-575-
CHAOTIC BEHAVIOUR IN A SIMPLE
MODEL OF URBAN MIGRATION
by
Erik Mosekilde, Steen Rasmussen,
Henrik Joergensen, Flemming Jaller,
and Claus Jensen
The Technical University of Denmark
ABSTRACT
By analysing the dynamics of a simple problem of urban migration, this paper
illustrates how chaotic behaviour can be internally generated even in a rela-
tively small (4-level) System Dynamics model.
Two different groups of minority families are considered to move around
between three sectors of a city. This migration occurs in response to changes
in certain social indicators which we take to be related to the number of
families already living in the respective sectors. Type I families, for
instance, prefer to live in areas with many households of the same kind and
tend to avoid neighbourhoods with many type II families. Type II families, on
the other hand, although also they like to live together, are at the same
time attracted to areas with many type I families.
For normal parameter values, this system has an unstable equilibrium point.
In base case it exhibits a limit cycle behaviour with the non-linear limiting
factors associated with a slowing down in the rate of emigration from a
certain sector as the number of remaining families approach zero. We show how
the system develops through a Feigenbaum cascade of period doubling bifurca-
tions as the inclination of type II families to move into areas with many
type I families is reduced by 15%. By calculating the largest Lyapunov expo-
nent for the system we finally show how the chaotic behaviour is quanti-
tatively distinguishable even from the most complicated limit cycle be-
haviour.
INTRODUCTION
In the field of non-linear dynamics chaos denotes a distinct mode of be-
haviour in the same way as exponential growth or damped oscillations are
characteristic modes of behaviour for linear systems.
Chaos may be described as a behaviour which is bounded in state space, and
which seems to have a certain recurrence. Each swing is unique, however, the
system never repeats itself, and the true period is infinite. Chaotic be-
haviour is also characterized by its extreme sensitivity to the initial
conditions. Trajectories started at points in state space which are infinite-
simally apart’will thus generally evolve in entirely different manners. As
as result, projection is no longer possible, even for systems which are
described by completely deterministic equations of motion.
-576-
In a way, the existence of such complicated trajectories even for relatively
simple systems should not come as a surprise to us. Tossing a coin, for
instance, must be considered an experiment with a simple dynamical system.
No one can doubt that the equations of motion governing the flight of a coin
are deterministic, and at least in principle, we can certainly specify the
forces acting upon the coin as functions of all relevant parameters, Never-
theless, the outcome of such an experiment is usually considered to be com-
pletely random, and in many cases coin tosses are even used as idealizations
of stochastic processes. Maintaining, however, that the flight of a coin is a
deterministic process, the apparently random nature of the outcome can only
be explained in terms of an extreme sensitivity to the initial conditions.
Even the slightest change in these conditions many reverse the result.
A closer examination of classical mechanics shows (Helleman, 1980) that
almost all the knowledge about the behaviour of dynamical systems that we
have acquired during the last centuries relates to (i) linear systems, (ii)
non-linear systems with one degree of freedom, or (iii) larger systems which
decompose into seperate one-dimensional systems. As soon as we move to prob-
lems which are a little more complicated, the equations of motion can no
longer be integrated, and in most cases the phase space becomes scattered
with regions where the trajectories evolve as unpredictably as the toss of a
coin. Chaos is therefore not a phenomenon which occurs only in exceptional,
Pathological systems, but almost all conservative systems with more than a
few degrees of freedom exhibit this type of behaviour.
In System Dynamics we deal with macroscopic systems in which large numbers of
identical elements (persons, machines, dollars, coffee bags etc.) flow and
accumulate, and for which it is impossible to follow the life curve of
individual elements in detail. Instead, the description is in terms of aggre-
gate rate and level variables. Such macroscopic systems are dissipative which
means that, in the absence of unlimited growth, the trajectories tend to
approach (be attracted by) certain points or curves in state space. The
simplest case, of course, is that of a stable equilibrium point which is
approached by all trajectories starting within a certain region of state
space. |
Besides in certain degenerate systems in which stable line or plane attrac-—
tors occur, the stable equilibrium point is the only attractor which can
exist in linear systems, and by the laws of thermodynamics it is also the
attractor which is bound to control the development of a system which cannot
exchange energy or resources with its surroundings. Social and biological
systems are neither linear nor thermodynamically isolated, however, and it is
therefore quite likely that more complicated attractors will occur.
