Þ

where

p2 ¼ 1þ P þ g ð24Þp1 ¼ gþ Pð1� r þ gÞ þ A2

H � B2HQ 13 þ C2

HQ13 þ CH P � rQ 13ð Þ; ð25Þp0 ¼ Pðgð1� rÞ þ A2

HÞ � B2HQ13 þ CH Pg� rgQ 13 þ 2AHBHQ 13ð Þ þ AHBH P � rQ 13ð Þ þ gC2

HQ13: ð26Þ

The stability of the stationary solution depends on the sign of f and the transition from stability to instability occurs whenthe real part of one or more of the eigenvalues passes through a zero from negative to positive.

In the case of Y1 the eigenvalues are f1 ¼ �g and 2f2;3 ¼ �1� P �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�1þ PÞ2 þ 4rP

q. The first and third ones are always

negative, while the second one changes sign from negative to positive at r ¼ 1. Since the marginal instability condition isf ¼ 0;Y1 becomes linearly unstable at rc ¼ 1, independent of the magnetic field. In the case of Y2 and Y3 the expressions

Fig. 4. Saturation value of the stream function amplitude A; jAjsat ¼ jAðs!1Þj, as a function of M1 for three different values of M3 for r ¼ 1:1; k ¼ p=ffiffiffi2p

andP ¼ 10. The orange (dashed), red (continuous), and dark-red (dashed-dotted) curves are for M3 ¼ 0:5;1:0;1:5, respectively. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. The upper frame shows the bifurcation diagram of A as a function of r. In the lower one the maximum Lyapunov exponent, kmax is plotted as afunction of r. The fixed parameters are k ¼ p=

ffiffiffi2p

; P ¼ 10;M3 ¼ 1:1 and M1 ¼ 20.



are much more complicated, which implies that analytical predictions are not tractable, but one can calculate numericallythe critical thresholds. Fig. 2 shows the critical reduced Rayleigh number, where the stationary convection state Y2 becomeslinearly unstable. This secondary instability threshold decreases with the magnetic field (M1 � H2

0), hence the magnetic fieldhas a destabilizing effect. This result is in agreement with the linear stability analysis obtained by Finlayson [20]. How theloss of stability of the stationary state is related to the onset of chaos, will be shown below.

5. Numerical simulations

In order to study numerically the dynamical behavior of our system we have integrated Eqs. (16)–(18) via a classical ex-plicit fourth order Runge–Kutta integration scheme with a fixed time step Mt ¼ 0:01 guaranteeing a precision of 10�8 for theamplitudes. For each set of parameters we let the numerical solution evolve for at least 106 time steps in order to excludetransient phenomena. In the plots, where the time dependence of a quantity is shown, we adjust the time window to therelevant dynamical properties under consideration. This system is a generalization of the Lorenz system, hence we expectthat the system can exhibit complex behavior.

In the stationary convection state, after some transient oscillations (Fig. 3), the stream function amplitude, A, takes itsconstant saturation value jAsat j ¼ jAðt !1Þj. The amplitudes of the transient oscillations as well as the saturation amplitudesincrease with the reduced Rayleigh number r (For r ¼ 1 the final amplitude is zero, of course). The saturation amplitudes arediscussed as a function of the magnetic parameters in Fig. 4. Just above the threshold (r ¼ 1:1) they increase strongly withthe magnetic field (M1) and only slightly with the magnetization nonlinearity M3.

In order to investigate how the system changes its dynamical behavior as a function of the control parameter, in partic-ular to find out what happens close and above the (secondary) instability of the stationary convection regime, we determinethe bifurcation diagram and calculate the largest Lyapunov exponent (LE). The bifurcation diagram (upper frame of Fig. 5) isobtained by taking repeatedly the maximum value of the stream function amplitude Amax in a given time interval; this isdone for a large range of different values of the control parameter r. If there is always the same Amax, then the system is peri-odic, while for finite continuous distribution of different Amax values, the behavior is either quasi-periodic or chaotic. To dis-criminate between the two latter possibilities, LEs ki defined by

ki ¼ lims!11s

lnkdYiðsÞkkdYið0Þk

� �;

are considered. LEs are numbers that quantify whether the distance between two initially close trajectories dYi of a vectorfield Y, subject to an evolution equation dYi=ds ¼ FiðY; sÞ, vanishes (LE negative) or diverges exponentially (LE positive). Thelatter is the hallmark of a chaotic behavior. Our basically 3-dimensional phase space carries 3 LEs [58–62], which can be or-dered in descending form, with the largest Lyapunov exponent denoted by kmax. The error Err in the evaluation of the LEs hasbeen checked by using Err ¼ r kMð Þ=max kMð Þ, where rðkMÞ is the standard deviation of kmax. In all cases studied here Err is ofthe order of 1%, which is sufficiently small for the purpose of the present analysis.

