Period-doubling bifurcations in measure-preserving flows

Volume 143, number 8

PHYSICS LETTERS A

29 January 1989

PERIOD-DOUBLING BIFURCATIONS IN MEASURE-PRESERVING FLOWS T. BOUNTIS and L. DROSSOS Department of Mathematics, University of Patras, Patras 261 10, Greece Received 25 April 1989; revised manuscript received 8 November 1989; accepted for publication 20 November 1989 Communicated by A.P. Fordy

We study period-doubling bifurcations in periodically driven two-dimensional flows, which are measure-preserving. We demonstrate on several examples, that this important bifurcation sequence proceeds very similarly and with the same universal constants, as in the Hamiltonian case of two degrees of freedom. This is explained by the fact that the Poincaré map of our models, upon return to any point of their periodic orbits, is (locally) area-preserving, a result which suggests an equivalence of all measures for such bifurcation phenomena in measure-preserving systems.

1. Introduction Period-doubling bifurcations have been extensively studied in Hamiltonian systems of two degrees of freedom, and their discrete analogue: area-preservingmappingsoftheplane [l—3].Thepreciseoccurrence of these bifurcations about an axis of symmetry and the universality of their scaling and parameter constants have been well established by numerical [2], as well as analytical (renormalization) methods [3]. We investigate here this important sequence of bifurcations in a larger class of flows gt, described by the equations

and study numerically their periodic orbits, on the surface of section ~T={(Xk, Yk)= (x(1k Y(tk))/tk =kT, keZ} (4) ,

under the Poincaré map P, PUk=Uk+I,

(1)

XEIR~

which are measure-preserving [4], in the sense that they conserve a smooth density M(x) >0, for all i.e. ~,

J

J

M(x) dx=

M(x) dx,

DcP’~.

(2)

t(D)

D

g

We consider, in particular, periodically driven flows with n=3, of the form

keZ.

(5)

We find that near every point Elk ofa periodic orbit of period m, satisfying ~ meN~J, (6) the return Poincaré map pm at that point is area-preserving, i.e. det

i=f(x),

Uk=(Xk,Yk),

fl

DP(Elk)=l

,

(7)

k=1

where DP(Elk) is the linearized Poincaré-map, taking small variations of Elk to the corresponding ones about Elk÷ We have found this to be true for all the periodic orbits we computed in our models and conjecture that this maybe generally valid for all measure-preserving systems. In any event, (7) certainly holds for all the orbits we followed in our period-doubling sequences and ~.

explains why the results are the same as in the area~=f 1(x, y, z),

j’=f2(x, y, z), 1=1

J(x, y, z) _—f(x, y, z+ T),

1= 1, 2,

,

(3a) (3b)

preserving case: Indeed, studying measure-preserving models, with and without obvious symmetries, we observed all the usual period-doubling phenom-

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379

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PHYSICS LETTERS A

.~=x—xy+qcoswt,

(8a) (8b)

2

j)=—y+xy+rcoswt, which in the case

~

is a reversible dynamical system [5] Indeed in that case it is easy to see that the transformation (x y )=R(x y)=(y x) (10)

-

1

~ 4/~’

,__

~

~

reverses the sign a0nf ~o

05

~s

~

15

2

~

k Fig. 1. Surface ofsection plot oforbits ofthe Lotka—Volterra model (8) at q= — r=0.225. Note at P~,P 2 the orbit of period 2 on the symmetry x=y axis, and the higher period orbits around it, near all ofwhich the return Poincaré map P’” is area-preserving.

ena, and computed universal constants <5, a and /i which appeared to converge to the well-known values of strictly area-preserving mappings and two-degrees-of-freedom Hamiltonian systems [1—31, for which of course, M(x) = 1, in (2). Reversible dynamical systems (i.e. systems which posses an involution in phase space that reverses the direction of the vector field [5]), near symmetric fixed points and periodic orbits, are qualitatively very similar to Hamiltonian systems [5—7] (see fig. 1). Recently, Quispel and Roberts have shown that symmetric periodic orbits of reversible mappingsperioddouble in exactly the same way as in area-preserving mappings [8,9]. In this Letter, we present evidence which suggests: (a) That similar results also hold for flows, and (b) That measure-preserving systems follow Hamiltonian period-doubling behavior, In fact, measure preservation might be used to establish further analogies with Harniltonian flows, in cases where reversibility and symmetry of periodic orbits cannot be easily demonstrated.

2. A measure-preserving Lotka—Volterra model Consider the periodically forced Lotka—Volterra equations 380

~

i~e

of fixed points x=y is an axis of symmetry of the

.-r. -~

29 January 1989

system. At q=0, (8), with (9), hastwo fixedpoints: A hyperbolic (unstable) one at (0, 0), and an elliptic (stable) one at (1, 1), which, for q 0, become fixed points of the Poincaré map P on the surface of section (4). As q becomes nonzero, the stable fixed point immediately turns unstable (still lying on the symmetry axis x=y) and bifurcates into a periodic orbit of period 2, whose points P1, P2 also lie on the x=y axis, see figs. 1 and 2a. This is the beginning of the period-doubling cascade. As q increases further, period 4 bifurcates out of perod 2 at q=q2, period 8 out of 4 at q=q3 and so on, as shown schematically in fig. 2a. The first few bifurcation values q, and the corresponding approximants of the universal constant 8.721...

