Quantized Dynamics in the Lorenz Equations

Quantized Dynamics in the Lorenz Equations

W. M. Schaffer
(11/06/2001)

Some years ago, I was invited to contribute a chapter to a book entitled Models in Population Biology which was slated for publication by the American Mathematics Society. The volume's title notwithstanding, my collaborator and I chose to submit some results we had recently obtained for the Lorenz equations,¹ which system, of course, has nothing to do with population biology. Dynamics, we reasoned, was dynamics. Even though our essay might be of little interest to population biologists, it would eventually be picked up by the larger dynamical systems community. In the event, our paper² sank without a trace. Indeed, so far as I am aware, it has never been cited, nor have I ever received correspondence regarding its contents. Ten years later, I remain convinced that the results are still relevant, reflecting, as they do, fundamental properties of chaotic sets. Accordingly, I here reproduce the paper's essentials (subject to corrections and editing for continuity) together with some retrospective commentary.

Excerpts From the Original.

The Lorenz Equations.

Of all the dynamical systems exhibiting chaotic behavior, perhaps none has been so intensively studied as the system of differential equations first derived by Saltzman ³ and later analyzed in greater detail by Lorenz.¹ The equations may be written as follows:

   dx/dt = s (y - x)

   dy/dt = rx - y -xz (1)

   dz/dt = xy - bz

Note that x, y and z are the state variables, while b, r and s are parameters. A good review can be found in the book by Sparrow.⁴ Some of the essentials are as follows:

1. The Lorenz equations can be derived⁵^, ⁶ from the Navier-Stokes equation for Bènard flow, i.e., convection in a fluid heated from below, via three-mode truncation.

Figure 1.

2. The resulting system of ODEs (1) is deceptively simple. Nonetheless, for values of r, the so-called "Rayleigh number," in excess of a critical value, they generate sustained fluctuations which are aperiodic. As reviewed by Sparrow,⁴ there exist open intervals of parameter values for which the dynamics are truly chaotic.

3. When x(t) or y(t) is plotted in the time domain (Figure 1), trajectories exhibit diverging oscillations about one of two values. At regular intervals, the trajectory jumps from one domain to the other. Let us call the values about which oscillations diverge X⁺ and X^- and the intervals between jumps as residence or first passage times. Figure 1 suggests that to first approximation, the residence times are inversely correlated with the trajectory's initial distance, i.e., just after a jump, from X⁺ or X^-.

Figure 2.

4. Figure 2 shows the motion in phase space. Apparently, and this is easily confirmed, the points, X⁺ and X^-, are non-stable saddle foci. Between them is a third fixed point, a saddle node, which sits at the origin.

5. The attractor itself is organized into sheets pinched together along the unstable manifold of the origin.⁷ The fact that Cantor-set like structure is not observed numerically reflects the extremely rapid rate at which volumes are contracted under the flow. As a consequence, calculations of the attractor's fractal dimension yield numbers slightly in excess of 2 (Figure 3).

Figure 3.

6. Eqs(1) exhibit both "deterministic" and "random" attributes. In particular:

Figure 4.

a. The sequence and timing of orbital excursions can be summarized by discrete mappings that are effectively one-dimensional¹^, ⁸^, ⁹. In particular, let Z_i be the i^th maximum in z(t). Plotting Z_i+1 vs. Z_i yields the picture first observed by Lorenz (Figure 4).¹

b. The relationship summarized in Figure 4 notwithstanding, correlations along solution curves decay rapidly.¹⁰ This reflects the splitting effect of the saddle at the origin on nearby trajectories.¹¹

7. There is an extensive mathematical literature,⁷^,¹²^,¹³^,¹⁴ on geometrical models of the Lorenz attractor. However, as discussed by Sparrow ⁴ and Guckenheimer and Holmes,⁵ the relevance of this work to Eqs (1) remains somewhat problematic.

Evolution of the Strange Attractor.

With s = 10 and b = 8/3, the genesis and subsequent evolution of the strange attractor with respect to variation in r can be summarized as follows:

1. For 1 < r < 24.74 ... , the origin is a saddle, while the saddle foci, X⁺ and X^-, are attracting. In this parameter range, trajectories tend rapidly to one of the two saddle foci.