Fig. 6. The top frame shows the 3D phase portrait of fA;B;Cg in the chaotic regime at r ¼ 15. The bottom frame shows the corresponding Fourier powerspectrum of A and, as inset, the appropriate time series. The fixed parameters are the same as in Fig. (5).



6. Results

In Fig. 5 the transition from the stationary to the chaotic behavior at r � 7:8 is clearly visible, since Amax becomes contin-uous and kmax positive above this secondary instability. The chaotic regime, however, is interrupted by (two smaller) regimes(at roughly r ¼ 45 and 50) and a larger regime (between r � 75 and r � 85), where the system is regular (e.g. kmax ¼ 0). For

Fig. 7. Color coded temperature profiles and streamlines as a function of the spatial coordinates x and z at three different times in the chaotic regime atr ¼ 15. Red/ upper half layer (blue/ lower half layer) mean hotter (colder) regions. The fixed parameters are k ¼ p=

ffiffiffi2p

; P ¼ 10;M3 ¼ 1:1, and M1 ¼ 20. Thefull time evolution is shown in a movie [63]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version ofthis article.)



very high values of the control parameter, r J 116, chaos is suppressed. However, there is the possibility that chaotic statesreappear for even higher r values, beyond those we have considered here, in a way similar to the Lorenz model [64–66]. Inthe following, we will discuss as examples the chaotic dynamic behavior at r ¼ 15 and the periodic one in the large regularwindow (r ¼ 82). Finally, we investigate transient chaotic regimes that are pronounced for intermediate magnetic fields.

Figs. 6–8 show the system in the chaotic regime. In the top frame of Fig. 6 the 3D phase portrait reveals a strange attractorof similar shape as the Lorenz attractor. In the bottom frame the Fourier power spectrum of the stream function amplitude Aand its corresponding time series (inset) is shown. The time dependence is aperiodic and, as a consequence, the Fourierpower spectrum is continuous, characteristic for chaotic behavior. To compute the Fourier spectrum in the chaotic regimewe have done the calculations for 50 different random initial conditions. In Fig. 7 the temperature profile and the streamlines

Fig. 8. Magnetic field lines appropriate to the temperature profiles and streamlines shown in Fig. 7.



in the chaotic regime are depicted as a function of the spatial coordinates x and z for three different times. A movie, includedas complementary material, demonstrates the chaotic nature of the temperature profile as a function of space and time [63].In Fig. 8 the appropriate magnetic field profiles are shown.

In Fig. 9 we show one example of behavior in the regular window at r ¼ 82. The top frame shows the 3D phase portraitand in the bottom one the Fourier power spectrum of A and its corresponding time series (inset) is presented. We observethat the trajectory is a closed orbit and there are only discrete peaks in the Fourier spectrum, which is expected for a regular(periodic or quasi-periodic) motion.

Fig. 9. The top frame shows 3D phase portrait of fA; B;Cg in the periodic window at r ¼ 82. The bottom frame shows the corresponding Fourier powerspectrum of A and, as inset, the time series of A. The fixed parameters are k ¼ p=

ffiffiffi2p

; P ¼ 10;M3 ¼ 1:1, and M1 ¼ 20.

Fig. 10. (Color online) Phase diagram displaying the largest Lyapunov exponent color coded as a function of the field amplitudes M1 and M3 fork ¼ p=

ffiffiffi2p

; P ¼ 10 and r ¼ 7:8. The resolution is 4M1 ¼ 0:1 and 4M3 ¼ 0:05.