(11)

a= urn a,= lim(dk_l/dk)=4.0l8...

(12)

<5=

lim

<5k

k-~oo

=

lim k-*oo

~

~lk—

—

q~1

and the scaling constant

of the Hamiltonian case, are given in table 1 (dk is the distance between the two points of the period 2k orbit lying on the symmetry axis, at q=q~~~). The convergence of these <5k, ak approxirnantS to the values (11) and (12) respectively is relatively slow. Due to accuracy limitations, we did not follow here the orbit of period 32 to its destabilization, in order to compute <55 and a5. This may be done using the Fourier series representation of these orbits and some recently developed techniques for solving nonlinear algebraic systems [10,11]. We do believe, however, that the results oftable I (and also of table

Volume 143, number 8

29 January 1989

PHYSICS LETTERS A

1.2

1.5

1.8

2.1

~

4\ /

Table I q= —r, w=2 in (8).

/

2.1

// ~

2

~-F~

‘~

1.8

8-..

0.2

k q,~ _____________________________________________ 1 2

0.0 0.7381028

21.627611

10.76454

3 4

0.7722306 0.7759941

9.068101 8.758436

3.83489 4.07397

5

0.7764238

41 •.815

I

~-.

2

8

0.1

1.2 (a)

~

_______________________________________ 0.1 0.2 ~ X~

lating the determinant of the linearized map DP, at some point Elk of a periodic orbit, we generally find detDP(Elk) ~ 1. However, for all the periodic orbits we checked (not only those of the period-doubling sequence) we verified that eq. (7) is satisfied, and hence that P is area-preserving upon return to one of the points of the orbit [9]. Similar results also hold for the case q~ r. Note, however, that here (8) no longer possesses the reversing involution (10) and is not symmetric about x=y. It may very well be reversible and have a symmetry curve in Z~,but it is not as easy any more to find it, or show that it does (or does not) exist. What is true, for all q and r, is that (8) is a mea—

2.1

2.5

2.3

,

0.7

~8

—~---~--.-

8

0.6 0.5

2

sure-preserving flow [12]. Indeed, writing (8) as a third order system of the form (3), we look for an invariant density M(x), which satisfies as a direct

—

.4

.

2

~i

8

—

D.3

I

I

.25\

.1

.15

.2

—

+(x—y)M+M2=0,

_______________________________________ .05

consequence of (2) the PDE [4] div(Mf)=(x—xy+qcoswz)M~ +(—y+xy+rcoswz)M~

~

X~

Fig. 2. (a) The first few penod-doublingbifurcations of (8) with q= —r. Note, as in the usual Hamiltonian case, the symmetry of these period 2” points, about the axis x=y of fixed points of involution (10). (b) Same as in (a), with q~—r=0. In spite of the obvious similarities between the two figures the (suspected) symmetries of this case are not easy to determine.

2, below) are convincing enough that these systems period-double just like Harniltonian ones (whose Poincaré map is everywhere area-preserving). Now, the Poincaré map P of our system certainly does not preserve areas everywhere. In fact, calcu-

(13)

where subscripts denote partial derivatives of M with respect to the indicated variable. It is not difficult to check now that eq. (13) has the solution M(x) =exp{

—

x— y+ [(q+ r) 1w)] sin wz} ,

(14)

which is certainly a density, since M(x) >0 for all x, and shows that (8) describes a measure-preserving flow for all q and r. Setting now r= 0, we compute the first few perioddoubling bifurcations of (8), as the value of q is increased. In table 2, we list the resulting approximants ofthe universal constants <5and a, which again appear to converge to the values (11) and (12) of the Hamiltonian case. 381

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Table 2 r=0, w=2 in (8).

_______________________________________________ k

q~

1

0.0 1.4062298 1.4977311 1.5084281 1.5096542

2 3 4 5

15.36842 8.55392 8.72441

29 January 1989

sents a true density for this flow. Now, with q = 0, (15) is a reversible dynamical system, with reversing involution (x’,y’)=R(x,y)=(x, —y) (18) 0. However, for q~0(just and of symmetry y=model as inaxis the Lotka—Volterra (8) with q~—r), reversibility can no longer be easily established and symmetry properties of periodic orbits of (15) are not evident.

11.74556 3.67238 4.09633

Here also we checked that the Poincaré map was area-preserving, only upon return to the starting point of a periodic orbit. Moreover, as in the previous (q= —r) case, the eigenvalues of the return map are on the unit circle and split off the circle at 1, at every period-doubling bifurcation. However, looking at fig. 2b, it is no longer clear about which symmetry curve (if indeed there is one!) these bifurcations occur in the r=0 case. There are definite similarities between figs. 2b and 2a, which allows us, e.g., to pick the right points for the calculation of the dk’s (and aks) in (12), and make us suspect that the periodic orbits of fig. 2b are also symmetric, albeit about an (as yet unknown) curve in the surface of section ~ —

Still, in both cases (q=0 and q~0),we found that the period-doubling bifurcations of the fixed point near the origin of (15), with f(x) = —x and f(x) = —x—x2, were in complete agreement with those of Hamiltonian two-degrees-of-freedom flows and area-preserving mappings of the plane.

