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Interactions of resonances and global bifurcations in Rayleigh-Benard convection
Volume 131, number 6
PHYSICS LETTERS A
29 August 1988
INTERACTIONS OF RESONANCES AND GLOBAL BIFURCATIONS IN RAYLEIGH-BENARD CONVECTION Robert E. ECKE Los Alamos National Laboratory, LosAlamos, NM 87545a, USA
and loannis G. KEVREKIDIS Princeton University, Princeton, NJ 08544, USA Received 24 November 1987; revised manuscript received 14 June 1988; accepted for publication 20 June 1988 Communicated by D.D. HoIm
We present detailed two-parameter experimental studies ofhysteresis and overlapping ofresonances and secondary Hopfbifur3He in superfluid 4He. The cations in the quasiperiodic regime of low-aspect-ratio Rayleigh—Benard convection in solutions of global bifurcations inferred from this experiment are similar to those observed in numerical studies of two-parameter families of maps of the plane with nonconstantjacobian.
The study of low-dimensional dynamic behavior and bifurcations in distributed systems through theory, computation, and experiment is an area of extensive research, particularly in the light of recent advances in the theory of dynamical systems. Rayleigh—Benard experiments in small-aspect-ratio convection cells put emphasis on the excitation of temporal degrees of freedom in a spatially coherent system and are thus ideal for the study of dynamical phenomena. In such systems, the experimental study of quasiperiodic states and the associated modelocking leading to chaos, has been of great recent interest [1—4].Theoretical interpretations of these experiments have been based primarily on 1 -D maps such as the standard circle map [51. The experimental results which we report here are incompatible with such 1 -D maps over a large region of parameter space. The results presented here and results of other experiments [3,41consist of complex dynamic behavior which necessitate comparisons with maps of two (or more) dimensions. We will show that there is good qualitative agreement between the bifurcation behavior of two-parameter families of maps of the plane [6—8] and the bifur344
cation behavior we observe experimentally in Rayleigh—Benard convection of 3He—superfluid 4He mixtures. For the two-dimensional maps these bifurcations involve global homoclinic and heteroclinic interactions of invariant manifolds of saddletype periodic solutions. Because of the close correspondence between the observed features of our experiment and predictions of numerical investigations we expect these low-dimensional mechanisms to also apply in our infinite-dimensional experimental system. Therefore, in the experimental regime under consideration here, the system follows no single standard route to turbulence [5,91.Instead, on any onedimensional path through the parameter space we expect, based on the results of Aronson et al. [61, to encounter a rapid alternation between regions where the attractor is smooth, where it has various degrees of roughness, and regions where it is downright strange. This rapid alternation between regular and chaotic regions is characteristic of the breakup of tori [101. We describe in this Letter a detailed experimental two-parameter investigation of the quasiperiodic regime of Rayleigh—Benard convection in dilute so-
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PHYSICS LETTERS A
lutions of 3He in superfluid 4He where the flow is characterized by two distinct frequencies [3]. This behavior is obtained after a sequence of transitions: from conduction to steady convection, to oscillatory convection (one frequency), to quasiperiodic convection via a Hopf bifurcation (two distinct frequencies) and finally through a hysteretic “jump” to a different quasiperiodic state. These transitions are described in detail by Haucke and Ecke [3]. The important qualitative dynamical phenomena in this regime include the existence of smooth invariant tori that either have quasiperiodic attractors winding around them (ratio of the two frequencies irrational) or are frequency-locked (ratio of the two frequencies rational). These periodic, frequency-locked solutions are referred to as resonances, and in a twoparameter study they exist in regions of parameter space called Arnol’d horns [11]. In this Letter we initially focus on the experimental observation and theoretical interpretation of hysteretic effects arising from the overlapping of resonances of different order. The overlapping of the horns implies the occurrence of global manifold interactions which break the quasiperiodic attractor we observe outside the resonance horns. We also experimentally observe secondary Hopf bifurcations (SHB) inside the Arnol’d horns, and discuss their occurrence in terms of two-dimensional maps. Finally, we use the analogy between the experiment and the map model to design further experiments for the elucidation of the system dynamics. The experiments were performed using a lowaspect-ratio convection cell (height h = 0.80 cm, length 2.00 x h and 1.40 x h) 4He filledat with a so3Hewidth in superfluid temperalution typically of 1.46% around 0.85 K. For the type of tures solutions that we examine, the system behaves as a single component fluid. The apparatus used in this experiment and the single component approximation for the solutions have been described in detail elsewhere [121. One dimensionless quantity underlying the development of instabilities in this system is the Rayleigh number R which is proportional to the temperature difference across the fluid layer. The second important dimensionless parameter in Rayleigh—Benard convection is the Prandtl number a, a ratio of the thermal to viscous diffusion times of the convecting fluid. In our experiment a is small
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region cally shaded labelled regions “S.H.B.” indicate refers where to the hysteresis secondary was Hopf observed. bifurcaThe tions discussed in the text.