2. Despite the stability of the saddle foci on the parameter interval, r = [1, 24.74 ...], the system undergoes a homoclinic bifurcation or "explosion" at r = 13.926 ... . This is reflected by the existence (at this parameter value only) of a homoclinic orbit involving the origin and produces what is called a "strange invariant set." The latter consists of a countable infinity of nonstable periodic orbits (saddle cycles), an uncountable infinity of aperiodic orbits and an uncountable infinity solution curves that terminate at the origin. This set is not an attractor. Rather, as in the case of the Smale horseshoe, trajectories in its vicinity leave via well-defined "escape hatches."

3. With increasing values of r, the escape hatches become smaller, and at r = 24.06 ... , they disappear. At this point, the strange invariant set becomes an attractor which coexists with the still stable saddle foci.

4. At r = 24.74 ... , the saddle foci lose stability via sub-critical Hopf bifurcations which destroy the simplest periodic orbits created by the aforementioned homoclinic explosion. Beyond this point, there is only the strange attractor to which nearly all initial conditions tend.

5. With further increases in r, the strange attractor undergoes continuing evolution in the sense that there is an open interval of r-values on which homoclinic explosions occur for each value. Each explosion removes or creates a countable infinity of periodic orbits¹⁵ with the consequence that there is not one, but infinitely many, Lorenz attractors, the topology of which changes continuously as r is varied. For r < 28.0, the attractor collapses monotonically with the maximum number of revolutions about one saddle focus or the other declining to 24. Beyond this point, new orbits are created. The latter are subsequently, i.e., at higher r-values, destroyed by period-halving bifurcations. Thus, the attractor continues to collapse, albeit in a more complicated fashion, with the maximum number of revolutions about X⁺ or X^- continuing to decline.

6. Beyond r = 313, there is a single stable periodic orbit.

First Passage Times.

Here we focus on the time between jumps. As noted above, first passage or residence times appear to increase as trajectories are re-injected closer to one of the saddle foci - essentially, because a greater number of revolutions about the focus in question are executed before trajectories jump. In fact (Figure 5a), the pattern is more complicated. Instead of a smooth function, we see what is essentially a series of discrete steps. Correspondingly, the frequency distribution of residence times (Figure 5b) is also discretized. We also note the following:

Figure 5.

1. Discretization of residence times in the Lorenz equations was first reported by Aizawa¹⁶ who computed the mapping, T: T_i(0) --> T_i+1(0), where T_i(0) and T_i+1(0) are successive residence times (Figure 5c).

2. Discretization of residence times is not a consequence of the distribution of first extrema (Figure 5d). Rather, discretization reflects the fact that each of the "steps" in Figure 5a corresponds to a different number of consecutive revolutions about a saddle focus prior to a jump. Each time the number of such revolutions increases, residence time increases abruptly.

We can further understand the distribution of residence times as follows: For any value of r, there will be a number, i*, which we can associate with the non-stable periodic orbits. This number is the maximum number of consecutive revolutions about one of the foci that such an orbit can evidence before jumping to the vicinity of the other focus. As pointed out by Sparrow, the maximum number of any orbit on the attractor cannot exceed i*. Hence, there is a natural relationship between the topology of the attractor and the number of discrete residence times. In particular, with increasing r, there is an overall decline in i* and thus the number of observed residence times (Figures 6, 7, and 8 ).

Retrospective.

It is worth emphasizing that the discretization of continuous dynamics is not peculiar to the Lorenz system. As discussed, for example, by Argoul et al.,¹⁷ similar phenomena are observable in systems such as Rössler's equations where chaotic orbits spiral out from a central saddle focus only to be re-injected into its vicinity from afar. This is the so-called "Shil'nikov scenario,"⁵^, ⁶ in which case discretization results from trajectories executing different numbers of revolutions about the saddle focus before temporarily leaving its vicinity. Interestingly, such dynamics have been observed experimentally¹⁷^, ¹⁸ in chemical systems such as the Belousov-Zhabotinskii and peroxidase-oxidase reactions, as well as in mathematical models thereof.¹⁹

Fundamentally, discretization reflects two facts:

1. Chaotic sets are "organized" about infinite numbers of non-stable periodic orbits (saddle cycles) in the sense that every point on a chaotic set is arbitrarily close to such a cycle. Indeed, as pointed out by Auerbach and others,²⁰^, ²¹ chaotic motion may be viewed as an elaborate choreography whereby an evolving trajectory (the lead dancer) successively shadows a sequence of periodic orbits (dances with a series of partners).