Above, we have discussed the dynamic behavior as a function of the applied temperature gradient. We now switch to thedependence on the external magnetic field. For somewhat smaller external fields and larger magnetic field nonlinearity, thetransition between the stationary regime and the chaotic one is more complicated. Fig. 10 shows a color-coded kmax phasediagram as a function of both magnetic numbers, M1 and M3 for a fixed value of r. For small values of M1 and M3 kmax is neg-ative and the system regular, while for larger values of these parameters there is chaos. However, the transition region is notsharp, but diffuse, presenting multiple transitions between chaotic to regular motion. In order to understand this transitionthree cuts of the phase diagram at different values of M1 are plotted in Fig. 11. For low M1 there is no transition, while forlarge M1 the transition at increasing M3 is almost direct from stationary to chaotic. In the intermediate case, M1 ¼ 10 there isa series of stationary-chaos transitions in a broad range of M3 values, before for larger M3 the chaotic state prevails. For oneof these intermediate stationary states at M3 ¼ 4:15, where kmax ¼ �0:0461;AðsÞ is shown in Fig. 12. There is a transient cha-otic behavior preceding the stationary state. Such transitions take place at a time scale that is by a factor of 20 smaller thanour maximum integration time. Changing slightly M3, a true chaotic state with positive kmax is found.

For the material parameters (M1 ¼ 20;M3 ¼ 1:1) chosen in the phase diagram Fig. 5 a higher resolution picture (Fig. 13)shows a very tiny regime (r � 7:7� 7:8) with a few transient chaotic solutions, before for r J 7:8 the chaotic regime isreached. Note, that this transition occurs well before the linear stability of the stationary state breaks down at rc ¼ 9:1, cf.Fig. 2. This phenomenon that a strange attractor appears at a reduced Rayleigh number smaller than the critical thresholdrc from the linear stability analysis, also happens in the Lorenz system [64].

Finally, we have looked at the Prandtl number dependence. Fig. 14 shows a color-coded kmax phase diagram as a functionof the magnetic number M1 and the Prandtl number P. Here, the chaotic region occurs in a compact pattern of a rather char-acteristic shape. For small P K 4 there is no chaos for any M1. Similarly, chaos is suppressed for high P values, where the rel-evant P decreases with decreasing M1.

Fig. 11. Largest Lyapunov exponent, kmax , as a function of M3 for M1 ¼ 5 (red, dotted curve), M1 ¼ 10 (blue, continuous curve), and M1 ¼ 20 (purple, dash-dotted curve), for k ¼ p=

ffiffiffi2p

; P ¼ 10, and r ¼ 7:8. (For interpretation of the references to colour in this figure legend, the reader is referred to the web versionof this article.)

Fig. 12. Time series of A in a chaotic transient regime for k ¼ p=ffiffiffi2p

; P ¼ 10; r ¼ 7:8;M1 ¼ 10 and M3 ¼ 4:15.

Fig. 13. Higher resolution version of the phase diagram Fig. 5 near the stationary to chaotic transition (k ¼ p=ffiffiffi2p

; P ¼ 10; r ¼ 7:8;M1 ¼ 10 and M3 ¼ 4:15).



7. Summary

We have studied the nonlinear convection of a ferrofluid in the two-dimensional spatial case. In particular we have de-rived a set of three ordinary nonlinear differential equations, which describe as a minimal model the complex dynamicbehavior in the presence of an external magnetic field. Without a magnetic field the classical Lorenz model is recovered.We have identified parameter regions, where stationary states or those with chaotic or regular dynamics occur, using theLyapunov exponent method, bifurcation diagrams, and phase portraits. We have performed intensive numerical simulationto get time series and power spectra of the stream function amplitude as well as spatial temperature profiles in the chaoticregime. We have found that the system has multiple transitions between regular and chaotic behavior in parameter space.Close to the transition from the stationary to a chaotic state, which occurs slightly below the linear stability boundary, thestationary states show transient chaotic behavior in time. Finally, we remark that our generalized Lorenz system has certainsimilarities with the one found for dielectric liquids subject to AC electric fields [17]; the complete comparison between bothsystems will be presented in future works.

Acknowledgments

D.L. acknowledges the partial financial support from FONDECYT 1120764, Millennium Scientific Initiative, P10-061F, Ba-sal Program Center for Development of Nanoscience and Nanotechnology (CEDENNA) and UTA-project 8750-12. P.G.S. isgrateful to Bangalore University for encouraging his research.

References

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Fig. 14. (Color online) Phase diagram displaying the largest Lyapunov exponent kmax color coded as a function of the field amplitude M1 and the Prandtlnumber P for k ¼ 3; r ¼ 15 and M3 ¼ 1:1. The resolution is 4P ¼ 0:2 and 4M1 ¼ 0:2.



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Chaotic convection in a ferrofluid

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