The next simplest form of an attractor is a limit cycle which is approached
by all trajectories starting within a certain region (the basin of attrac-
tion) on both sides of the cycle. John Sterman’s simple Kondratieff wave
model (Sterman, 1983), and Jack Homer's model of worker burn out (Homer,
1984) both show examples of this kind of behaviour. For certain parameter
values, one can also obtain limit cycle behaviour in Dennis Meadow’s commodi-
ty cycle model (Meadows, 1970) as well as in Nathaniel Mass’ business cycle
model (Mass, 1975). Until recently at least, there has been a clear tendency
for System Dynamics practitioners to ‘tune’ their models for damped oscilla-
tions rather than for limit cycle behaviour. This reflects a particular view
on the stability of social systems, a view which may not necessarily be
correct.
-577-
For models with 3 or more state variables, competition between a basic growth
tendency and nonlinear limiting factors may cause the attractor to change
form. As some parameter B in the system is increased, the simple 1-cycle
which closes to itself after a single revolution in phase space may suddenly
become unstable to be replaced by an attractor with twice the original period
of revolution, a 2-cycle. This phenomenon is referred to as period doubling
(Feigenbaum, 1980). The 2-cycle almost closes to itself after one revolution
in phase space. It misses a little, however, and precisely returns to its
initial point only after two revolutions.
At a slightly higher value of B, a new period doubling will occur, so that
the attractor only returns to its starting point after 4 revolutions in phase
space. If the behaviour of the system is spectrally analysed, the first
period doubling corresponds to the generation of a component at the half
subharmonic frequency of the original limit cycle, with the fundamental and
its subharmonic component being phase-locked together. The second period
doubling similarly corresponds to the generation of subharmonic signals at
1/4 and 3/4 the original frequency. (In physics and engineering this phenome-
non has previously been termed parametric subharmonic generation (Zemon,
1968)).
The remarkable and very powerful result is now that once this route has been
initiated, the period doubling bifurcations will continue until at a finite
parameter value B,, the period becomes infinitely long, and the attractor
turns chaotic. Moreover, Feigenbaum (Feigenbaum, 1980) has shown that this
route to chaos asymptotically develops in quantitatively the same manner for
all dissipatice systems independent of their nature (social, biological,
Physical or technical) or number of state variables, and independent of the
precise form of the equations of motion.
Two universal constants a = 2.5029078... and & = 4.6692016... exist such that
the value of the parameter B at the n'th period doubling is given by
B, = B, - ab? (1)
and such that the split between loops decreases by a factor a from period
doubling to period doubling. a is here a (positive or negative) constant,
depending on the actual system.
In a recent paper (Rasmussen, 1985) we have demonstrated how John Sterman's
simple Kondratieff wave model (Sterman, 1983) through a number of bifurca-
tions finally turns chaotic if the model is driven by a relatively weak
sinusoidal signal, representing a business cycle variation in demands for
goods. It is the purpose of the present study to illustrate how a similar
phenomenon can be endogeneously generated in a relatively simple System
Dynamics model.
Preliminary results also indicate that chaotic behaviour can develop if two
commodity cycle models are connected via demand cross elasticities. In this
ease, however, the system follows an alternative route to chaos which invol-
ves a competition between the periodicities of the two sub-models. As long as
the coupling between the two sub-models is relatively weak, phase locking and
quasi periodic behaviour is produced. As the coupling becomes stronger,
however, the system suddenly switches into a chaotic state. Alternative
routes to chaos have also been identified for physical systems (Eckmann,
1981).
-578-
A MODEL OF URBAN MIGRATION
The problem to be studied is related to the migration of two minority popula-
tions in the town of Waycross which is situated approximately 400 miles north
of Promise Gorge. Figure 1 illustrates the outlay of the downtown area with
its three main districts: Richmond (1), Jonesboro (2), and Camden (3). The
downtown area houses about 60000 families out of which 3000 are Italian and
3000 Puertorican. The total number of families is assumed to remain constant
over time, and so is the distribution between different ethnic groups. For
the individual district, however, the number of families of each of the two
minority populations may vary as a result of migration from one area to
another.
TOWN OF WAYCROSS
PROMISE GORGE
Figure 1. The downtown area of Waycross with its three main districts. For
various social reasons, minority families migrate between these districts in
response to changes in the relative population sizes.
Italian families naturally prefer to live in Italian neighbourhoods, which
implies that these families tend to move from districts with a small to
districts with a larger Italian population. Similarly, Puertorican families
tend to move to areas where the Puertorican population is already relatively
large. For various reasons, however, Puertorican families also like to live
in Italian neighbourhoods, and once some Puertoricans start to move into
predominantly Italian areas others will follow. The Italians on the other
hand do not particularly appreciate Puertoricans, and when Puertorican fami-
lies start to move into ‘their’ areas, they usually prefer to move out.