2. The saddle cycles can be viewed as belonging to families distinguished by a common base period. In the Lorenz equations, these families are created by the homoclinic explosions⁴ discussed above. So long as there are at least two such families, discretization of the dynamics will result. Thus, each of the "steps" in Figure 5a correspond to motion in the vicintity of saddle cycles of different base periods.

In sum, discretization is a consequence of the topology of chaotic sets which is what distinguishes chaos from fluctuations which are truly random.

References and Notes.

1. Lorenz, E. N. 1963. Nonperiodic deterministic flow. J. Atm. Sci. 357: 282-291.

2. Schaffer, W. M. and Truty, G. L. 1989. Chaos versus noise-driven dynamics. Pp. 77-98, In, Hastings, A. (ed.) Models in Population Biology. Lectures on Mathematics in the Life Sciences. Volume 20. The American Mathematics Society. Providence, RI.

3. Saltzman, B. 1962. Finite amplitude free convection as an initial value problem. J. Atmos. Sci. 19: 329-341.

4. Sparrow, C. S. 1983. The Lorenz Equations: Bifurcations, Chaos and Strange Attractors. Springer-Verlag. New York.

5. Guckenheimer, J. and P. Holmes. 1983. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York.

6. Lichtenberg, A. and M. Lieberman. 1982. Regular and Stochastic Motion. Springer-Verlag. New York.

7. Williams, R. F. 1980. Structure of Lorenz attractors. Publ. Math. I.H.E.S. 50: 59-72.

8. Yorke, J. A. and E. D. Yorke, 1979. Metastable chaos: the transition to sustained chaotic behavior in the Lorenz model. J. Stat. Phys.21: 263-277.

9. Shaw, R. 1981. Strange attractors, chaotic behavior and information flow. Z. f. Natürforsch. 36a: 80-112.

10. Lücke, M. 1979. Statistical dynamics of the Lorenz model. J. Stat. Phys. 15: 455-475.

11. Farmer, D., Crutchfield, J. Froehling, H., Packard, N. and R. Shaw. 1980. Power spectra and mixing properties of strange attractors. Ann. N. Y. Acad. Sci. 357: 453-472.

12. Williams, R. F. 1977. The structure of Lorenz attractors. In, Bernard, P. (Ed.) Lecture Notes in Mathematics. 615: 93-113. Springer-Verlag. New York.

13. Williams, R. F. 1979. The bifurcation space of the Lorenz attractor. Ann. N.Y. Acad. Sci. 316: 393-399.

14. Guckenheimer, J. and R. F. Williams. 1980. Structural stability of the Lorenz attractor. Publ. Math. I.H.E.S. 50: 73-100.

15. Periodic orbits in the Lorenz system can be distinguished by the numbers of revolutions they make about each saddle focus before repeating. For example, an orbit which circles only X⁺ could be labeled "+", while an orbit about X^- could be labeled "-". Similarly, an orbit first circling X⁺ and then X^- could be labeled "+-", etc.

16. Aizawa, Y. 1982. Global aspects of the dissipative dynamical systems. I. Prog. Theor. Phys. 68: 64-84.

17. Argoul, F., Arneodo, A. and P. Richetti. 1991. Symbolic dynamics in the Belousov-Zhabotinskii reaction: From Rössler's intuition to experimental evidence for Shil'nikov's homoclinic chaos. Pp. 79-118. In, Baier, G. and M. Klein (eds.) A Chaotic Hierarchy. World Scientific. Singapore.

18. Hauser, M. J. B. and L. F. Olsen.1996. Mixed mode oscillations and homoclinic chaos in an enzyme reaction. J. Chem. Soc. Faraday Trans. 92: 2857-2863.

19. Schaffer, W. M., Bronnikova, T. V. and L. F. Olsen. 2001. Nonlinear dynamics of the peroxidase-oxidase reaction: II. Compatibility of an extended mechanistic model with previously reported model-data correspondences. J. Phys. Chem. B. 105: 5331-5340.

20. Auerbach, D., Cvitanovic, P., Eckmann, J.-P., Gunaratne, G. and I. Procaccia. 1987. Phys. Rev. Lett. 58: 2387-

21. Gunaratne, G. H. and I. Procaccia. 1987. Phys. Rev. Lett. 59: 1377-