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Fig. 2. Inverse Prandtl number 1/u versus no~a1izedRayleigh number R/RC. Various locking regions are labelled. The verti-
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R/RC Fig. 1. Rotation number W—_f
2/f1 versus normalized Rayleigh number R/RC. Each curve represents a scan at fixed Prandtl number a (1/u is used for better representation) with values of Wfor increasing R marked by ., while those with decreasing R are marked by ~. The curves are offset for clarity and the W scale is indicated on the right-hand side. The largest locking is W= 2/13; all other major lockings are labelled,
tained by starting with a low value of R, increasing R incrementally to a specified high value, and then incrementallydecreasing R back to its original value. In fig. 1 measurements made for increasing R are differentiatedfrom those made for decreasing R. This distinction is not made for the runs at a> 0.0683 (l/a< 14.64) since in those cases there is no discernible difference between the measurements for increasing or decreasing R. For a~0.0683 (1 / a~14.64) there are hysteresis effects indicated by the vertical arrows in fig. 1; at some points the value of W depended upon whether it was obtained for increasing or decreasingR. Fig. 2 summarizes the results of thirteen runs (including those described in fig. 1), showing the regions in the R/RC— 1/a parameter space (R~= 2000) where W has various rational values. We will refer 346
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to these regions as Arnol’d horns, or W-horns when we want to emphasize the specific value of W. The hysteresis shown in fig. 1 corresponds to the overlapping of horns illustrated in fig. 2. Fig. 2 also shows that the distribution of the W-horns appears to be in accord with the Farey sequence [3,13]. For example, the 7/46-horn occurs between the 5/33- and 2 / 13-horns. This distribution has also been observed in convection experiments where the circle map provides an adequate description of system dynamics [2]. An additional feature of the data which is not revealed in the winding number plots of fig. 1 is a seeondary instability of the 2/13 resonance horn. This instability is observed in the power spectrum where the apperance of a new spectral peak corresponds to the development of small invariant tori surrounding the stable frequency-locked periodic points. This phenomenon, first observed for convection in experiments by Haucke and Ecke [3], is best described through Poincaré section analysis. Recall that, according to the theory ofPackard et al. and of Takens [141, if the attractor for the öT( t) -flow is finite dimensional then we can represent it in a phase space with coordinates given by öT(t), öT(i—t1) öT( t— IN) for a suitable number N of delays. The fact that the power spectra derived from 8T(t) can be completely described in terms of two independent frequencies indicates that two delays will suffice here. .
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ondary torus shrinks again to give a stable 2/13 periodic solution. In the hysteretic regime adjacent to the right-hand boundary of the 2/13-horn this 2/13 periodic solution (point I, fig. 4e) is shown to coexist with a 5 / 32 periodic solution (point J, fig. 4f). Finally, we recover an attractor similar to the smooth torus observed outside the left-hand bound-
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I 2/13 I
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_________________________________________ R/R~ Fig. 3. Rotation number W versus R/R, for a=0.0667 (1 /u= 14.99). The nature of the attractor at representative P~.’ rameter values (indicated by capital letter A—J) is presented in figs. 4 and 5 and discussed in the text.
The intersections of the resulting three-dimensional trajectories with a transverse plane generate a Poincare section. Fig. 3 shows a one-parameter bifurcation diagram across the 2/13 resonance horn obtained for a=0.0667 (1/a= 14.99). Figs. 4 and 5 show Poincaré sections (constructed using the above procedure with sample time 50 ms, 11= 1 s and 12 = 2 s) obtained at representative parameter values labeled in fig. 3. Outside the resonance horn (point A, fig. 4a) we observe a quasiperiodic attractor winding around a torus whose Poincaré section appears smooth. As R increases we obtain two coexisting attractors in the hysteretic regime: a frequency-locked periodic solution (W=7/46, point B, fig. 4b) coexisting with the periodic solution belonging to the 2/13 resonance horn (point C, fig. 4c). As R/RC increases the hysteretic region comes to an end. This occurs probably through global interactions of the invariant manifolds of the saddle-type (unstable) frequency-locked periodic solutions, as suggested by numerical experiments in simple maps or periodically forced oscillators in similar regimes. The stable, frequency-locked 2/13 periodic solution undergoes a secondary Hopf bifurcation (as two of its characteristic multipliers cross the unit circle in the cornplex plane) and a secondary torus surrounding it appears (point E, fig. 4d). Eventually a reverse seeondary Hopf bifurcation is observed, and the see-
ary of the 2/13-horn. Fig. 5 present a more detailed view of the secondary Hopf bifurcation regime. The stable 2/13 periodic solutiongiving (point C, fig. undergoes a Hopf bifurcation, birth to a 5a) stable secondary torus (point D, fig. Sb). This secondary torus is initially small in amplitude and surrounds the now unstable 2/13 periodic solution. It subsequently grows in amplitude (point E, fig. Sc) and exhibits its own frequency lockings (point F, fig. Sd) and possibly its own Arnol’d horn structure. The secondary tori then .
become smaller in amplitude (point G, fig. Se) and we observe a reverse secondary Hopf bifurcation where they shrink back to yield stable 2/13 periodic solutions (point H, fig. Sf). The nature of this secondary Hopf bifurcation is elucidated further in fig. 6, where we present a sequence of Poincaré sections showing the transient dynamic behavior of the system before it relaxes to the attractor. Three such Poincaré sections are shown, restricted for clarity around one of the periodic points of the 2/13 periodic solution. The smooth curves through every third point on the transient illustrate the order of iteration of the points and are not trajectories ofthe system. We connect every third point because the average rotation per iterate is approximately 2it/3. Remember that this average rotation rate is proportional to the imaginary part of the (complex) Floquet multipliers of the periodic solution. In fig. 6a the attractor observed is a torus resulting from a secondary Hopfbifurcation within the 2/13 resonance horn. As we increase R we cross the right-hand-side boundary of the SHB curve, and the attractor becomes periodic. Close to this transition boundary a slower transient spiralling approach to the periodic orbit is observed (see fig. 6b). This is an indication that the norm of the principal Floquet multipliers is close to the unit circle, signalling the proximity of a (secondary) Hopf bifurcation. In fig. 6c, far from the SHB boundary the attractor remains a periodic point. The transient approach to this pe347
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(e)
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Fig. 4. Experimental Poincaré sections of the attractor reconstructed with two time delays at representative points of the bifurcation diagram in fig. 3: (a) quasiperiodic attractor, (b) locked, W=7/46, (c) locked, W=2/l3, (d) secondary tori, (e) locked, W=2/13, and (f) locked, W=5/32.