4. Concluding remarks Theory and practice concerning the dynamics of two-degrees-of-freedom Hamiltonian systems and area-preserving mappings of the plane are reasonably well established [13]. It is, therefore, natural to ask which aspects of this dynamics and in what form survive in the more general framework of measure-preserving systems. —

—

3. Other examples

On the other hand, of great interest is also the notion of reversibility, and its connection with the dy-

We have obtained results similar to the above following the period-doubling bifurcations of other measure-preserving flows, like e.g. the forced anharmonic oscillators

namical properties of flows and/or planar maps. Now, almost all area-preserving mappings studied to date have been reversible, while not all reversible

= y+

q cos wt,

= — v2

+f(x) + r cos wt,

(1 5)

with f(x) = —x—Kx2. Note that the corresponding eq. (13) for system (15), with ~ = 1, is (y + q cos wz ) M~+ —

2yM+ M~= 0.

[— ~2 +f(

x) + r cos wz ] M~ (16)

Eq. (16), clearly, has a solution of the form M(x) = exp [ 2x —2 (q/w) sin wz ,

(17)

demonstrating that (15) is indeed measure-preserving, since M(x)>0, for all x, and thus (17) repre382

systems behave like Hamiltonian ones: As has proved for flows [5], and Quispel andDevaney Roberts have demonstrated on several examples of planar maps [9], reversibility requires also theHamiltonian existence of symmetric periodic orbits to reproduce behaviour. In this Letter, we have presented evidence in support of the following conjecture: The universal dynamical properties of measure-preserving systems are the same as those of Hamiltonian ones. In other words, all measures M(x) are equivalent to M(x) = 1, the measure of the volume-preserving case. Of course, our evidence to date comes from the study of period-doubling bifurcations of several examples of measure-preserving flows. To test the more general validity of our conjecture, one should also

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PHYSICS LETTERS A

study the break-up of KAM curves in these models. Work is currently in progress in this direction, and results are expected to appear in future publications [14]. Our measure-preserving flows also behave as if they were all reversible, having symmetric periodic orbits; and, indeed, for some parameter values they are. If this could be proved for all choices of the parameters, then, according to recent results on reversible maps [8,9], KAM curves would also be expected to break-up exactly as they do in the Hamiltonian case. Finally, we point out that all our results to date concern periodically driven two-dimensional flows. It would be very interesting to examine how (and whether!) these results extend to the more general case of three-dimensional (and higher) measurepreserving systems.

Acknowledgement We wish to thank one of the referees for pointing out that Darboux’s theorem may be used to show that all 3-D measure-preserving flows are equivalent under suitable changes of coordinates. We do not think, however, that this is a widely appreciated fact when the coordinates are not canonical, as is the case with the examples treated in this paper. We also wish to thank Dr. R. Quispel, Professor H. Capel and Professor C. Zagouras for many illuminating discus-

29 January 1989

sions on reversibility, measure-preserving systems and periodic orbits. This work was partially supported by a grant from the Greek Ministry of Industry, Energy and Technology, Division of Research and Technology.

References [1] R.H.G. Helleman, in: Universality in chaos, ed. P. Cvitanovic (Hilger, Bristol, 1984) p. 420. [2] T.C. Bountis, Physica D 3 (1981) 577. [3] J.M. Greene, R.S. MacKay, F. Vivoldi and M.J. Feigenbaum, Physica D 3 (1981) 468. [41V.1. Arnol’d, ed., Encyclopaedia of mathematical sciences, Vol. 3. Dynamical systems III (Springer, Berlin, 1988) p. 131. [5] R.L. Devaney, Trans. Am. Math. Soc. 218 (1976) 89. [6] J.K. Moser, Math. Ann. 169 (1967)136. [7] M.B. Sevryuk, Lecture notes in mathematics, Vol. 1211. Reversible systems (Springer, Berlin, 1986). [8] G.R.W. Quispel and J.A.G. Roberts, Phys. Lett. A 132 (1988) 161. [9] G.R.W. Quispel and J.A.G. Roberts, Phys. Lett. A 135 (1989) 337. [lO]M.N. Vrahatis and T.C. Bountis, On a convergenceimproving method for computing periodic orbits near bifurcation points, J. Comput. Phys. (1989), to be published. [11] N. Budinsky, Ph.D. Thesis, Department of Mathematics, University (1983). [121 Clarkson G.R.W. Quispel, private communication. [13] R.S. MacKay and J.D. Meiss, eds., Hamiltonian dynamical systems (Hilger, Bristol, 1987). [14] L. Drossos and T.C. Bountis, in preparation.

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