bifurcations in a model of magnetoconvection...

Physica 9D (1983) 379-407 North-Holland Publishing Company

BIFURCATIONS IN A MODEL OF MAGNETOCONVECTION

E. KNOBLOCH Department of Physics, University of California, Berkeley, CA 94720, USA

and

N.O. WEISS Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, England

Received 17 January 1982 Revised 15 December 1982

Bifurcations from oscillatory solutions are studied in a truncated model of two-dimensional Boussinesq magnetoconvection. The fifth order system of nonlinear differential equations is integrated numerically and in certain parameter regimes there is a bifurcation from symmetrical to asymmetrical oscillations followed by a period-doubling cascade. After the accumulation point there is a semiperiodic cascade leading to chaotic behaviour. Then the semiperiodic cascade is repeated in reverse, followed by a period-halving cascade and a bifurcation back to symmetry. Finally, the branch of oscillatory solutions terminates with a symmetrical heteroclinic orbit that connects two saddle-foci. The interval with aperiodic solutions contains many pairs of narrow windows with asymmetrical or symmetrical periodic solutions, each with its own cascade. This pattern of behaviour is likely to occur when a periodic orbit approaches a symmetrical pair of saddle-foci with eigenvalues that satisfy Shil'nikov's inequality.

1. Introduction

Recent developments, both theoretical and experimental, have stimulated widespread interest in nonlinear fluid dynamical problems that exhibit a transition from periodic oscillations to behaviour that is apparently chaotic. In particular, much attention has been focussed on systems in which a sequence of period-doubling bifurcations is followed by aperiodic oscillations. In this paper we explore the behaviour of one such system, derived as a model of convection in the presence of an imposed vertical magnetic field. In order to understand the complicated structures that appear in our numerical solutions it is first necessary to relate this problem to other, simpler systems that have al- ready been investigated.

Sensitivity and accuracy in fluid dynamical experiments have improved with the advent of low temperature techniques, as in experiments on

Rayleigh-Brnard convection in liquid helium [30, 32], and laser-doppler techniques, as in experiments on convection in water [18, 17] and in studies of the Taylor-Couette problem [14]. An excellent review of the possible routes to aperiodicity, together with supporting experimental evidence, has recently been given by Eckmann [12]. Of his possible scenarios, the transition to aperiodicity by way of a period doubling cascade is typical of many systems and has been demon- strated most convincingly in the small aspect ratio convection experiments, using liquid helium and mercury, of Libchaber and his colleagues [31, 32, 29, 28]. These experiments reveal unambiguously a succession of four period-doubling bifurcations of the basic oscillatory solution as the Rayleigh number is increased. The bifurcations appear to be converging (cf. also experiments with water [17]); beyond the accumulation point the oscillations are no longer strictly periodic although a basic period-

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380 E. Knobloch and N.O. Weiss~Bifurcations in a model of magnetoconvection

icity persists. Lorenz [34] has called such oscillations semiperiodic: they manifest themselves by the appearance of noise skirts superposed on top of the sharp spectral lines present in the oscillation spectrum before the accumulation point. The semiperiodic bands apparently undergo an inverse period-halving cascade, at the end of which the oscillations are fully aperiodic and the sharp lines disappear from the spectrum. If the Rayleigh number is further increased, narrow windows with periodic oscillations appear within the chaotic regime and each such window contains its own, new period-doubling sequence [28].

Such behaviour is well known from studies of one-dimensional maps [36, 13, 8, 47, 19, 34, 9, 11, 6]. In particular, it has been shown that the successive period-doubling bifurcations accumulate asymptotically at a geometric rate [13, 8, 47] and that the immediately following backwards cascade of semiperiodic bands accumulates at the same rate[19, 34, 9, 11]. The development of this pattern of bifurcations as a stability parameter ~ is increased is shown schematically in fig. 1. If #, is the value of # at the bifurcation to period 2" in the Feigenbaum sequence, then

6 = lim #" -- #.--1 ~ 4.66920.. . (1) n ~ / t n + 1 - - / t .

is a universal constant. Beyond the aperiodic range other periodic solutions appear via saddle-node

~// /" "~

I a

Fig. 1. Sketch showing the development of a Feigenbaum sequence. As the stability parameter # is increased there is a cascade of period-doubling bifurcations, followed by an inverse cascade of semiperiodic bifurcations.

bifurcations [36]. Thus a new periodic solution is heralded by intermittency [42, 12] before the periodic solution achieves stability. Each basic period- icity undergoes a Feigenbaum cascade and the process repeats with the next periodic solution. The ordering of the basic periodicities is the same for all unimodal maps [40, 44, 37, 36, 20]. A theorem due to Sharkovsky [44, 20] implies that periodic orbits first appear in the order (2, 4 . . . . . 5 x 2 r, 3 × 2', . . . . 10, 6 . . . . . 7, 5, 3). Suppose that an orbit of period N appears at/~ = p(m; then there is at least one value of/~ (m such that #(2) < #(m <//2(3) for all N >t4. Sharkovsky's theorem does not, however, make any statement about the stability of these periodic solutions or about the number of different solutions with the same period. In fact, solutions with periods 2 or 3 only appear once but all other periodicities appear also for #(~0 > #(3) and the number of different values of #tin increases rapidly with N [37, 36]. Moreover, it is possible to ascertain the precise order in which the periodic windows appear [40, 37, 36]. Metropolis et al. [37] provide a list, in order of appearance, of all stable periodic solutions with N ~< 11; for N ~< 7 the ordering is (2, 4, 6, 7, 5, 7, 3, 6, 7, 5, 7, 6, 7, 4, 7, 6, 7, 5, 7, 6, 7), which is consistent with Shark- ovsky's theorem.

The experiments indicate that the fundamental period-doubling phenomenon occurs in continuous systems as well. The appropriate physical circumstances appear to be those where the boundary conditions effectively limit the number of degrees of freedom that can be excited, e.g. convection in a box whose horizontal dimensions allow no more than five rolls side by side. With larger aspect ratios the pattern seems never to settle down completely, resulting in the presence of low frequency noise [30] and qualitatively different behaviour.

It is important to clarify the extent to which results for one-dimensional mappings can be car- ried over first to ordinary and then to partial differential equations. It has been conjectured [e.g. 41] that period doubling with the same value of 6 will occur also in ordinary differential equations

E. Knobloch and N.O. Weiss/Bifurcations in a model of magnetoconvection 381

such as the Lorenz equations [33]. Feigenbaum cascades have indeed been found in this [15] and other [35, 16, 10] systems. The story may, however, be much more complicated: even in the ~wo-dimensional H~non mapping [23], a period 6 limit cycle can appear and complete its period- doubling cascade before the accumulation point of the fundamental period-doubling sequence [3]. Ap- parently Sharkovsky's theorem does not neces- sarily extend to higher dimensional mappings or to differential equations.

The Lorenz equations [33] are the system most studied from this point of view and their properties have recently been summarised in a very readable account by Sparrow [46]. His extensive computations illustrate the transition from periodic to aperiodic behaviour as the Rayleigh number, r, is decreased. For large r the oscillations are symmetrical. A bifurcation to asymmetry is followed by a cascade of period-doubling bifurcations and an inverse cascade of semiperiodic bifurcations. Within the chaotic regime there are many narrow windows with periodic orbits. In each case the appearance of a different basic periodic solution at a saddle-node (or loop-loop) bifurcation is an- nounced by intermittency: nearly periodic solutions are maintained for many cycles, interspersed with episodes of chaos. After the period-doubling cascade has accumulated, banded trajectories appear, becoming increasingly chaotic until the next periodic window intervenes. Finally, all the unstable periodic orbits are consumed in a sequence of homoclinic explosions [46].