riodic point spirals much less than the transient in fig. 6b. The experimental technique of utilizing Poincaré sections of transients to extract quantitative stability information for the periodic points and elucidate the secondary Hopf bifurcations may be applicable to the study of the more complex dynamics described below, 348
The experimental results given in figs. 1 and 2 are strikingly similar to results obtained by Aronson et al. [6] and Johnson [7] in their analytical and nurnerical studies oftwo-parameter families of maps of the plane, and by Kevrekidis et al. [81 in their numerical studies of stroboscopic maps generated by forced oscillators. These similarities strongly suggest
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(b)
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Fig. 5. Experimental Poincaré sections of the attractor reconstructed with two time delays showing the gradual development and closing of the secondary Hopf bifurcations along the bifurcation diagram in fig. 3: (a) locked, W= 2/13, (b) small secondary tori, (c) large secondary tori, (d) locked secondary tori, (e) small secondary tori, and (f) locked, W= 2/13.
that in the experimental regime considered here, the system can be modelled by a two-parameter family of planar maps with nonconstant jacobian. The experimental Poincaré sections of the reconstructed attractors (presented in figs. 4 and 5) as well as those of the transient approach to an attractor (presented in fig. 6) are also consistent with the behavior of the diffeomorphisms studied in refs. [6—81.A suitably parameterized family of standard circle maps would also exhibit some of the qualitative features we have observed in our experiments [3}. Circle maps, however, do not exhibit secondary Hopf bifurcations or the frequency lockings of these secondary tori and cannot be diffeornorphisms in the parameter region
where hysteresis occurs. Moreover, Boyland has proved that an Arnol’d horn for a circle map cannot contain both boundaries ofan overlapping horn [15]. This would rule out the overlap of the 2/13-horn and the 5/32-horn shown in fig. 2 for l/a~i15. Thus, the experimental data are better represented by a planar map model than by 1 -D circle maps. The planar map model not only describes many features of the experimental data, but also makes, as we will see, specific important predictions about systern dynamics in the hysteresis and secondary Hopf bifurcation regions of parameter space. Although there are some differences between model and experiment in the simple regions of the horn, we ob349
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(a)
X(t+T)
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Fig. 6. Poincaré sections showing transientdecay to attractor: (a) approach to secondary tori, R/R~=11.15, (b) transient approachto period-two (W=2/13) attractor, R/RC = 11.18, and (c) transient approach to period-two (W= 2/13) attractor, R /R~= 11.21.
serve that the analogy applies well in the resonance overlap region. Consequently we consider a twoparameter family of diffeomorphisms of the plane {Mab} each of which has a unique fixed point Pa,b which undergoes a Hopf bifurcation on a curve r in the a—b parameter space. That is, the eigenvalues of the differential DMa,b (Pat,) cross from the inside to the outside of the complex unit circle as the parameter point (a, b) crosses F in the appropriate direction. If the crossing occurs at (a*, b*) e F and if the eigenvalues of DMa~,b*are p/q-th roots of unity (p, q relatively prime integers with q ~ 5) then, as shown by Arnol’d and by Takens [6], there is a cusped region (Arnol’d horn) emanating from the point (a*, b*) in the parameter plane in which all points on the attractor have rotation number p/q. Moreover, in a sufficiently small neighborhood of (a*, b*) none of the horns starting on F overlap, and inside each of these horns the attractor is a smooth invariant circle consisting of q sinks and q saddles connected by the unstable manifolds of the saddles (see figs. 7a and 7b). The Arnol’d—Takens theory only establishes the existence of the horns in a small neighborhood ofthe 350
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(b)
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•
&
Fig. 7. Schematic drawings of: (a) several Arnol’d horns emanating from a bifurcation curve F close to which the horns do not overlap (shaded region); (b) a smooth invariant circle involving three sinks and three saddles; (c) an invariant set involving three sinks and three saddles showing homoclinic crossings of the stable and unstable manifolds of the saddles. Representative Poincare sections in the left- (d) and right-hand-side (f) hysteretic regions aswell as in the nonhysteretic interior (e) of a resonance horn for a model periodically-forced chemical oscillator.
bifurcation curve r. The horns can, however, be extended numerically. As the parameter point moves into the horn away from F the invariant set which is originally a smooth circle progressively loses its smoothness and ultimately loses its topological type (i.e. it is no longer a topological circle) [6—8].The evolution ofthe invariant set as the parameter point moves through a fixed horn is studied in detail in ref. [6] where it is shown that one of the main mechanisms involved in the breakup of the invariant circle is the formation of heteroclinic and homoclinic crossings. Schematic homoclinic crossings of the stable and unstable manifolds of periodic saddles are shown in fig. 7c. In the subset ofthe horn where these homoclinic crossings occur, other horns, conesponding to rotation numbers distributed according to the Farey sequence may overlap the given horn, and hysteretic behavior is observed numerically.
