ROSSLER BANDS IN ECONOMIC AND BIOLOGICAL SYSTEMS
B.G. Hald, C.N. Laugesen, C. Nielsen and E. Mosekilde
Physics Laboratory III
The Technical University of Denmark
2800 Lyngby, Denmark
E. R. Larsen
Institute of Computer and System Sciences
Copenhagen School of Economics and Business Administration
1925 Frederiksberg C, Denmark
J. Engelbrecht
Department of Dairy Science
Royal Veterinary and Agricuitural University
1870 Frederiksberg C, Denmark
Abstract
A Réssler band is presumably the simplest of all chaotic attractors.
It can develop in systems with only three state variables if two of
these produce an outward spiralling trajectory, and the third folds
this trajectory back towards its center when the amplitude of the
expanding oscillation becomes sufficiently large. In the present
paper we show how Réssler bands can develop through slight modifica-
tions of well-established economic and biological models.
1. Introduction
Non-linear dynamic phenomena are increasingly recognized as essential
for the function of normal biological systems. Besides the beating of
the heart and the ovarian cycle, the classical examples of self-
sustained physiological oscillations, investigations performed during
the last decade have revealed the existence of a great variety of
rhythmic phenomena with periods ranging from fractions of a second to
several hours or even days. Examples are hormonal regulation and
neuronal function, with secretion of insulin (1), growth hormone, and
luteinizing hormone (2) all showing periods of 2-3 hours. A somewhat .
similar period is observed for the production of enzymes in certain
bacteria (3).
In the macroeconomic realm, the economic long wave as depicted in the
Sterman model (4) is a typical example of a self-sustained oscilla-
tion. It is also clear that other modes of the macroeconomic system
can turn unstable and produce highly non-linear dynamic phenomena.
The ordinary 3-7 year business cycle thus appears to become destab-
lized during the later stages of a long-wave upswing (4).
510
We have worked with this type of phenomena for a couple of years. In
particular, we have developed a model for the regulation of pressures
and flows in the nephrons of the kidney (5). For normal parameter
values, this model reproduces experimentally observed self-sustained
oscillations in the nephron pressure. However, if the delay of the
main negative feedback loop is increased, deterministic chaos deve-
lops through a Feigenbaum cascade of period-doubling bifurcations
(6). This corresponds to observations for hypertensive rats and for
rats which have received.a light dose of furosemide, a drug which is
known to influence the function of the kidney. A similar cascade of
period-doubling bifurcations has been found in a model of resource
allocation in a managerial system (7), and we have also ‘shown how the
economic long-wave model can become chaotic when perturbed by a sinu-
soidal variation in the demand for capital to the goods sector (8).
Both for the kidney model and for the managerial resource allocation
model, the observed chaotic attractor resembles a Réssler band (9).
This is presumably the simplest type of a strange attractor that one
can think of, and the purpose of the present paper is to show how
similar attractors can develop in other economic and biological
models. The examples that we shall consider are slightly modified
versions of the commodity market model developed by Meadows (10) and
of a model of a classical microbiological control system, the trypto-
phan operon (3). Our aim is primarily to clarify the type of struc-
ture which can lead to Réssler-like attractors.
In its original version, the Réssler model consists of the follow-
ing set of three coupled non-linear differential equations (8):
ax ay dz
— = iy ~ © (la) —=x+ay (1b) —=b+ xz - cz (1c)
at at dat
where a, b and c are parameters. Often, a and b are kept constant at
a=b = 0.2 while c is varied between 2 and 10. A typical Réssler
band. exists, for instance, for c = 5.7. As illustrated in figure 1,
this band may be considered to be generated by:a trajectory which for
small z spirals outwards immediately over the xy-plane. As the ampli-
tude of this expanding oscillation becomes sufficiently large, the
term xz in the equation for dz/dt causes z to increase, and the
trajectory then moves away from the xy-plane. The increase in z gives
rise to a reduction of the amplitude of oscillation such that the
511
trajectory is folded back towards the z-axis. The reduction in Ix] in
turn causes z to relax towards its initial small value whereafter a
new outward spiralling movement is begun.
Figure 1. Rdssler band
obtained from Eqs. la-c for
a = b = 0.2 and c = 5.7. The
chaotic attractor arises
through a Feigenbaum cascade
of period-doubling bifurca-
tions as c is increased.
2. The Commodity Market Model
Figure 2 shows the flow diagram for a slightly modified version of
Meadows' commodity market model (10). Let us for a moment consider
the production capacity PC to be constant. The model then contains
two negative feedback loops which regulate production and consumpt-
ion, respectively. In accordance with classical cobweb theory, the
stability of the model is controlled by the elasticities of the
demand and supply curves in the equilibrium point. These curves are
represented by the table functions for indicated per capita consump
tion (IPCC) and for desired production (DP). A high demand elasticity
and a low supply elasticity produce stable behavior. It is quite
likely, however, that a commodity market in certain periods can
operate under unstable conditions. Our first modification to the ori-
ginal model has therefore been to reduce the demand elasticity, until
growing oscillations were obtained.