Although the Lorenz equations provide a fascinating demonstration of the range of behaviour that one may expect, these bifurcations occur in a parameter range for which the system is no longer reliable as a model of the convection problem from which it was originally derived [10]. Other artificially truncated models of continuous systems have also been investigated [5, 16] but it is important to construct simple systems whose solutions are at any rate qualitatively similar to those of the partial differential equations from which they were derived. Doubly-diffusive convection

provides examples for which this is possible. The best studied is thermosolutal convection [24, 10 and references therein], where a destabilizing thermal gradient is opposed by a stabilizing (bottom- heavy) solute gradient. In such a system convection can set in as overstable oscillations that increase in amplitude as the Rayleigh number is increased. Numerical solutions to the full two-dimensional partial differential equations [24, 38] show a clear transition from symmetrical to asymmetrical oscillations, followed by period doubling and chaos. Within the chaotic regime, narrow windows with periodic solutions can be identified, though the sequence of bifurcations is more complicated than in simple maps. It is possible to construct a fifth order model of thermosolutal convection that is exact at small amplitudes and seems to be qualitatively correct at larger amplitudes [10]. Indeed, the model suggested that the transition to chaos took place by period doubling, and provided a fuller understanding of the bifurcation structure of the partial differential equations.

In this paper we consider another example of doubly-diffusive convection: motion driven by thermal buoyancy in an electrically conducting fluid in the presence of an externally imposed vertical magnetic field. The field inhibits the onset of convection and in the absence of other effects, it would support hydromagnetic oscillations. Pro- vided that the ratio of magnetic to thermal diffusivity is sufficiently small (i.e. the electrical conductivity is sufficiently high) convection sets in through a Hopf bifurcation from the static state and oscillations grow increasingly nonlinear as the Rayleigh number r is increased. We confine our attention to two-dimensional behaviour, which is physically unrealistic but mathematically con- venient. Detailed numerical investigations of two- dimensional magnetoconvection [48] have revealed a richer variety of behaviour than is present for thermosolutal convection. A recent review [43] catalogues the complex range of possibilities and describes the astrophysical significance of this problem. Knobloch, Weiss and Da Costa [27] have constructed a fifth order model that successfully

382 E. Knobloch and N.O. Weiss~Bifurcations in a model o f magnetoconvection

mimics qualitative behaviour of the full two- dimensional problem. The truncated system makes it possible to establish details of the bifurcation structure and, in particular, of the transition from oscillatory to steady motion. This approach can succeed where the observed behaviour is typical of a two-dimensional system and if the appropriate subset of all the possible degrees of freedom has been identified. A rigorous identification procedure focusses on the dynamics near multiple bifurcations [25, 21, 22, 7]. Ref. 27 adopts a more intuitive approach and attempts to identify the relevant modes on physical grounds.

The present paper is devoted to a detailed study of the bifurcations from oscillatory solutions in the model system. This is scarcely feasible for the partial differential equations, which cannot be solved with adequate precision. We find a rich variety of behaviour while r is still of order unity. At such small values of r only a few modes are excited and we may expect the model to be qualitatively correct. M a n y - b u t not a l l - fea tures of solutions to the model problem do indeed correspond to properties of the original system. Much of their fascination derives from the fact that it is not yet clear which features are specific to the model and which are generically valid.

The fifth order system and its basic properties are summarized briefly in the following section. Numerical computations which reveal the pattern of bifurcations from the oscillatory solutions, and the transition from periodic to aperiodic behaviour, are described in section 3. The dynamical structure of these solutions is far more complex than hitherto expected. In conclusion, we sum- marize our results and try to establish the conditions that are needed for such behaviour to occur. Our calculations have yielded many details that are fascinating and amazing in themselves but this elaborate treatment of a special system is justified by its relevance to a wider class of problems.

2. The model system

Boussinesq fluid is described by three scalar fields, the stream function, the temperature and the magnetic flux function. If periodic boundary conditions are asumed, these fields can be expanded as Fou- rier series in the spatial co-ordinates and the lowest non-trivial truncation yields the system

d = a [ - a +rb -~qd{1 + ( 3 - 03)e}], (2)

6 = - b + a( l - c ) , (3)

( = 03 ( - c + ab), (4)

d = - ( d + a(l - e ) , (5)

k = -- (4 - 03)(e + 03ad, (6)

discussed by Knobloch, Weiss and Da Costa [27, 43]. Here the stability parameter r is a normalized Rayleigh number (a dimensionless measure of the thermal driving force) while q is a normalized Chandrasekhar number (a dimensionless measure of the imposed magnetic field); tr is the Prandtl number, ( is the ratio of the magnetic to the thermal diffusivity and 03 (0 < 03 < 4) is a geo- metrical factor related to the width of a convection cell; the dots denote derivatives with respect to the dimensionless time ~.

Eqs. (2)-(6) possess the trivial solution a = b = c = d = e = 0, corresponding to a static state with no convection. In the limit a2~0, the fifth-order system can be derived as an asymptotic approximation to the partial differential equations; then a measures the first order velocity perturbation, while b, c and d, e are measures of the first and second order perturbations to the temperature and to the magnetic flux function, respectively. The system (2)-(6) possesses an important symmetry, for it is invariant under the transformation

(a, b, c, d, e)--*(-a, - b , c, - d , e). (7)

Two-dimensional magnetoconvection in a Moreover, the equations describe a contraction

E. Knobloch and N.O. Weiss~Bifurcations in a model of magnetoconveetion 383

mapping in the five-dimensional phase space, since

aa o6 o~ od do aa+~+~ +Sd+ae

= - [,r + ( 1 + ~ ) + ~ ( 5 - ~ ) 1 < o . (8)

Trajectories may be attracted to fixed points, to limit cycles or, possibly, to a strange attractor.

The stability of steady solutions has been discussed in detail [27]. As r is increased from zero, the static solution (0, 0, 0, 0, 0) eventually becomes unstable. We are interested in the case when

< I , q > ((1 + a)/a(1 + ~) , (9)

so that instability sets via a Hopf bifurcation when

r = r ¢°) = (tr + ()[(1 + ()/tr + (q/(1 + tr)]. (10)

A branch of oscillatory solutions bifurcates from the static solution at r = r ~°) and there is also a branch of steady solutions that bifurcates at

r = r ~¢~= 1 + q > re°). ( l l )

Along this steady branch,

r = (1 + a2)[1 +/~q(/i + a2)-2{/i + (4 -- 03)a2}],

(12)

where/2 = ( 4 - e3)(2/o3; thus the number of turn-

ing points may be two, one or zero. We shall confine our attention to choices of parameters such that there is at least one turning point and steady (subcritical) finite amplitude solutions exist for r ~> r~,, r~. < r ~e). Then the upper portion of the steady branch is stable for r > r=. (if a possible Lorenz bifurcation at large r is excluded) while the entire lower portion is unstable. (We assume that a is sufficiently large for the solution to lie in region e of fig. 3(b) of [27].)

When the Hopf bifurcation at r (°~ is supercritical, finite amplitude oscillatory solutions per-

sist over the range r ~°~ < r < r ~ , rrm n < rraax < r (e).

In order to establish how the oscillatory branch terminates, we may consider the case when r<°)~r ~e) and there is a single turning point on the steady branch. Knobloch and Proctor [25] showed

that in that case

a(z ) oc sn(z*lm ) , z* oc z , (13)

where m increases from zero to unity as r is increased from r (0) to rma x. Thus the period of the Jacobian elliptic function becomes infinite as r approaches rma x and, in the limit, the oscillatory branch terminates on the unstable portion of the steady branch.