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PHYSICS LETTERSA
Poincaré sections representative of this regime obtamed from a periodically-forced brusselator [8] are also shown in fig. 7. Fig. 7d is obtained in the hysteretic regime adjacent to the left-hand-side boundary of a resonance horn and shows the coexistence of two attractors (periodic points and a smooth circle). Fig. 7e is obtained in the interior of the resonance horn, where no hysteresis is observed and the attracting set consists ofthe periodic points. Finally, fig. 7f is obtained in the hysteresis region adjacent to the right-hand-side boundary of the resonance horn, where again two attractors (periodic points and a smooth circle) coexist. The transition between states illustrated in figs. 7d and 7e and the transition between states in figs. 7e and 7f involve homoclinic crossings of the stable and unstable manifolds of the periodic saddles, similar to those schematically depicted in fig. 7c. Aronson et al. [6] prove that under quite general hypotheses the existence of homoclinic crossings such as those shown in fig. 7c, implies the existence of an interval I such that for every real number re I there is at least one point on their attracting set whose rotation number is r. Moreover, there are also points on this attractor whose rotation number does not exist. Thus, the theory of parameterized planar maps predicts the existence of subsets of the Arnol’d horns where the dynamics are quite complicated. In these subsets Arnol’d horns overlap so that the attractor contains periodic points with countably infinite different rational rotation numbers, as well as possibly smooth invariant circles and chaotic subsets. We expect that most of these states would have vanishingly small stability intervals and would not be observable experimentallydue to the presence of noise. Nevertheless, multiple (greater than two) hysteretic states might be seen in experiment and we plan to investigate this prediction ofthe planar map model. In a recent publication [16] Brunsden and Holmes predict the power spectral density for strange attractors in a system possessing transverse homoclinic orbits. It may prove possible to establish the existence of such orbits by examining the qualitative characteristics of the power spectrum close to the inner hysteresis boundaries of the resonance horns. ,The theory ofpararneterized planar maps also prediets that the invariant circles resulting from seeondary Hopf bifurcations may break due to
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homoclinic interactions. Experimental Poincaré seetions are consistent with that prediction but more data, in particular, data employing the transient techniques described above, are necessary for verification of the predictions. These results will be presented elsewhere. In summary, our experimental observations of hysteresis, overlapping of Arnol’d horns and secondary Hopf bifurcations are consistent with numerical observations and theoretical predictions obtained from two-parameter maps of the plane with nonconstant jacobian. We distinguish two main features of the proposed model of our experimental system in terms of a family of maps. The first is the qualitative prediction by the model of the fine structure of the Arnol’d horns, including the hysteretic regime and the secondary Hopf bifurcations, along with suggested mechanisms underlying these bifurcations, especially the existence of homoclinic tangles. This correspondence implies that in the parameter region examined here the dynamic behavior of this infinite dimensional system is low-dimensional. The second important feature is that in our parameter region the system follows no single standard route to turbulence. Indeed, in agreement with the predictions of Aronson et al. [6] on any one-dimensional path through the parameter space, we generally encounter a rapid alternation between regions where the attractor is smooth, where it has various degrees of wrinkling, and regions where it is chaotic. This work was supported by funds provided by the U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Science. We would like to acknowledge the experimental assistance of Hans Haucke and many valuable discussions with Donald Aronson and Richard McGehee.
References [l]J.P.
Gollub and S.V. Benson, J. Fluid Mech. 100 (1980)
[2] J. Stavans, F. Heslot and A. Libchaber, Phys. Rev. Lett. 55 (1985) 596. [3] H. Haucke and R.E. Ecke, Physica D 25 (1987) 307. [4] M. Dubois and P. Berge, Phys. Scr. 33 (1986)159. [5] Si. Shenker, Physica D 5 (1982) 405; M.H. Jensen, P. Bak and T. Bohr, Phys. Rev. Lett 50 (1983) 1637.
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[6] D.G. Aronson, M.A. Chory, G.R. Hall and R.P. McGehee, Commun. Math. Phys. 83 (1982) 303. [7] J.P. Johnson, Ph.D. Thesis, University of Minnesota (1985); D.G. Aronson, J.P. Johnson and R.P. McGehee, in preparation. [8]1G. Kevrekidis, R. Aris and L.D. Schmidt, Chem. Eng. Sci. 41(1986)905; 1G. Kevrekidis, Am. Inst. Chem. Eng. J., to be published. [91J.-P. Eckmann, Rev. Mod. Phys. 53 (1981) 643. [10] S. Ostlund, D. Rand, J. Sethna and E. Siggia, Physica D 8 (1983) 303. [111VI. Arnol’d, Funct. Anal. Appl. 11(1977) 2-1;
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F. Takens, Commun. Math. Inst. Rijksuniversiteit Utrecht 3 (1974) 1. [12] Y. Maeno, H. Haucke, R.E. Eckeand S.C. Wheatley, J. Low Temp. Phys. 59(1985)305. [13]H. Rademacher, Lectures on elementary number theory (Blaisdel, New York, 1964). [14]N.H. Packard, J.P. Ci-utchfield, iD. Farmerand R.S. Shaw, Phys. Rev. Lett. 45 (1980) 712; F. Takens, Lecture notes in Mathematics, Vol. 898 (Spnnger, Berlin, 1981). [15] Boyland, and Commun. Math.Phys. Phys.Rev. 106 (1986). [16] P.L. V. Brunsden P. Holmes, Lett. 58 (1987) 1699.