To restrain these expanding oscillations we have introduced a second
modification: it is assumed that depreciation of production capacity
depends upon capacity utilization. Thus, if the capacity utilization
factor CUF averaged over the capacity adjustment time CAT is unac-
ceptably low, the capacity depreciation. rate CDR is increased by a
factor DMF which typically takes a value of 2 to 6. The associated
reduction in production capacity forces the model into a region
512
FOP
gpolation
/
Po
proat. ad
deiay’ [> inventory
car
capital
acquisitios
Pc
product.
capacity
COR
capital
repreciationfi+"} “cprecst
multip.
or
Figure 2. Slightly modified version of the classical commodity
market model.
where capacity utilizaton is nearly complete and where the effective
supply elasticity therefore is small. However, as the production
capacity is gradually rebuilt, a new growing spiral in production and
consumption is started.
Figures 3-6 present a series of simulation results obtained with the
above model as the capacity adjustment time CAT gradually is in-
creased from 1 to 5 months. Each figure shows a projection of the
phase space trajectory into the plane expanded by inventory and
production capacity, together with a plot of simultaneous values for
18.00
\
\
|
i
16.00
PRODUCTION CAPACITY
exrrcnerererem - 14.00, eererity tape enmnsparamen
ae cra BRS ERE TRIE SS eo sare aa ce ae
Figure 3. Transient approach to the limit cycle existing for CAT=
1 month and DMF = 4. s
513
production and sale. Figure 3 includes the transient behavior as the
model is initiated with a relatively high production capacity. Note
how production and sale during this transient spiral outwards towards
the stable limit cycle attractor. In the subsequent figures the
transient has been left out, retaining only the stationary long-term
behavior.
Figures 4 and 5 show the stable period-2 and period-4 cycles existing
for CAT = 1.75 and 3.5 months, respectively. In both cases DMF = 4.
It is interesting to note in figure 4 how the production capacity
builds up during two oscillations of the inventory cycle until sud-
denly the capacity utilization becomes too low, and the production
capacity is reduced by about 10% within a relatively short period of
time. Figure 5 shows a somewhat similar picture except that four
inventory oscillations must be completed before the model reproduces
itself. As CAT is further increased, the model exhibits the usual
cascade of period-doubling bifurcations to reach a chaotic regime for
CAT ® 4,2 months. Figure 6 shows a characteristic example of a
chaotic attractor. In this mode, the model is sensitive to the ini-
tial conditions. It is therefore impossible to make predictions over
longer time horizons even though the equations of motion are comple-
tely deterministic.
A more complete overview of the behavior of the model is provided by
the bifurcation diagram in figure 7. This diagram was obtained for
DMF = 4.25 by slowly scanning CAT from 1 to 8.5 months while plotting
for each oscillation of the model the maximum production attained.
For low values of CAT we observe the characteristic Feigenbaum cas-
cade of period-doubling bifurcations (6) with a transition to chaos
at CAT = 3 months. In the chaotic regime we note the periodic win-
dows, particularly the very pronounced period-3 window existing for
CAT > 5.7. months.
To complete. our investigation of the commodity market model, figure 8
shows the distribution of stationary solutions in the parameter plane
expanded by CAT and DMF. Besides the already mentioned periodic
solutions this figure shows the positions in parameter plane where
one can find stable period-8, period-16, period-6, period-12, and
period-5 solutions. These findings are very similar to those pre-
viously reported for the managerial resource allocation model (7).
514
j i
.
E resol
8
a
3 reco 200
Zz
S isso A aooo 4
E ~~
B iso it
3 i
me
Be t4s0 16.00
1400 dom 14.00, bo
20.00 40.00 60.00 80.00 100,00 120.00 140.00 5.00 10,00" 18.00. 20.00. 25.00” 30.00 "35.00.
INVENTORY PRODUCTION
Figure 4. Stable period-2 cycle observed for CAT = 1.75 months.
17.00 26.00 2
Ff i
>»
& i i
Bisse 3 24.00 3
ai 3
a 3 i
Sreoo na00 4
z i ae
Sisso S 20.0 4
a 4 a
5: i
3 is.00 2 1e.00 3
6 i 3
a i :
Peiaso 3 1600 4
3 FI
j 4.
00 . . x
140898 "4000 GOOD BOOS TOGD T2800 Ta0.00 1400 IS TEGO Se S000 BSUS SOE 35.00
INVENTORY PRODUCTION
Figure 5. Stable period-4 cycle observed for CAT = 3.5 months.
17.00 26,00
5 isso 24.00
=
Sss.00 22.00
z
Basso so
5 a
B 15.00 18.00
o
7
Be taso 16.00
1400950 4ObO BOG BEBE TODS TIOGS” V90.00 40 GS TOON Teas Oe aOD BOGS 35100
INVENTORY PRODUCTION
Pigure 6. Chaotic attractor observed for CAT = 5 months.