The behaviour of nonlinear periodic solutions can be investigated more generally by integrating eqs. (2)-(6) numerically on a computer [27]. We have used a program for solving ordinary differential equations that was kindly supplied by Colin Sparrow, incorporating a standard NAG- LIB routine based on a fourth-order Runge- Kut ta-Merson scheme with variable time steps. Fig. 2a shows a as a function of r for a case with q = 5 , a3=2 , ~ = 0 . 4 and a = l . Only positive values of a are shown on the steady branch, which has two turning points; stable steady solutions exist for r /> r=, ~ 4.55. Root mean square values of a are shown along the branch of oscillatory solutions. When r is only slightly greater than r 10) the periodic solutions are almost harmonic and a limit cycle projected onto the ad plane deviates only slightly from an ellipse. At larger values of r oscillations become more obviously nonlinear. Fig. 3a shows a periodic orbit in the ad-plane, described anticlockwise, when r = 5.22. The velocity a rises to a maximum, then drops and hovers near its value on the unstable steady branch before revers- ing; the magnetic perturbation d falls to a subsidiary minimum so that the trajectory turns a sharp corner in the neighbourhood of the unstable steady solution. The projection of the orbit onto the de-plane, shown in fig. 3b, exhibits less structure. Note that the limit cycles are perfectly symmetrical, as might be expected from (7), and that e varies with a period ½P, while a and d have period


1"0:

(b) r~O)

( a )

rmi n r (e)

e

\

\

IO0

8O

6O

4O

2O

0 3 0

0 I I i I 3 0 4"0 5'10 6"0 7"10 4"0 510

r r

2 .0 a

-0.8

I I I -0.8 0.8

I I I 0.8 0.8

1

1 I I I I I

o.o d

Fig. 3. Cusps and loops with a = 1. Limit cycles for (a), (b) r = 5.22 and (c), (d) r = 5.255, projected onto the ad and de planes.

0.8

I I I I I 0.0

e

0 . 8 -

d

a d b 0.8

I I a -0.8 0.8

0.8

d c d

0.8-

0.8 0.8

Fig. 2. (a) The oscillatory and steady branches for tr = 1; a as a function of r, with the unstable portion of the steady branch denoted by a broken line. (b) Variation of the period P with r along the oscillatory branch.

E. Knobloch and N.O. Weiss~Bifurcations in a model of magnetoconvection 385

P. Fig. 2b shows P as a function of r; as expected, P increases very rapidly as r ~rma x ~ 5.25580, while the r.m.s, value of a approaches the value of the unstable portion of the steady branch. Thus the branch of oscillatory solutions terminates with a symmetrical heteroclinic orbit [27, 43].

The corners in fig. 3a are real and not just a consequence of inappropriate projection. As r approaches r~x, however, the form of the orbit changes: the corners develop into cusps and then into loops around the unstable singular points, as shown in figs. 3c and d. The smooth transition between corners and loops shows that there is no real topological distinction between them. Never- theless, alternating corners and loops are an important feature of the solutions to be described in the next section. As trajectories approach the singular points, their behaviour depends on the eigenvalues at those points. For tr = 1, r = 5.256 the five eigenvalues are (0.360, - 0 . 5 5 0 _ 1.052i, - 1.715, -2.745). As we shall see in section 4, we expect one real positive eigenvalue and a pair of complex conjugate eigenvalues with negative real parts. Trajectories that are inclined to the plane containing the eigenvectors corresponding to complex eigenvalues will slow down as they approach a singular point, turn a sharp corner and move away towards the eigenvector that corresponds to the positive eigenvalue. Trajectories approaching the point in a direction close to the complex eigenplane will describe one or more loops around the singular point and then depart. Although the real part of the complex eigenvalues is less in magnitude than the two negative eigenvalues, it is only when r is very close to rmax that more than one loop can be seen in practice.

3. Period-doubl ing and aperiodicity

The pattern of behaviour described above ap- plies provided that tr is not too large; if a is increased (while q, ( and 03 are held constarrt) other families of solutions bifurcate from the oscillatory branch before it terminates. We have system-

atically investigated time-independent behaviour with q = 5 , ( = 0 . 4 , o 3 = 2 and 1~<a~<20, for values of r between 5.2 and 5.4. These computations yielded a bewildering variety of behaviour: bifurcations from symmetry to asymmetry were followed by period-doubling sequences which led to aperiodic motion; within these chaotic regions, new periodic orbits appeared through saddle-node bifurcations and went through their own period-doubling sequences; finally, all these bifurcations were repeated in reverse, culminating in a symmetrical heteroclinic limit cycle at the end of the oscillatory branch. In this section we offer an illustrated guide to the jungle that we have explored. Our aim is not just to catalogue the exotic species that we found but rather to distinguish between qualitatively different patterns of behaviour. All nontrivial periodic solutions that we have found are, however, listed in the appendix.

A sequence of bifurcations followed by a corresponding sequence in reverse has a bubble-like structure [26] (see fig. 9). In the first of three parts into which this section is divided, we describe the evolution of the bubble as tr is increased. The complete Feigenbaum sequence is found only for sufficiently large tr and for yet larger tr the forward and backward Feigenbaum sequences are separated by chaotic solutions, as sketched in fig. 10b. In the second part, we investigate the sequence of windows containing different periodicities within this region. We can distinguish between two kinds of window, loosely called asymmetrical and symmetrical, depending on whether successive split- tings occur in the vicinity of one or both saddle points. The symmetrical windows appear typically in the centre of the bubble. A schematic diagram, showing the periodic windows that we have found, is provided in fig. 10c. Finally, in the third part, we demonstrate that, for sufficiently large tr, the bubble loses stability in the centre. Thus the branch of oscillatory solutions appears divided into two pieces, one of which is disconnected from the rest of the stable oscillatory solutions (see fig. 10c). The separate pieces apparently terminate in homoclinic or heteroclinic orbits. In reading this section it may

386 E. Knohloch and N.O. Weiss/Bifurcations in a model o f magnetoconvection

a 0.8~ b 0.8-

r 1.5

-0.8

1.5

0.8

t 1.5

C 0.8- d

-1.5 I 1.5

0.8-

-0.8 -0.8

q 1.5

Fig. 4. Appearance of the bubble, (r = 6. Limit cycles in the ad plane for (a) r = 5.15 (symmetrical); (b) r = 5.23 (asymmetrical); (c) r = 5.285 (symmetrical with cusps); and (d) r = 5.285 (with loops).

prove helpful to refer to fig. 10 in order to relate specific details to the overall picture.

3.1. The bubble bursts

For ~ <~ 3 symmetrical periodic solutions persist to the end of the oscillatory branch. When a = 5 this symmetry is, however, broken: for 5.15~<r~<5.21 there are two (slightly) asymmetrical solutions, which transform into each other under the symmetry operation (7). The appearance and disappearance of asymmetry when a = 6 is illustrated in fig. 4. The symmetrical limit cycle shown in fig. 4a loses stability at r ~ 5.18. After this bifurcation there are two asymmetrical limit cycles; fig. 4b shows one of these, with a corner

near one of the unstable singular points, for r = 5.23. As r is increased, a corner gradually develops near the other singular point until, at r ~ 5.265, there is another bifurcation after which symmetry is restored. Fig. 4c shows a symmetrical orbit, with two corners, at r = 5.285. Thereafter, the corners develop into loops and the period becomes infinite, as before. The bifurcation pattern is shown schematically in fig. 9a: asymmetrical solutions exist within the bubble, between a pair of oppositely directed pitchfork bifurcations, where the symmetrical solution is unstable.

Similar behaviour is found with (r = 8 but another pair of bifurcations appears when a = 9. Fig. 5a shows a periodic solution for r = 5.28: the trajectory closes on itself after two cycles round the


CI e a

- i . s

b /£

Fig. 5. Period doubling, a = 9, r = 5.28: (a) limit cycle; and (b) detail showing structure near the upper singular point.

C1

b

origin, so the period has doubled. Fig. 5b shows a

detail o f structure near the upper singular point. In

this ne ighbourhood successive cycles describe cor-

ner and loop-like orbits. Period doubl ing appears

at a bifurcation a round r ~ 5.275 and disappears

at a subsequent bifurcation a round r ~ 5.305; the

corresponding bifurcation pat tern is shown schematically in fig. 9b.

The bubble is virtually unchanged for a = 9.1

but at a = 9.2 there is a further bifurcation at r ~ 5.285 after which the period doubles again, so

that the trajectory repeats after four cycles round

Fig. 6. Period doubling, a = 9.5: (a) r = 5.29 (period 4); (b) r = 5.291 (period 8).

the origin; there is a bifurcation back to period 2

at r ~, 5.297, followed by further bifurcations to

asymmetrical and then to symmetrical orbits. The splitting with period 4 is relatively slight but fig 6

shows details o f solutions with period 4 and period 8 for a = 9.5. The basic period 2 orbit, with a corner and a loop, splits for period 4 and then

splits again for period 8 etc. As expected, the splitting becomes less at each successive bifurcation.

