Volume 57A, number 5 PHYSICS LETTERS 12 July 1976
AN EQUATION FOR CONTINUOUS CHAOS
O,E. ROSSLER
Institut für physikalische und theoretische Chemie der lfniversitlit Tubingen, Germany
Received 27 May 1976
A prototype equation to the Lorenz model of turbulence contains just one (second-order) nonlinearity in one
variable. The flow in state space allows for a “folded” Poincard map (horseshoe map). Many more natural and
artificial systems are governed by this type of equation.
Continuous chaos has, under the name of deter- y
ministic nonperiodic flow, been first described by
E.N. Lorenz in a model of turbulence [1]. The same
model has recently been found to apply to lasers as
well, explaining the phenomenon of irregularly spiking
lasers in this case [2]. The Lorenz equation consists of
three coupled ordinary differential equations which
contain two nonlinear terms (of second order, xz and
xy): Z
8(1) x ____________
~=lO(y x), 5’”x(28—z)—y, zxy -
3z.
The flow of trajectories in state space (fig. 1) shows Fig. 1. Trajectories of the Lorenz model (eq. 1). Stereoscopic
two unstable foci (spirals) suspended in an attracting view. (Parallel projections; the left-hand picture is meant for
surface each, and mutually connected in such a way the right eye and vice versa.) Numerical simulation on a
HP9820A calculator with peripherals, using a standard
that the outer portion of either spiral is “glued” Runge-Kutta-Merson integration routine (adapted by F.
toward the side of the other spiral, whereby the outer- Göbber). Axes: —29 ... +29 for x andy, 0 ... 58 for z. Initial
most parts of the first spiral map onto the more inner conditions assumed: x(O) = 2.9, y(O) = 1.3, z(O) = 25.
parts of the second, and vice versa. Unexpectedly, the Final values: tend = 31.668, x(end) = 4.451, y(end) =
qualitative behavioi of eq. (1) is still insufficiently 2.3833, z(end) = 30.933.
understood, mainly because the usual technique for
analyzing oscillations — to find a (Poincaré) cross- generated flow (fig. 2) is that of a (disk-embedded)
section through the flow which is a (auto-) diffeo- single spiral. The outer portion returns, after an apmorphism
[3] is not applicable. A trick which propriate twist (so that the formation of a Möbius
exploits the inherent (though imperfect) symmetry band is involved [4]), toward the side of the same
between the two “leaves” of the flow (see fig. 1), so spiral, with the outermost parts again facing the more
that in effect only a single leafneeds to be considered, central parts.The trajectorial convolute looks much
has yet to be found. like that on a single leafof fig. 1. This time, however,
Therefore, a simpler equation which directly a qualitative understanding of the “chaotic flow”
generates a similar flow and forms only a single spiral (a term coined by Yorke for analogous discrete systems;
may be of interest, even if this equation has, as a see ref. [5] and below) is easier to obtain.
“model of a model”, no longer an immediate physical By drawing an unwindingspiral on a transparent
interpretation. The proposed equation is: sheet of paper, foldingthe sheet over, and gluing the
(2)
= —(y +z), j~= x + O.2y, I = 0.2 + z(x 5.7). outer part of the spiral onto the inner one, an analog
to the flow of fig. 2 is obtained. When carefully folThere
is only a single nonlinear term (zx) now. The lowing-up the prescribed course of a trajectory within
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Volume 57A, number 5 PHYSICS LFTTERS 12 July 1976
[4]. Thus, the limit set is a so-called strange attractor
[61 whose cross-section is a two-dimensional Cantor
set~the flow is nonperiodic and structurally stable [6],
even though all trajectories are unstable [1]. Thus,
most of the results which have been conjectured about
L~Lc~eq. (1) [1] turn out to be true for eq. (2). The simplic-ity (not to say: triviality) of eq. (2) has the additionalasset that some further results that one would like toknow about strange attractors in general (basin struc-x ture; emergence through hard and soft bifurcation;behavior of the monostable variant; behavior under
Fig. 2. Trajectorial flow of eq. (2). Stereoplot as in fig. 1. time reversal) may be easier to obtain with this equaAxes:
14... +14 forxandy, 0... 28 forz. Assumed initial tion.
conditions: x(0) — 0,y(O) — 6.78, z(0) = 0.02. Final values: Eq. (2) incidentally illustrates a more general printend
= 339.249, x(end) 7.8366, y(end) — 4.1803,
z(end) — 0.014385. ciple for the generation of “spiral type” chaos [71:
combining a two-variable oscillator (in this case x and
y) with a switching-type subsystem (z) in such a way
this “trap”, one comes up with a picture very much that the latter is being switched by the first while the
like that of fig. 2. Ifone then varies the degree of flow of the first is dependent on the switching state of
overlap, it is apparent that nonperiodic behavior is the latter. Eq. (2) has in fact been derived from a more
obtained if and only if at least two successive increases complicated equation for which this “building-block
of amplitude are possible for the outermost trajectory, principle” has been shown to apply strictly [41.The
after it has become the innermost trajectory. Most named design principle not only enables the construerecently,
a proof of this result has been described tion of an unlimited number of artificial chaotic sys(under
the suggestive title “period 3 implies chaos”) tems, but at the same time can be used as a guideline
for one-dimensional “cap-shaped” maps [5]. Such a for the identification of further natural systems showmap
will indeed be found along any cross-section ing the same behavior (by suggesting to probe into
through the desired paper-sheet flow, if the reentry their parameter space). The field of possible applicapoint
through the cross-section is plotted as a function tions of equations of the type of eq. (2) thus ranges
of the entry point. (The converse is also true: every from astrophysics, via chemistry and biology, to
cap-shaped map gives rise to a paper-sheet flow pos- economics [71.
sessing this map as a Poincaré cross-section.) To conlude, continuous chaos is “stangely attracCloser
inspection of fig. 2 reveals, however, that the tive” as a physical phenomenon (cf. [8]).
flow actually is not confined to a (folded) twodimensional
surface, but rather to a ~folded) disk of This work has been supported by the Stiftung
finite width. Every cross-section through the flow is Volkswagenwerk. I thank Professor H. Haken for
therefore two-dimensional (rather than one-dimensio- discussions.
nal). It assumes the form of a horseshoe between one
transition and the next. This becomes evident if one
follows the course of one (at first) rectangular cross- References
section as it is “stretched” and then “folded” before Ill EN. Lorenz, J. Atmos. Sci. 20 (1963) 130.
it is mapped backonto itself. 12] II. Haken, Phys. Lett. 53A (1975) 77.
As it turns out, the properties of such “folded” 131 S. Samle, Bull. Amer. Math. Soc. 73 (1967) 747.
diffeomorphisms, called horseshoe maps [3], are well- l~lO.E. Rössler, Z. Naturforsch. 31a (1976) 259.
known in the theory of dynamical systems, and so is 151 T.Y.LiandJ.A.Yorke, Amer. Math. Monthly 82 (1975) 985.
161 D. Ruelle and F. Takens, Comm. Math. Phys. 20(1971)167.
the fact that each of them can give rise to a three- 171 O.E. Rdssler, Chaos in Abstract Kinetics: Two Prototypes,
dimensional “suspended” flow [3]. Only a simple Bull. Math. Biol.,to be published.
(three-dimensional) example had been lacking so far [81R.M. May, Nature, London 256 (1975) 165.
398