515
33.00 3
)
&
8
29.00
MAX(PRODUCTION
8 8
28.00 4
27.00 rT r i r r r r 1
0.00 1.00 2.00 3.00 4,00 5.00 6.00 .7.00 800 9.00
CAT
Figure 7. Bifurcation diagram for the commodity market model.
6.00 3° —
5.00 4
4.00 4.
DMF
3.00
2.00
PERIOD=1
1.00 4
0.00 r r 1
0.00) 200 4.00 6.00 8.00 10.00 12.00
CAT
Figure 8. Distribution of stationary solutions for various values of
the capacity adjustment time and the depreciation muitiplier.
3. The Bacterial Operon
As discussed in the introduction to this paper, it becomes increas~
ingly clear that many biological control systems operate in an un-
stable mode. There may be a number of advantages associated with such
operation, and attempts to understand these are presently at the
center of our interests. At this place, however, it suffices to note
that many technical control systems including those of ordinary
refrigerators, freezers, airconditioners, oil burners, etc. are con-
structed to operate in a bistable mode where they switch between an
516
on state and an off state.
As an example of an oscillatory bacterial control system, Bliss et
al. (3) have studied the tryptophan operon in E. coli. Experimentally
they have observed that the gene expression from this operon becomes
unstable if the normal feedback inhibition is reduced, and that self-
sustained oscillations with a period of the order of 30 min result.
These oscillations can be observed both in the intracellular concen-
tration of tryptophan and in the concentration of the enzyme anthra-
nilate synthetase which is involved in control of the tryptophan
production.
Figure 9 shows a flow diagram for the model suggested by Bliss et al.
(3). mRNA is produced by transcription of the bacterial DNA molecule.
It is assumed that this process requires approximately 30 sec, and
that the produced mRNA molecules have a lifetime of 60 sec. The con-
centration of mRNA molecules controls the production of the enzyme
anthranilate synthetase through a sigmoidal relation. This implies
that there is a threshold concentraton of mRNA below which very
little enzyme production takes place, a control region in which the
enzyme production increases significantly with the RNA concentration,
and a saturation region in which changes in the RNA concentration
have little influence on the enzyme production. The translation
process is assumed to take 30 sec, and the produced enzyme molecules
are assumed to have a lifetime of 500 sec.
transcription ;
x Figure 9. Flow
s degradation diagram for the
pa ee tryptophan
operon.
degradation
C.
——
[>| enzyme
innibition
\ »
tryptophan
cell metabolism production
517
The concentration of anthranilate synthetase controls the production
of tryptophan, an amino acid which plays an important role for the
cell metabolism. Also in this case, the control is via a sigmoidal
relation. An advantage of this type of cascaded system is that a
considerable amplification can be attained such that a few mRNA
molecules can give rise to a significant number of tryptophan mole-
cules.
Similar cascaded systems are found in human hormonal regulation. The
production of testosteron (the male sex hormone), for instance, is
regulated by the production of luteinizing hormone in the pituitary
gland, and this production is again controlled by the production of
luteinizing hormone releasing hormone tn hypothalamus. The endocrine
gland has a tendency to oversecrete its hormone. Because of this
tendency, the releasing hormone exerts more and more of its control
effect on the target cells in the pituitary gland which also overpro-
duce. To restrain the hormone production, the cells at the final
stage of the cascade usually produce a factor which feeds back to the
first stage and causes the cells here to decrease their secretion
Likewise, for our enzymatic control system in figure 9, the end
product feeds back to the rate of mRNA transcription to reduce the
transcription rate when the concentration of tryptophan becomes high
enough. However, due to the involved time delays and phase shifts,
the negative feedback regulation may become unstable and produce
self-sustained oscillations.
The modification that we have introduced is to assume that besides
the feedback from the concentration of tryptophan to the rate of RNA-
transcription there is a parallel feedback from the concentration of
anthranilate synthetase. Such a nested feedback is also characte-
ristic of many biological systems. With this additional loop, the
interaction between mRNA and anthranilate synthetase can become un-
stable all by itself. If started close to equilibrium this will
produce an expanding oscillation in the enzyme concentration. How-
ever, when this concentration becomes sufficiently high an increased
production of tryptophan is triggered. This activates the second
negative feedback loop which brings the system back towards its
equilibrium point. With this modification, the operon model produces
the same types of behavior as the modified commodity market model: a
cascade of period-doubling bifurcations leading to a chaotic attrac-
tor of Réssler type.
518
4. Conclusion
It has been the purpose of the present paper to illustrate how easily
well-established economic and biological models can be modified to
produce bifurcation cascades and Réssler like attractors. At the same
time we have emphasized that endogeneously generated chaos is likely
to occur in many biological systems. We have no reason to assume that
managerial or macroeconomic systems should be more stable.
Acknowledgments
We would like to thank Margot Wisborg for her assistance in preparing
the manuscript.
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