388 E. Knobloch and N.O. Weiss/Bifurcations m a model of magnetoconvection

Q

b

Fig . 7. S e m i p e r i o d i c t r a j e c t o r i e s , a = 9 .5: (a ) r = 5 .292 ; (b)

r = 5 .293 .

By ~r =9 .5 there is a complete sequence of period-doubling bifurcations, culminating at r ~5.2915, followed by a sequence of period- halving bifurcations, starting at r ~ 5.2985. The bubble has burst and throughout most of the intervening region the solutions are aperiodic. Fig. 7a shows details of a run at r = 5.292: at first sight this might appear to be an example with period 8 but under greater magnification it can be seen that the trajectories never repeat exactly but wander within a very restricted range. The solution is semiperiodic, of period 8, with a characteristically banded structure. As r is increased, this banded structure becomes more diffuse. In fig. 7b, for

r = 5.293, the solution is semiperiodic of period 2 with a hierarchically banded structure that one associates with the appearance of a strange attractor. As a is increased further, the same basic pattern persists: a bifurcation to asymmetry is followed by a complete sequence of period- doubling bifurcations; then, after an interval with aperiodic, or chaotic, behaviour, there is an inverse sequence of bifurcations, ending in a symmetrical orbit which eventually forms a heteroclinic connection between the singular points. The bifurcation pattern is shown schematically in fig. 9c. S.J. Geard (private communication) has confirmed that the unstable symmetrical solution for a = 10 exists throughout the region between the first and last bifurcations; the unstable asymmetrical solution exists similarly between the first period-doubling and the last period-halving bifurcation. As ~ is increased the aperiodic solutions become more obviously chaotic. The trajectory for a = 10, r = 5.2932, shown in fig. 8, is more spread out and

has more structure than those in fig. 7. Note, however, that asymmetry is maintained, with all interesting structure confined to the neighbourhood of one of the two unstable singular points.

From these results we can deduce how the bubble evolves. In a canonical Feigenbaum sequence, like that sketched in fig. 1, the positions of the bifurcations are determined by a single parameter/~ and the period-doubling cascade is followed by an inverse cascade of semiperiodic orbits. In the bubbles sketched in fig. 9, each period-doubling bifurcation is matched by a bifurcation at which the period is halved. The Feigenbaum parameter /~ = ~(r, a) and, for fixed #, r is a multiply valued function of ~. The incidence of bifurcations as r is varied for a given value of a is sketched in fig. 10a: as r increases, # rises to a maximum and then falls again. Thus all bifurcations from one periodic orbit to another are matched by inverse bifurcations back. For sufficiently large values of a, the trajectory rises beyond the accumulation point of the Feigenbaum sequence and goes through a sequence of semiperiodic bifurcations and beyond;


Q (a) (b) 0.8- ~ _ _ _ _

?,Is -0.8

b

Fig. 8. Chaos, a = 10, r = 5.2932. (a) Asymmetrical orbits; (b) detailed structure.

I !

then it comes through a matching semiperiodic I

sequence in reverse and so I?ack through the Feigenbaum sequence. Note that since dlz/ar = 0 at the top of the bubble it is particularly easy to investigate behaviour when # is near its maximum but rather more difficult to isolate solutions when /z varies rapidly with r, particularly on the de- scending limbs of the (asymmetrical) curves. The loci of bifurcations in the ra-plane are sketched in fig. 10b. Exotic behaviour is apparently confined to a restricted region, where /z(r i a) exceeds some critical value.

(c)

b Fig. 9. Sketches of the development of the bubble, showing the appearance of (a) asymmetry; (b) period doubling; and (c) aperiodic orbits.

3.2. Order peeps out of chaos

Within the shaded regions in fig. 10, which lie beyond the accumulation point of the period- doubling sequence, trajectories are typically aperiodic. There are, however, numerous windows within which stable periodic solutions can be found. Those that we have located are shown schematically in fig. 10c. Any description is bound to emphasize the isolated regions within which order is observed but it must be borne in mind that these regions form a set of small, though finite, measure and that they can only be located through a combination of luck and persistent application. Most of the runs made for parameters corresponding to the shaded regions yielded solutions that appeared to be chaotic.

The appearance of a periodic window is most

readily detected if it occurs near the top of a trajectory in the r~t-plane. A new periodic solution appears as a result of a saddle-node (more strictly, a loop-loop) bifurcation, which is heralded by intermittency. When a = 9.5, for instance, solutions are aperiodic at r =5.29498 but at r = 5.29502 an orbit with period 12 appears, re- peating several times before it drifts off to become aperiodic and then returning again for a while. Similar intermittent behaviour extends over the range 5.295 ~< r ~< 5.296, with an apparently stable solution with period 12 around r = 5.2953, flanked


(a)

Po A

y

r r

(b)

0

(c)

\ /

\ ~ J / , Q (~)_,

14 -- ~ ~

12

8 r

Fig. 10. Sketches showing the location of bifurcations (a) in the r/~-plane; and (b) in the rot-plane; (c) schematic diagram showing location of periodic windows in the ra-plane. The asymmetrical and symmetrical windows that were found are indicated by squares and circles respectively. The curves represent the accumulation points of the Feigenbaum sequence for the cycle of period 1 and the transition from asymmetrical to symmetrical behaviour; the broken lines mark the ends of the oscillatory branch.

by intermittency on either side. The reality of this solution is confirmed by proceeding to a = 9.501. Fig. 1 la shows a detail of the orbit for r = 5.2953; for 5.2954 ~< r ~< 5.2956 the trajectories split to give period 24, with a bifurcation back to give period 12 at r = 5.2957. Figs. 1 lb and c show details of the right-hand group of trajectories in fig. 1 la but

now for a = 9.502, with periods 24 and 48. Finally, at a = 9.52, there are semiperiodic solutions, as shown in fig. l id . This pattern helps one to recognize a window containing a new period- doubling sequence. The periodic orbits are pre- ceded by intermittency, and followed by banded semiperiodic trajectories as in fig. 1. The bi-

391

CI

C

b

C

E. Knobloch and N.O. Weiss/Bifurcations in a model of magnetoconvection

d

C

Fig. 11. T h e w i n d o w wi th pe r iod 12: (a) a = 9.501, r = 5.2953 (pe r iod 12); (b) r = 5.2955, tr = 9 .502 (pe r iod 24); (c) r = 5.2957 (pe r iod

48); (d) a = 9.52, r = 5.2955 (semiper iodic) .

furcations are then repeated in reverse but in general the period-doubling and period-halving cascades are well separated, with other pairs of windows in between them. Such windows are numerous but difficult to locate; the fortunate feature of period 12 was that, by varying r with a = 9.5, we just reached the saddle-node bifurcation near the maximum in the r/~-plane.

With a = 12 we found six different windows with asymmetrical period-doubling sequences. The basic periodic orbits are illustrated in fig. 12 and further details are listed in the appendix. In each

case at least one period doubling was obtained. The initial bifurcation from symmetry to asymmetry and the final bifurcation back occur at r ,~ 5.238, 5.338 and the accumulation points of the basic period-doubling sequences are at r ~ 5.295, 5.328. Fig. 12a shows an orbit of period 3, with a magnified picture in fig. 12b; this periodic window extended over the range 5.3028<~ r ~< 5.3034. It was followed by windows with periods 7 and 5 (in that order) illustrated in figs. 12c and d. Next came period 4, shown in fig. 12e and then a second, different period 7, shown in fig. 12f.