PHYSICS LETTERS A
29 August 1988
INTERACTIONS OF RESONANCES AND GLOBAL BIFURCATIONS IN RAYLEIGH-BENARD CONVECTION Robert E. ECKE Los Alamos National Laboratory, LosAlamos, NM 87545a, USA
and loannis G. KEVREKIDIS Princeton University, Princeton, NJ 08544, USA Received 24 November 1987; revised manuscript received 14 June 1988; accepted for publication 20 June 1988 Communicated by D.D. HoIm
We present detailed two-parameter experimental studies ofhysteresis and overlapping ofresonances and secondary Hopfbifur3He in superfluid 4He. The cations in the quasiperiodic regime of low-aspect-ratio Rayleigh—Benard convection in solutions of global bifurcations inferred from this experiment are similar to those observed in numerical studies of two-parameter families of maps of the plane with nonconstantjacobian.
The study of low-dimensional dynamic behavior and bifurcations in distributed systems through theory, computation, and experiment is an area of extensive research, particularly in the light of recent advances in the theory of dynamical systems. Rayleigh—Benard experiments in small-aspect-ratio convection cells put emphasis on the excitation of temporal degrees of freedom in a spatially coherent system and are thus ideal for the study of dynamical phenomena. In such systems, the experimental study of quasiperiodic states and the associated modelocking leading to chaos, has been of great recent interest [1—4].Theoretical interpretations of these experiments have been based primarily on 1 -D maps such as the standard circle map [51. The experimental results which we report here are incompatible with such 1 -D maps over a large region of parameter space. The results presented here and results of other experiments [3,41consist of complex dynamic behavior which necessitate comparisons with maps of two (or more) dimensions. We will show that there is good qualitative agreement between the bifurcation behavior of two-parameter families of maps of the plane [6—8] and the bifur344
cation behavior we observe experimentally in Rayleigh—Benard convection of 3He—superfluid 4He mixtures. For the two-dimensional maps these bifurcations involve global homoclinic and heteroclinic interactions of invariant manifolds of saddletype periodic solutions. Because of the close correspondence between the observed features of our experiment and predictions of numerical investigations we expect these low-dimensional mechanisms to also apply in our infinite-dimensional experimental system. Therefore, in the experimental regime under consideration here, the system follows no single standard route to turbulence [5,91.Instead, on any onedimensional path through the parameter space we expect, based on the results of Aronson et al. [61, to encounter a rapid alternation between regions where the attractor is smooth, where it has various degrees of roughness, and regions where it is downright strange. This rapid alternation between regular and chaotic regions is characteristic of the breakup of tori [101. We describe in this Letter a detailed experimental two-parameter investigation of the quasiperiodic regime of Rayleigh—Benard convection in dilute so-
0375-960l/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 131, number 6
PHYSICS LETTERS A
lutions of 3He in superfluid 4He where the flow is characterized by two distinct frequencies [3]. This behavior is obtained after a sequence of transitions: from conduction to steady convection, to oscillatory convection (one frequency), to quasiperiodic convection via a Hopf bifurcation (two distinct frequencies) and finally through a hysteretic “jump” to a different quasiperiodic state. These transitions are described in detail by Haucke and Ecke [3]. The important qualitative dynamical phenomena in this regime include the existence of smooth invariant tori that either have quasiperiodic attractors winding around them (ratio of the two frequencies irrational) or are frequency-locked (ratio of the two frequencies rational). These periodic, frequency-locked solutions are referred to as resonances, and in a twoparameter study they exist in regions of parameter space called Arnol’d horns [11]. In this Letter we initially focus on the experimental observation and theoretical interpretation of hysteretic effects arising from the overlapping of resonances of different order. The overlapping of the horns implies the occurrence of global manifold interactions which break the quasiperiodic attractor we observe outside the resonance horns. We also experimentally observe secondary Hopf bifurcations (SHB) inside the Arnol’d horns, and discuss their occurrence in terms of two-dimensional maps. Finally, we use the analogy between the experiment and the map model to design further experiments for the elucidation of the system dynamics. The experiments were performed using a lowaspect-ratio convection cell (height h = 0.80 cm, length 2.00 x h and 1.40 x h) 4He filledat with a so3Hewidth in superfluid temperalution typically of 1.46% around 0.85 K. For the type of tures solutions that we examine, the system behaves as a single component fluid. The apparatus used in this experiment and the single component approximation for the solutions have been described in detail elsewhere [121. One dimensionless quantity underlying the development of instabilities in this system is the Rayleigh number R which is proportional to the temperature difference across the fluid layer. The second important dimensionless parameter in Rayleigh—Benard convection is the Prandtl number a, a ratio of the thermal to viscous diffusion times of the convecting fluid. In our experiment a is small
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region cally shaded labelled regions “S.H.B.” indicate refers where to the hysteresis secondary was Hopf observed. bifurcaThe tions discussed in the text.