C1

C

b

-0.8

0 . 8 -

>/ e

392 E. Knobloch and N.O. Weiss~Bifurcations in a model of magnetoconvection

f

;/

Fig . 12. A s y m m e t r i c a l w i n d o w s cr = 12: (a), (b) r = 5 .3028 ( p e r i o d 3); (c) r = 5.304 ( p e r i o d 7); (d) r = 5 .3045 ( p e r i o d 5); (e) r = 5.306 ( p e r i o d 4); ( f ) r = 5.30671 ( p e r i o d 7).

E. Knobloch and N.O. Weiss/Bifurcations in a model of magnetoconvection 393

C[

-0 .8

b

r -1 .5 1.5

-o .8

Fig. 13. Transi t ion f rom asymmetrical to symmetrical aperiodic trajectories, a = 12: (a) r = 5.3252; (b) r = 5.3250.

In all these windows the periods doubled as r was increased. The corresponding inverse sequences are more difficult to find (cf. fig. 10a) but that for period 3 was located in the range 5.3271 ~<r ~< 5.3273. A more painstaking search would doubtless reveal many more windows.

We believe that there is a countable infinity of windows, each with a period-doubling sequence based on some integer N, as suggested by the results for one-dimensional maps (cf. section 1). For a = 12 the ordering of periodic solutions found was (2, 4, 8, 3, 6, 12, 7, 14, 5, 10, 4, 8, 7, 14), which is consistent with the results of Metropolis et al. [37] that were listed in section 1. As expected, it is difficult to detect windows lying between those

for N = 2 and N = 3 but we have located some of the windows with N ~< 7 that lie beyond that range.

When the asymmetrical windows are exhausted a further transition takes place. The aperiodic trajectories become (apparently) symmetrical, with similar structure in the vicinity of each saddle- point, and there is, naturally, a corresponding transition from symmetry back to asymmetry. For a = 12, these two transitions occur at r ~ 5.313, 5.3251. Fig. 13a shows an aperiodic solution for r = 5.3252. It is asymmetrical, with all structure in the vicinity of the upper singular point. At r = 5.325 the trajectory remains asymmetrical for a long while but eventually flips over to give structure at the other singular point. The solution continues to flip irregularly from one asymmetry to the other, so that a sufficiently long run appears statistically symmetrical though a finite run, like that in fig. 13b, is not quite symmetrical. As r is further decreased the number of cycles between each reversal becomes less and the trajectory becomes more obviously symmetrical, as well as being more noticeably chaotic than it was before the transition.

A new family of symmetrical periodic solutions appears within the region where chaotic solutions are symmetrical. For a = 12, two windows with period 5 were discovered; for 5.31-6 ~< r ~< 5.3173 the period doubled while for 5.31895 ~< r ~< 5.3206 the period was halved as r increased. Fig. 14a shows an exactly symmetrical solution of period 5 for r = 5.32. The trajectory describes a corner, followed by a loop, followed by three smooth passages near each singular point. By r = 5.31925 there has been a bifurcation to asymmetry, a necessary precursor to period doubling. Figs. 14b and c show details of structure near the two singular points; the asymmetry is very slight but is visible at the corners. This is followed by period- doubling bifurcations and, eventually, by banded semiperiodic trajectories. Fig. 14d shows the splitting for period 10, while figs. 14e and f give magnified views of the vicinity of the corners for periods 20 and 40.

As a is increased, more periodic windows ap-

394 E. Knobloch and N.O. Weiss/Bifurcations in a model of magnetoconvection

El

• -0.8 -J

1.5

C

/

b

f

d

e t f

Fig. 14. Symmetr ica l w indows , a = 12. (a) r = 5.32 (symmetrical cycle o f period 5); (b), (c) details o f top and b o t t o m for r = 5.31925 (slightly asymmetrical period 5); (d) r = 5.317 (period I0); details for (e) r = 5.31725 (period 20) and (f) r = 5.3173 (period 40).

Q b

E. Knobloch and N.O. Weiss~Bifurcations in a model o f magnetoconvection 395

c d

J , ',

Fig. 15. Symmetrical windows: (a) tr = 13, r = 5.3218 (period 7); (b) a = 15, r = 5.3395 (period 9); (c), (d) a = 15, r = 5.336 (period 3).

pear. Fo r tr = 13, per iods 5 and 7, i l lustrated in fig. 15a, were found. Fig. 15b shows an orbi t with

per iod 9 for a = 15. Finally, a symmetr ica l per iod per iod 3 was found. Figs. 15c and d show this solut ion for tr = 15. With a = 14, 15 there are bifurcat ions to a (slightly) asymmetr ica l solution and back, indicating tha t this window lies at the top o f the pa th in the r / t -plane o f fig. 10a. F o r a = 16 two windows with per iod 3 were found,

with semiperiodic solut ions in between. These re-

suits suggest that symmetr ica l periodic windows appea r in the order (5, 7, 9, 3), though others are doubtless present. W h a t happens beyond per iod 3 is unclear, since the oscil latory branch becomes unstable.

3.3. Heteroc l in ic and homocl in ic orbi ts

Period doubl ing in fig. 5 is associated with an a l ternat ion between corners and loops in the neigh-

396

Cl

E. Knobloch and N.O. Weiss/Bifurcations in a model of magnetoeonvection

b

C

Fig. 16. D e v e l o p m e n t o f a sp i ra l , a = I: (a ) r = 5.25; (b) r = 5.255; (c) r = 5 .25575; (d) de ta i l f o r r = 5 .25579.

bourhood of a singular point and both corners and loops were found even when a = 1 (cf. fig. 3). As r was increased, the corners of fig. 3a developed tiny loops (like a ~) at r = 5.23 and these loops had become significant by r = 5.25, as shown by the detail in fig. 16a. At r = 5.255, the loop has developed a hump, shown more clearly in fig. 16b, which develops into a second looplet, as in fig. 16c. Finally, the second looplet itself repeats the process, as shown by the magnified detail for r = 5.25579 in fig. 16d. Apparently the trajectory becomes a spiral in the immediate neighbourhood of the singular point, which behaves as an unstable saddle-focus, as expected from the eigenvalues at the end of section 2.

This pattern persists wherever the oscillatory branch terminates in a heteroclinic orbit. For

a = 6, there is some hysteresis: the symmetrical corners persist until r = 5.286, when they develop into finite loops. As r is decreased the loops remain stable until r = 5.282, when they dwindle into corners. For r = 5.283 ~< r ~< 5.285 either solution is possible, as shown in fig. 4c and d. Moreover, the symmetrical loops themselves bifurcate briefly to asymmetry in the range 5.2905 ~< r ~< 5.293, as though another bubble were developing. One of the loops develops an elongated form, as shown in fig. 17a. After the bifurcation back to symmetry both loops are elongated and spiral in towards the saddle-focus, until the oscillatory branch becomes unstable and the trajectory eventually spirals in towards the stable singular point on the upper steady branch, as shown in fig. 17b. The bifurcation pattern for a = 6 is shown schematically in


i

£1

- l l s

0.8

-0.8

115

b 087

-lt.5 2.0

-0.8

Fig. 17. Asymmetrical loops, tr = 6: (a) r = 5.29 (asymmetrical loops); (b) r = 5.295 (trajectory spirals in to stable singular point).

fig. 18a, with two bubbles and an episode of hysteresis. A similar pat tern, with a transit ion f rom corners to loops which spiral in to the unstable singular point , is found at the end of the oscil latory

branch for a = 12, 16 and 20. By then, however, things have become more

compl ica ted within the original bubble. The first bifurcat ion to per iod 2, as shown in fig. 5, has a corner and a tight loop, which split a t each successive bifurcation. This s tructure continues up to

= 13.5. At t7 = 14 the initial bi furcat ion f rom asymmet ry to symmet ry at r ~ 5.21 is followed by

(a)

( b )

©

~- -~ '~ '~ - -~ '~ '~ - -~ -~ \ '~ - .~ , ,~ - -~ \~ : o = 13

t " " ,. - .1%.~_%.\%..%~ 0 =14

S--6

O=16

Fig. 18. Sketches illustrating (a) behaviour for tr = 6, and (b) relationships between trajectories with tight loops (heavy lines) and elongated loops (light lines). Dashed lines indicate unstable solutions and shading indicates aperiodicity.

successive per iod-doubl ing bifurcat ions leading to aperiodic solutions at r = 5.3055; th roughou t this sequence, f lattened corners al ternate with tight loops, as shown in fig. 19a for per iod 2, at r = 5.303, and fig. 19b for per iod 4, at r = 5.304. At r = 5.307 the (aperiodic) tight loops are, however, unstable and develop into an elongated loop with period 2. Moreover , there is a region of hysteresis. Fig. 19c shows a solution with per iod 2 and an elongated loop a t r = 5.304; as r is decreased this loses stability and relaxes at r = 5.303 to the orbit with a tight loop and period 2 that was illustrated in fig. 19a. As r is increased above 5.307 the loops become progressively more elongated until r ~ 5 . 3 1 1 7 ; for r = 5 . 3 1 1 7 2 the trajectory seems to be intermit tent , as shown in fig. 19d and for r = 5.31175 the solution is aperiodic. The bifurcat ion d iagram is apparent ly as sketched in fig.

18b. Fur thermore , ano ther compl ica t ion appears

C

398

(1

E. Knobloch and N.O. Weiss~Bifurcations in a model of magnetoconvection

b

Fig. 19. Hysteresis with tr = 14. Solutions with tight loops for (a) r = 5.303 (period 2); (b) r = 5.304 (period 4). Elongated loops for (c) r = 5.304 (period 2); and (d) r = 5.3172 (intermittent).

within the region with aperiodic symmetrical solu-

tions. When tr = 14, the transit ion f rom asymmetrical to symmetrical chaos occurs a round r ~ 5.318. At r = 5.319 there is, however, an asymmetrical orbit o f period 8, followed at r = 5.31902 by the orbit o f period 4 that is shown in fig. 20a. This window is an obvious interloper: asymmetrical solutions are embedded in a region with symmetrical solutions and the per iod-doubl ing sequence develops in the wrong direction. Moreover ,

tight loops alternate with elongated loops in fig.

20a and the elongated loops are precursors o f instability.

When a = 15, the oscillatory branch loses stability and regains it before losing it irrevocably in the end. The solution o f period 2 begins with a tight loop which gradually swells to become more elongated, without any hysteresis. The period doubles at r ,~ 5.3025 and fig. 20b shows a solution o f period 4. There are no further bifurcations until

b CI


C d

Fig. 20. Bifurcations of elongated loops: (a) tr = 14, r = 5.31902 (period 4); tr = 15: (b) r = 5.304 (period 4); (c) r = 5.3078 (period 4); and (d) r = 5.308 (period 2).

r ~ 5.3066, when there is a b i fu rca t ion back to

pe r iod 2. A t r ~ 5.3078 there is a b i furca t ion to

pe r iod 4, bu t now a small co rner and loople t

appea r on the e longated loop itself, as shown in fig.

20c. By r = 5.308, the o rb i t has rever ted to per iod

2, with a single looplet , as i l lus t ra ted in fig. 20d.

This develops into a spiral , f rom which the t ra-

j ec to ry escapes a t r ~ 5.3081 and spirals into a

s table fixed point . The osc i l la tory b ranch can be

recovered with symmetr ica l aper iod ic solut ions,

shor t ly before the appea rance o f pe r iod 3 at

r ~ 5.332. As r is decreased, the solu t ions are lost,

once more , f rom the loop at r ~ 5.326. A p p a r e n t l y

there is a homocl in ic orb i t a t r ,~ 5.308, fol lowed

by 2 dis t inct heterocl inic orbi ts at r ~ 5.326, 5.358.

Wi th tr = 16 there is again a g radua l t rans i t ion

f rom a t ight l oop to an e longated loop with pe r iod

2. This is fol lowed by a comple te pe r iod -doub l ing

sequence, with semiper iodic orbi ts a t r = 5.3055,

5.3058, as though a new bubble had burst . Then at

r = 5.306 a window with pe r iod 6 appears , as

shown in fig. 21a; this is fol lowed by pe r iod 12 at


r = 5.3065 and semiperiodic solutions at r = 5.307.

Around r = 5.308 there seem to be intermittent solutions of period 4, followed by a periodic

solution at r = 5.3081, resembling that in fig. 20c, after which the oscillatory branch is lost at r = 5.30811. This complex pattern of bifurcations is shown schematically in fig. 18b. As before, symmetrical oscillatory solutions are recovered at r ~ 5.334, with period 3.

For a = 20 the bubble seems to have collapsed. At r = 5.3 there is a solution of period 2 with a tight loop, which splits at r ~5.303 to give a

solution of period 4 with one tight and one elongated loop. The period doubles again at

Q

b

Fig . 21. O r b i t s w i t h e l o n g a t e d l o o p s : (a ) a = 16, r = 5 . 3 0 6 ( p e r i o d 6); (b) a = 20, r = 5 . 3 0 6 2 ( p e r i o d 4) .

r ~ 5.3061. This orbit is shown in fig. 21b for r = 5.3062; at r = 5.30645 the trajectory escapes from a spiral to the stable singular point, and the oscillatory branch was only recovered at r ~ 5.353, with an aperiodic symmetrical solution. As a increases, therefore, chaotic solutions are stable only

in a very narrow range of r, bounded by homoclinic and heteroclinic orbits.

4. Conclusion

The numerical experiments described in the pre- ceding section yielded a profuse variety of solutions to the truncated model of magnetoconvection

represented by eqs. (2)-(6). In this final section we at tempt to understand and to interpret these re- suits in a broader context. After summarizing the behaviour that was found, we seek, in particular, to answer two important questions: why does this model exhibit chaotic behaviour and is such behaviour generic or specific to this system?

The results are conveniently summarized by delineating different regions in the ra-plane. Fig. 22a shows the extent of the bubble confined between the first bifurcation from symmetrical to asymmetrical solutions and the bifurcation back to symmetrical solutions of period 1. Within this bubble is the region where aperiodic solutions exist, lying between the accumulation point of the first period-doubling sequence and that of the corresponding period-halving sequence at the end. Thus this diagram corresponds to the schematic pictures in fig. 10. The shaded region can be divided into two parts with asymmetrical and with (nearly) symmetrical solutions; both parts contain windows with appropriate periodic orbits. For a > 14.5 they are progressively eaten away by a region in which the oscillatory solutions disappear.

The heavy lines mark the loci of heteroclinic or homoclinic solutions, at which the period of an oscillation becomes infinite. Beyond the heavy line bounding the figure from the right, found by following the spiralling-in of the solution until it passes through the saddle-focus (cf. section 3.3), no


0

20

18

16

14

12

10

8

4

El

2 I

5-2 5 .3 r 5 .4

I S I 11 A r//A 12A-//A 3 '~ V/.,~ 7~//.4 J V//.~ I//~ r///~ ¢///#

0 I~o I~1

b

S S S S 5 A ~¢//~,~ 4 A I~'/~I 7A'f,~'///// 5~,i/,~ 7 ~'//~9 I~'//A3 ~///^ / r..-/.,A ~-//.j v/~,'/1"A ~.,'7.,I ¢'//.,0 V/ . ,O Y.//7.,4N

P2 P3

Fig. 22. (a) Location of period-doubling and aperiodic solutions in the rtr-plane. The lightly shaded region lies between the first bifurcation to asymmetry and the accumulation point of the initial cascade, and the corresponding inverse bifurcations; the heavily shaded region contains aperiodic solutions. The heavy lines denote homoclinic or heteroclinic orbits. Shil'nikov's inequality (15) is satisfied above the broken line. The gradient of/~ is parallel to the line denoting the centre of the bubble; (b) Sketch showing the occurrence of bifurcations as ~ is increased.

osci l la tory so lu t ions are found. N o t e tha t this

curve s tar ts f rom r = r ~e) = 6 when tr ~ 0.153. F o r

a < 0.153 there a re no osc i l la tory so lu t ions and as

a is increased, r (0) decreases f rom r (e) to a m i n i m u m

value o f 3.2795 for ~ = 2.156, af ter which r ( ° )~3 .4

as tr---, ~ . Thus the end o f the osc i l l a to ry b r anch

moves up the uns tab le pa r t o f the s t eady b ranch ,

shown in fig. 2a, as t~ is increased, unti l a ~ 2.