I•_
14.20
14.14
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Fig. 2. Inverse Prandtl number 1/u versus no~a1izedRayleigh number R/RC. Various locking regions are labelled. The verti-
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R/RC Fig. 1. Rotation number W—_f
2/f1 versus normalized Rayleigh number R/RC. Each curve represents a scan at fixed Prandtl number a (1/u is used for better representation) with values of Wfor increasing R marked by ., while those with decreasing R are marked by ~. The curves are offset for clarity and the W scale is indicated on the right-hand side. The largest locking is W= 2/13; all other major lockings are labelled,
tained by starting with a low value of R, increasing R incrementally to a specified high value, and then incrementallydecreasing R back to its original value. In fig. 1 measurements made for increasing R are differentiatedfrom those made for decreasing R. This distinction is not made for the runs at a> 0.0683 (l/a< 14.64) since in those cases there is no discernible difference between the measurements for increasing or decreasing R. For a~0.0683 (1 / a~14.64) there are hysteresis effects indicated by the vertical arrows in fig. 1; at some points the value of W depended upon whether it was obtained for increasing or decreasingR. Fig. 2 summarizes the results of thirteen runs (including those described in fig. 1), showing the regions in the R/RC— 1/a parameter space (R~= 2000) where W has various rational values. We will refer 346
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to these regions as Arnol’d horns, or W-horns when we want to emphasize the specific value of W. The hysteresis shown in fig. 1 corresponds to the overlapping of horns illustrated in fig. 2. Fig. 2 also shows that the distribution of the W-horns appears to be in accord with the Farey sequence [3,13]. For example, the 7/46-horn occurs between the 5/33- and 2 / 13-horns. This distribution has also been observed in convection experiments where the circle map provides an adequate description of system dynamics [2]. An additional feature of the data which is not revealed in the winding number plots of fig. 1 is a seeondary instability of the 2/13 resonance horn. This instability is observed in the power spectrum where the apperance of a new spectral peak corresponds to the development of small invariant tori surrounding the stable frequency-locked periodic points. This phenomenon, first observed for convection in experiments by Haucke and Ecke [3], is best described through Poincaré section analysis. Recall that, according to the theory ofPackard et al. and of Takens [141, if the attractor for the öT( t) -flow is finite dimensional then we can represent it in a phase space with coordinates given by öT(t), öT(i—t1) öT( t— IN) for a suitable number N of delays. The fact that the power spectra derived from 8T(t) can be completely described in terms of two independent frequencies indicates that two delays will suffice here. .
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ondary torus shrinks again to give a stable 2/13 periodic solution. In the hysteretic regime adjacent to the right-hand boundary of the 2/13-horn this 2/13 periodic solution (point I, fig. 4e) is shown to coexist with a 5 / 32 periodic solution (point J, fig. 4f). Finally, we recover an attractor similar to the smooth torus observed outside the left-hand bound-
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I 2/13 I
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_________________________________________ R/R~ Fig. 3. Rotation number W versus R/R, for a=0.0667 (1 /u= 14.99). The nature of the attractor at representative P~.’ rameter values (indicated by capital letter A—J) is presented in figs. 4 and 5 and discussed in the text.
The intersections of the resulting three-dimensional trajectories with a transverse plane generate a Poincare section. Fig. 3 shows a one-parameter bifurcation diagram across the 2/13 resonance horn obtained for a=0.0667 (1/a= 14.99). Figs. 4 and 5 show Poincaré sections (constructed using the above procedure with sample time 50 ms, 11= 1 s and 12 = 2 s) obtained at representative parameter values labeled in fig. 3. Outside the resonance horn (point A, fig. 4a) we observe a quasiperiodic attractor winding around a torus whose Poincaré section appears smooth. As R increases we obtain two coexisting attractors in the hysteretic regime: a frequency-locked periodic solution (W=7/46, point B, fig. 4b) coexisting with the periodic solution belonging to the 2/13 resonance horn (point C, fig. 4c). As R/RC increases the hysteretic region comes to an end. This occurs probably through global interactions of the invariant manifolds of the saddle-type (unstable) frequency-locked periodic solutions, as suggested by numerical experiments in simple maps or periodically forced oscillators in similar regimes. The stable, frequency-locked 2/13 periodic solution undergoes a secondary Hopf bifurcation (as two of its characteristic multipliers cross the unit circle in the cornplex plane) and a secondary torus surrounding it appears (point E, fig. 4d). Eventually a reverse seeondary Hopf bifurcation is observed, and the see-
ary of the 2/13-horn. Fig. 5 present a more detailed view of the secondary Hopf bifurcation regime. The stable 2/13 periodic solutiongiving (point C, fig. undergoes a Hopf bifurcation, birth to a 5a) stable secondary torus (point D, fig. Sb). This secondary torus is initially small in amplitude and surrounds the now unstable 2/13 periodic solution. It subsequently grows in amplitude (point E, fig. Sc) and exhibits its own frequency lockings (point F, fig. Sd) and possibly its own Arnol’d horn structure. The secondary tori then .
become smaller in amplitude (point G, fig. Se) and we observe a reverse secondary Hopf bifurcation where they shrink back to yield stable 2/13 periodic solutions (point H, fig. Sf). The nature of this secondary Hopf bifurcation is elucidated further in fig. 6, where we present a sequence of Poincaré sections showing the transient dynamic behavior of the system before it relaxes to the attractor. Three such Poincaré sections are shown, restricted for clarity around one of the periodic points of the 2/13 periodic solution. The smooth curves through every third point on the transient illustrate the order of iteration of the points and are not trajectories ofthe system. We connect every third point because the average rotation per iterate is approximately 2it/3. Remember that this average rotation rate is proportional to the imaginary part of the (complex) Floquet multipliers of the periodic solution. In fig. 6a the attractor observed is a torus resulting from a secondary Hopfbifurcation within the 2/13 resonance horn. As we increase R we cross the right-hand-side boundary of the SHB curve, and the attractor becomes periodic. Close to this transition boundary a slower transient spiralling approach to the periodic orbit is observed (see fig. 6b). This is an indication that the norm of the principal Floquet multipliers is close to the unit circle, signalling the proximity of a (secondary) Hopf bifurcation. In fig. 6c, far from the SHB boundary the attractor remains a periodic point. The transient approach to this pe347
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Fig. 4. Experimental Poincaré sections of the attractor reconstructed with two time delays at representative points of the bifurcation diagram in fig. 3: (a) quasiperiodic attractor, (b) locked, W=7/46, (c) locked, W=2/l3, (d) secondary tori, (e) locked, W=2/13, and (f) locked, W=5/32.
riodic point spirals much less than the transient in fig. 6b. The experimental technique of utilizing Poincaré sections of transients to extract quantitative stability information for the periodic points and elucidate the secondary Hopf bifurcations may be applicable to the study of the more complex dynamics described below, 348
The experimental results given in figs. 1 and 2 are strikingly similar to results obtained by Aronson et al. [6] and Johnson [7] in their analytical and nurnerical studies oftwo-parameter families of maps of the plane, and by Kevrekidis et al. [81 in their numerical studies of stroboscopic maps generated by forced oscillators. These similarities strongly suggest
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(b)
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Fig. 5. Experimental Poincaré sections of the attractor reconstructed with two time delays showing the gradual development and closing of the secondary Hopf bifurcations along the bifurcation diagram in fig. 3: (a) locked, W= 2/13, (b) small secondary tori, (c) large secondary tori, (d) locked secondary tori, (e) small secondary tori, and (f) locked, W= 2/13.
that in the experimental regime considered here, the system can be modelled by a two-parameter family of planar maps with nonconstant jacobian. The experimental Poincaré sections of the reconstructed attractors (presented in figs. 4 and 5) as well as those of the transient approach to an attractor (presented in fig. 6) are also consistent with the behavior of the diffeomorphisms studied in refs. [6—81.A suitably parameterized family of standard circle maps would also exhibit some of the qualitative features we have observed in our experiments [3}. Circle maps, however, do not exhibit secondary Hopf bifurcations or the frequency lockings of these secondary tori and cannot be diffeornorphisms in the parameter region
where hysteresis occurs. Moreover, Boyland has proved that an Arnol’d horn for a circle map cannot contain both boundaries ofan overlapping horn [15]. This would rule out the overlap of the 2/13-horn and the 5/32-horn shown in fig. 2 for l/a~i15. Thus, the experimental data are better represented by a planar map model than by 1 -D circle maps. The planar map model not only describes many features of the experimental data, but also makes, as we will see, specific important predictions about systern dynamics in the hysteresis and secondary Hopf bifurcation regions of parameter space. Although there are some differences between model and experiment in the simple regions of the horn, we ob349
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Fig. 6. Poincaré sections showing transientdecay to attractor: (a) approach to secondary tori, R/R~=11.15, (b) transient approachto period-two (W=2/13) attractor, R/RC = 11.18, and (c) transient approach to period-two (W= 2/13) attractor, R /R~= 11.21.
serve that the analogy applies well in the resonance overlap region. Consequently we consider a twoparameter family of diffeomorphisms of the plane {Mab} each of which has a unique fixed point Pa,b which undergoes a Hopf bifurcation on a curve r in the a—b parameter space. That is, the eigenvalues of the differential DMa,b (Pat,) cross from the inside to the outside of the complex unit circle as the parameter point (a, b) crosses F in the appropriate direction. If the crossing occurs at (a*, b*) e F and if the eigenvalues of DMa~,b*are p/q-th roots of unity (p, q relatively prime integers with q ~ 5) then, as shown by Arnol’d and by Takens [6], there is a cusped region (Arnol’d horn) emanating from the point (a*, b*) in the parameter plane in which all points on the attractor have rotation number p/q. Moreover, in a sufficiently small neighborhood of (a*, b*) none of the horns starting on F overlap, and inside each of these horns the attractor is a smooth invariant circle consisting of q sinks and q saddles connected by the unstable manifolds of the saddles (see figs. 7a and 7b). The Arnol’d—Takens theory only establishes the existence of the horns in a small neighborhood ofthe 350
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Fig. 7. Schematic drawings of: (a) several Arnol’d horns emanating from a bifurcation curve F close to which the horns do not overlap (shaded region); (b) a smooth invariant circle involving three sinks and three saddles; (c) an invariant set involving three sinks and three saddles showing homoclinic crossings of the stable and unstable manifolds of the saddles. Representative Poincare sections in the left- (d) and right-hand-side (f) hysteretic regions aswell as in the nonhysteretic interior (e) of a resonance horn for a model periodically-forced chemical oscillator.
bifurcation curve r. The horns can, however, be extended numerically. As the parameter point moves into the horn away from F the invariant set which is originally a smooth circle progressively loses its smoothness and ultimately loses its topological type (i.e. it is no longer a topological circle) [6—8].The evolution ofthe invariant set as the parameter point moves through a fixed horn is studied in detail in ref. [6] where it is shown that one of the main mechanisms involved in the breakup of the invariant circle is the formation of heteroclinic and homoclinic crossings. Schematic homoclinic crossings of the stable and unstable manifolds of periodic saddles are shown in fig. 7c. In the subset ofthe horn where these homoclinic crossings occur, other horns, conesponding to rotation numbers distributed according to the Farey sequence may overlap the given horn, and hysteretic behavior is observed numerically.