Thereaf ter , the locus o f the symmetr ica l hetero-

clinic o rb i t moves s lowly back down the curve.

Moreover , for a t> 15 there are three values o f r,

co r r e spond ing to three po in ts on the uns tab le

s teady branch , for which homocl in ic or hetero-

clinic orb i t s can occur.


A useful way of representing the bifurcation sequence is in terms of a stability parameter/~ (r, a) whose gradient is parallel to the line dividing the doubling cascades from the halving cascades (cf. fig. 10). This line is shown in fig. 22a. It terminates at a heteroclinic orbit, when the solutions lose stability. We may set # ( r (° ) ,a )=0; then for 0 </~ < #0 there is a symmetrical solution of period 1, bifurcating at Y0 to give asymmetrical solutions of period 1 which undergo a cascade of period- doubling sequences whose accumulation point is at #l, as shown schematically in fig. 22b. Immediately beyond Y1 lie semiperiodic solutions. In the interval ~t~ < / t </t2 the solutions remain asymmetrical; they are typically chaotic, with narrow periodic windows, with small but finite #-measure, interspersed among the aperiodic bands. The selection of windows that we have found is shown in fig. 22b. At Y2 there is a further bifurcation from asymmetry to symmetry. For/~ > Y2 each periodic window has an initial, basic solution that is exactly symmetrical and bifurcates to become slightly asymmetrical before indulging in a period- doubling sequence. We have not found any windows with basic solutions that have structure near both singular points and yet are asymmetrical.

The symmetrical solutions listed in fig. 22b all have odd periodicities. This is to be expected. For suppose the basic periodic solution is such that the structure around one singular point transforms to the same structure around the other singular point if the solution is run for l(2m - 1) cycles, where m is a positive integer. Then symmetry implies that there must be ( 2 m - 1) cycles in a period. This restriction does not apply to windows with basic solutions that are asymmetrical. The symmetrical windows do not seem to follow the ordering that holds for simple one-dimensional maps [40, 44, 37, 20], though the last window that we found did have period 3. We surmise that, as in the Lorenz equations [46], the symmetrical periodic solutions are born in one or more heteroclinic bifurcations and that the branch of oscillatory solutions terminates at /~ =/~3 such a bifurcation. Similarly, the asymmetrical periodic solutions are presumably

born in one or more pairs of homoclinic bifurcations. The order in which these orbits appear and disappear can in principle be determined by the methods used by Sparrow [46].

The complicated structures that were exhibited in section 3 are evidently closely associated with the unstable singular points. In their vicinity behaviour must be largely determined by the linear- ization of the equations about them. In order to explain this behaviour we have computed the five eigenvalues characterizing the stability of the steady branch, which satisfy a quintic dispersion relation F(s)= 0 [27]. The results are best under- stood with reference to fig. 23. The shape of the steady branch, sketched in fig. 23a, is independent of tr. It has two turning points and there is always

(a)

(b)

(c)

(d)

E r

F(S) I /

Fig. 23. (a) Sketch of the steady branch. Sketches of the quintic function F(s) at: (b) the turning point A; (c) B; and (d) points in BC.


a zero eigenvalue at E, A and M. For a < 0.153 there are five-real, negative eigenvalues in EA, one of which passes through zero at E and A. For 0.153 < tr < 0.292 there are two real positive eigenvalues near E, which merge to give a pair of complex conjugates; there is a Hopf bifurcation in EA, after which the roots become real, and negative, and one passes through zero at A [27]. When tr = 0.292 the Hopf bifurcation is at A, where there are two zero roots, and for a > 0.292 there are five real roots in EA, three negative and two positive. Of the latter, one passes through zero at E and A. At A, therefore, F(s ) is as sketched in fig. 23b. In AB the eigenvalues are real, with one positive and four negative. At B two eigenvalues merge, as shown in fig. 23c, and in BM there is a pair of complex conjugate eigenvalues with negative real part, together with one real positive and two real negative eigenvalues, so that F(s ) behaves as in fig. 23d. In the region BC the magnitude of the real part of the complex pair is less than the positive eigenvalue, as we should expect from fig. 23b and c. So the eigenvalues are ordered as follows

- s s > - s4 > Sl > - R e s 2 = - R e s 3 > 0. (14)

Since the positive eigenvalue must decrease to pass through zero at M, there will be a point C at which s~ = - Re s2 and in CM - Re s2 > sl > 0.

This situation resembles that discussed by Shil'nikov [45] for a three-dimensional system. He considered a .saddle-focus with a positive real eigenvalue Sl and a pair of complex conjugate eigenvalues s2, s3 such that

(a~

(b)

Fig. 24. Sketches showing orbits (a) for a saddle-focus with a homoclinic connection; and (b) for two symmetrical saddle-foci with heteroclinic connections.

of saddle foci connected by a heteroclinic orbit like that sketched in fig. 24b [7]. These results can be extended to a fifth order system provided that - s5 >> Sl, - s4 ~> s~, so that behaviour in the neighbourhood of the singular points can be described by three order variables.

In our problem only one of the five eigenvalues can be regarded as large and negative and the real parts of the others are typically all of the same order of magnitude. For example, at r = 5.3, tr = 10 we have

(Sl, Re s2, S4, Ss)

(0.613, --0.580, -- 1.640, -- 12.013). (16)

sl > - Re s2 > 0 , (15)

and with a homoclinic connection between its unstable and stable manifolds, as sketched in fig. 24a. Then it can be proved that the dynamics in the neighbourhood of the homoclinic orbit contains an infinite number of horseshoes, i.e. mixing aperiodic orbits as well as a countable number of unstable periodic orbits [45, 21, 1, 2, 7]. Similar results are likely to hold for a symmetrical system with a pair

Thus the situation may be more complicated than that envisaged by Shil'nikov. Nevertheless, the inequality (14) still seems to be related to the onset of chaos. In fig. 22a we have plotted the line along which st = - Re s2. In the absence of other eigenvalues of similar magnitude, Shil'nikov's theorem would predict complex dynamics (if stable) to the fight of this line, in the neighbourhood of a heteroclinic connection. In fact chaotic behaviour was found close to the end of the oscillatory


branch but not in the immediate neighbourhood of the heteroclinic orbit, and appears only at a ~ 9.4 while the inequality (15) is satisfied for saddle-foci with a heteroclinic connection if a > 5.8. The trajectories are affected both by the presence of the real eigenvalue s4 and by nonlinearities but it seems likely that the bifurcations described in section 3 are a consequence of the existence of eigenvalues that satisfy (15).

A recurrent feature of the computed solutions is the alternation between corners and loops in the vicinity of the unstable singular points. The corners presumably form when a trajectory approaches the fixed point in a direction close to that of the eigenvector corresponding to s4, while the loops form when the trajectory lies predominantly in the plane spanned by the eigenvectors corresponding to the complex eigenvalues. In the immediate neighbourhood of the singular points trajectories spiral in towards the saddle-focus, as shown in fig. 16, but elsewhere the real eigenvalue seems to have a larger basin of attraction. It is evident that a trajectory approaching one singular point on an orbit with a corner may emerge on the unstable manifold and connect up to trajectories with corners or loops at either of the singular points. Thus many different types of periodic orbits can develop as the parameters are varied, leading to different homoclinic or heteroclinic orbits in the limit. The links between them might be determined by the methods of symbolic dynamics that have been applied successfully to the Lorenz equations [46].