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Poincaré sections representative of this regime obtamed from a periodically-forced brusselator [8] are also shown in fig. 7. Fig. 7d is obtained in the hysteretic regime adjacent to the left-hand-side boundary of a resonance horn and shows the coexistence of two attractors (periodic points and a smooth circle). Fig. 7e is obtained in the interior of the resonance horn, where no hysteresis is observed and the attracting set consists ofthe periodic points. Finally, fig. 7f is obtained in the hysteresis region adjacent to the right-hand-side boundary of the resonance horn, where again two attractors (periodic points and a smooth circle) coexist. The transition between states illustrated in figs. 7d and 7e and the transition between states in figs. 7e and 7f involve homoclinic crossings of the stable and unstable manifolds of the periodic saddles, similar to those schematically depicted in fig. 7c. Aronson et al. [6] prove that under quite general hypotheses the existence of homoclinic crossings such as those shown in fig. 7c, implies the existence of an interval I such that for every real number re I there is at least one point on their attracting set whose rotation number is r. Moreover, there are also points on this attractor whose rotation number does not exist. Thus, the theory of parameterized planar maps predicts the existence of subsets of the Arnol’d horns where the dynamics are quite complicated. In these subsets Arnol’d horns overlap so that the attractor contains periodic points with countably infinite different rational rotation numbers, as well as possibly smooth invariant circles and chaotic subsets. We expect that most of these states would have vanishingly small stability intervals and would not be observable experimentallydue to the presence of noise. Nevertheless, multiple (greater than two) hysteretic states might be seen in experiment and we plan to investigate this prediction ofthe planar map model. In a recent publication [16] Brunsden and Holmes predict the power spectral density for strange attractors in a system possessing transverse homoclinic orbits. It may prove possible to establish the existence of such orbits by examining the qualitative characteristics of the power spectrum close to the inner hysteresis boundaries of the resonance horns. ,The theory ofpararneterized planar maps also prediets that the invariant circles resulting from seeondary Hopf bifurcations may break due to
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homoclinic interactions. Experimental Poincaré seetions are consistent with that prediction but more data, in particular, data employing the transient techniques described above, are necessary for verification of the predictions. These results will be presented elsewhere. In summary, our experimental observations of hysteresis, overlapping of Arnol’d horns and secondary Hopf bifurcations are consistent with numerical observations and theoretical predictions obtained from two-parameter maps of the plane with nonconstant jacobian. We distinguish two main features of the proposed model of our experimental system in terms of a family of maps. The first is the qualitative prediction by the model of the fine structure of the Arnol’d horns, including the hysteretic regime and the secondary Hopf bifurcations, along with suggested mechanisms underlying these bifurcations, especially the existence of homoclinic tangles. This correspondence implies that in the parameter region examined here the dynamic behavior of this infinite dimensional system is low-dimensional. The second important feature is that in our parameter region the system follows no single standard route to turbulence. Indeed, in agreement with the predictions of Aronson et al. [6] on any one-dimensional path through the parameter space, we generally encounter a rapid alternation between regions where the attractor is smooth, where it has various degrees of wrinkling, and regions where it is chaotic. This work was supported by funds provided by the U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Science. We would like to acknowledge the experimental assistance of Hans Haucke and many valuable discussions with Donald Aronson and Richard McGehee.
References [l]J.P.
Gollub and S.V. Benson, J. Fluid Mech. 100 (1980)
[2] J. Stavans, F. Heslot and A. Libchaber, Phys. Rev. Lett. 55 (1985) 596. [3] H. Haucke and R.E. Ecke, Physica D 25 (1987) 307. [4] M. Dubois and P. Berge, Phys. Scr. 33 (1986)159. [5] Si. Shenker, Physica D 5 (1982) 405; M.H. Jensen, P. Bak and T. Bohr, Phys. Rev. Lett 50 (1983) 1637.
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[6] D.G. Aronson, M.A. Chory, G.R. Hall and R.P. McGehee, Commun. Math. Phys. 83 (1982) 303. [7] J.P. Johnson, Ph.D. Thesis, University of Minnesota (1985); D.G. Aronson, J.P. Johnson and R.P. McGehee, in preparation. [8]1G. Kevrekidis, R. Aris and L.D. Schmidt, Chem. Eng. Sci. 41(1986)905; 1G. Kevrekidis, Am. Inst. Chem. Eng. J., to be published. [91J.-P. Eckmann, Rev. Mod. Phys. 53 (1981) 643. [10] S. Ostlund, D. Rand, J. Sethna and E. Siggia, Physica D 8 (1983) 303. [111VI. Arnol’d, Funct. Anal. Appl. 11(1977) 2-1;
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F. Takens, Commun. Math. Inst. Rijksuniversiteit Utrecht 3 (1974) 1. [12] Y. Maeno, H. Haucke, R.E. Eckeand S.C. Wheatley, J. Low Temp. Phys. 59(1985)305. [13]H. Rademacher, Lectures on elementary number theory (Blaisdel, New York, 1964). [14]N.H. Packard, J.P. Ci-utchfield, iD. Farmerand R.S. Shaw, Phys. Rev. Lett. 45 (1980) 712; F. Takens, Lecture notes in Mathematics, Vol. 898 (Spnnger, Berlin, 1981). [15] Boyland, and Commun. Math.Phys. Phys.Rev. 106 (1986). [16] P.L. V. Brunsden P. Holmes, Lett. 58 (1987) 1699.