The above discussion suggests that the complex range of behaviour we have found is not specific to the system studied. Similar behaviour may be expected in a variety of systems of order n ~> 3 provided that there is a saddle-focus with a homoclinic or heteroclinic connection and eigenvalues that satisfy the inequality (15). The structure of the attractor is simplest when n = 3 [2, 7] and, as we have seen, there will be complications if n > 3 and the real parts of the other eigenvalues are of the same order as s~. Yet more structure is likely to appear as the number of order variables increases. For partial differential equations there is an infinite

set of eigenvalues; it is known, however, that the solutions can be adequately represented by Galer- kin expansions and the eigenvalues can be ordered so that those of highest order are sufficiently negative to be ignored. The nature of the dynamics in the vicinity of unstable singular points will be determined by the first few eigenvalues and we would therefore expect behaviour resembling that for low order systems.

The only example where a correspondence between the solutions of a low order model and the full partial differential equations has been estab- lished is provided by thermosolutal convection [10, 24, 38]. In this case the model is a fifth order one, quite similar to eqs. (2)-(6), and has been used to suggest that the chaotic behaviour found in the partial differential equations originates in a Shil'nikov-type mechanism [38]. The same mechanism is responsible for the aperiodic oscillations in a model of overstable convection due to Moore and Spiegel [39, 4]. This model is a third order system, and is closely related in structure to the model discussed in this paper when the amplitude of oscillation is quite small [25]. Unfortunately, numerical experiments on two-dimensional magnetoconvection [48, 43] have not so far yielded any aperiodic solutions, though bifurcations to asymmetry have been found. Further progress can be made by investigating higher order truncations to see what structures survive as more terms are included in the model equations, and by searching for a parameter range in which solutions of the partial differential equations are chaotic. We may, however, be confident that behaviour governed by the Shil'nikov condition at a heteroclinic orbit will generalize to the more realistic three-dimensional problem. Such a problem still contains a subcritical branch of saddle-points with the possibility of heteroclinic (or homoclinic) orbits. Only mild conditions on a few of the dominant eigenvalues are necessary to produce complex dynamics. If three- dimensional modes are unimportant then the eigenvalues will remain close to their two- dimensional values. Similarly,. more realistic boundary conditions will affect the presence of


chaos only in so far as they affect the dominant

eigenvalues. In conclusion, it may be useful to compare the

behaviour discussed in this paper with that of solutions to the Lorenz equations. In the latter system there is a homoclinic explosion from which there emerge a countable number of unstable periodic orbits; one of these is identical with the orbit absorbed by the subcritical Hopf bifurcation from the branch of nontrivial steady solutions [46]. When 03 is sufficiently small a heteroclinic bifurcation is also found [46]. For the system (2)-(6) there is a supercritical Hopf bifurcation from the trivial solution which may lead eventually to a heteroclinic bifurcation. The symmetry is crucial in both systems and they provide models for the behaviour to be anticipated wherever there is a Hopf bifurcation in a symmetrical system. We have

seen, moreover, how rich and complicated the dynamics may become when the Shil'nikov condi-

tion is satisfied.

Acknowledgements

This research was stimulated by discussions with participants in the 1981 G F D Summer Program at Woods Hole Oceanographic Institution. We are particularly grateful to J. Guckenheimer and C.T. Sparrow for their advice and help, and to F. Cattaneo, P. Coullet, V. Franceschini, M.R.E. Proctor, A.J. Roberts and P. Swinnerton-Dyer for comments and suggestions. This work was done while E.K. was visiting Cambridge; he thanks St. John's College for its kind hospitality and the Alfred P. Sloan Foundation for financial support.

Note added in proof

In a recent preprint P.A. Glendinning and C.T. Sparrow provide a detailed description of the Shil'nikov mechanism. They explain the origin of bubble structures and multiple solution branches, and predict the existence of subsidiary homoclinic orbits. A.J. Bernoff has confirmed numerically that such orbits are

present in the system (2)-(6) when a = ~ .

Appendix A

The results described in section 3 are drawn from a total of about 870 runs made for different values of tr and r, over dimensionless time ranges 0 ~< z ~< T. One cycle round the origin was normally completed with T ~ 20; for a typical run, T ~ 400 though some runs were continued to T = 2000. Most runs yielded solutions that appeared to be chaotic and periodic windows were generally found by chance, when the solutions for r equal to a round number were identified as periodic, intermittent or semiperiodic. In this appendix we list only nontrivial periodic solutions, excluding the basic period doubling sequence.

Symmetrical, or almost symmetrical, solutions are indicated by an S. The program, written by Dr. C.T. Sparrow, used the NAGLIB routine DO2BAF over intervals

Az = 0.1, with a variable step length chosen so that the error was less than 0.0001. Reducing AT or the error parameter displaces the bifurcation pattern slightly, without qualitatively affecting it. Some runs were checked by obtaining more accurate solutions with Az = 0.05.

Per iod i c w i n d o w s

tr = 9.5 Period 12: r = 5.2952, 5.2953 (possibly still intermittent).

406 E. Knobloch and N.O. Weiss/Bifurcations in a model o f magnetoconvection

a = 9.501

a = 9.502

a = l O

o - = 1 2

a = 1 3

P e r i o d

P e r i o d

P e r i o d

P e r i o d

P e r i o d

P e r i o d

P e r i o d

P e r i o d

12: r = 5.2953; p e r i o d 24: r = 5.2954, 5.2955, 5.2956; p e r i o d 12: r = 5.2957.

24: r = 5.2955; p e r i o d 48: r = 5.2956, 5.2957.

5: r = 5.2964; p e r i o d 10: r = 5.2965.

3: r = 5.3028; p e r i o d 6: r = 5.303; p e r i o d 12: r = 5.3034.

7: r = 5.304; p e r i o d 14; r = 5.30401.

5: r = 5.3045; p e r i o d 10: r = 5.3046.

4: r = 5.306; p e r i o d 8: r = 5.3061.

7: r = 5.30671, 5.306711; p e r i o d 14: r = 5.306712.

P e r i o d 5S: r = 5.316, 5.3162, 5.3164, 5.3166, 5.3168, 5.3169, 5.31692; p e r i o d 10S: r = 5.31694,

5.31696, 5.31698, 5.31699, 5.317, 5.31702, 5.31704, 5.3171; p e r i o d 20S: r = 5.3172, 5.31725;

p e r i o d 40S: r = 5.3173.

P e r i o d 20S: r = 5.31895, 5.319; p e r i o d 10S: r = 5.3191; p e r i o d 5S: r = 3192, 5.31925, 5.3193,

5.3195, 5.32, 5.3205, 5.3206.

P e r i o d 6: r = 5.32715; p e r i o d 3: r = 5.3272.

P e r i o d 3: r = 5.3064; p e r i o d 6: r = 5.3065, 5.30655, 5.3066; p e r i o d 24: r = 5.30665.

P e r i o d 5S: r = 5.3172, 5.3174, 5.3176, 5.3178, 5.318, 5.3182, 5.3185, 5.319, 5.3192; p e r i o d 10S:

r = 5.3193, 5.31935, 5.3194, 5.3195, 5.31952.

Pe r iod 7S: r = 5.32168, 5.3217, 5.3218, 5.32185; p e r i o d 14S: r = 5.32187, 5.32188.

o = 14 P e r i o d

P e r i o d

P e r i o d

P e r i o d

a = 15 P e r i o d

P e r i o d

a = 16 P e r i o d

P e r i o d

P e r i o d

8: r = 5.319, 5.31901; p e r i o d 4: r = 5.31902.

5S: r = 5.3232, 5.3233, 5.32332; p e r i o d 10S: r = 5.32333.

3S: r = 5.33, 5.3305, 5.331, 5.3315, 5.332, 5.3325, 5.333.

5S: r = 5.3371 ( s emipe r iod i c a t r = 5.33705).

3S: r = 5.332, 5.333, 5.3335, 5.334, 5.336.

18S: r = 5.33948(?); p e r i o d 9S: r = 5.3395, 5.33953, 5.33956.

6: r = 5.3059, 5.306, 5.3062, 5.3064; p e r i o d 12: r = 5.3065.

3S: r = 5.334, 5.3342, 5.3343; p e r i o d 6S: r = 5.3344.

12S: r = 5.3384; p e r i o d 3S: r = 5.3385, 5.339, 5.34.

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