Altogether, the pattern of migration can be represented through the diagram
on figure 2. Note, however, that since we have assumed the total number of
both Italian and Puertorican families to remain constant, the flowdiagram
corresponding to the detailed equations of motion only has two state variab-
les for each type of minority population.
-579-
PL
Puertorican
. families in
Richnond
~
RICHMOND maiz
i Migration
n —_——_—a rate for
Ttalian a Puertoricans
families in
Richnond \
P3 . P2 7
Puertorican |g Puertorican
families in families in /
camden AA sonesboro /
- /
Migration
CAMDEN JONESBORO ra linitation factor
7
13 12 Aa
Ttalian Italian
families in families in
Camden: Jonesboro
“
“wv
o 1 2 3 4 = »/PO
Relative population remaining
Figure 2. Basic migration pattern in downtown Waycross. The insert shows the
factor which slows down emigation from a t given district as the number of
remaining families approaches zero.
As a first approximation we have taken the inclination of Puertorican fami-
lies to migrate from Richmond (1) to Jonesboro (2) to be given by
IMP12.K=A(P2.K-P1.K)+B(12.K-11.K)
with similar expressions for the inclination of Puertoricans to move from
Jonesboro to Camden (3), and from Camden to Richmond.
-580-
A and B are here constants. P2 and Pl denote the number of Puertorican
families already living in Jonesboro and Richmond, respectively while I2 and
I1 represent the number of Italian families in each of the two districts.
Note, that the inclination to migrate may change sign, depending upon the
relative population sizes in the various districts. In the simulations to be
discussed in the next section, we have taken A=1, while B gradually is
reduced from 2 in the base case run to 1.1 in the final example. Expressing
the relative strength by which Puertorican families are attracted to Italian
neighbourhoods, B is the bifurcation parameter of our problem.
Correspondingly, the inclination of Italian families to migrate for instance
from Jonesboro to Camden is given by
IMI23.K=C(13.K-I2.K)+D(P3.K-P2.K)
where the constants C=1 and D=-4.5 remain unchanged in all simulations. I3
and I2 represent the number of Italian families already living in Camden and
Jonesboro, while P3 and P2 give the corresponding number of Puertorican
families. To obtain the rates of migration we divide the inclinations to
migrate by a constant AMD (average migration delay). Rather arbitrarily. we
have chosen AMD=50 months, the purpose of this constant being primarily to
define a scale on the time axis.
At this stage, the model is still linear. The equilibrium point is unstable,
however, and the model will therefore produce a growing oscillatory be-
haviour. At a certain amplitude this will lead to the population sizes be-
coming negative, which is clearly a meaningless result. It is therefore
necessary to introduce nonlinearities which slow down the rate of emigration
out of a certain district as the number of remaining families approaches
zero. The rate of emigration of Puertorican families out of Richmond is
therefore multiplied by a limitation factor
DP1.K = 1-EXP(-P1.K/PO)
with the scaling population P0=400 families.
As long as the number of Puertorican families in Richmond P1 is well above
PO, the limitation factor is only slightly less than unity. For P1=P0, DP1 is
reduced to 0.63, and as P1 becomes much smaller than PO, DP1 approaches zero.
The detailed relation is sketched in the insert of figure 2. It is necessary
to stress, however, that any limitation factor which will serve a similar
function as DP1, be it a DYNAMO-table function or an alternative analytical
expression, can be expected to give fundamentally the same simulation re-
sults. In accordance with our usual paradigm, the basic behaviour of a dyna-
mical system is little sensitive to the precise form of the table functions.
When bifurcations can occur, we only have to give the term ‘basic behaviour’
@ somewhat more general meaning than we are otherwise used to. Introducing an
alternative limitation factor may change the dynamics from say limit cycle
behaviour to chaos. The ability, however, to generate both periodic and
chaotic behaviour is conserved. The threshold for chaotic behaviour (B,) is
just moved to another parameter value.
Together with the limitation factors we have also introduced a number of
shift functions which, depending on the direction of migration, apply limita-
tion factors corresponding to the populations under reduction. The total
DYNAMO-program hereafter reads:
~581-
MIGRATION OF MINORITY POPULATIONS IN WAYCROSS
NOTE
NOTE PUERTORICAN POPULATION
NOTE
L
AHO rr rere weep yr Paes
P1.K=P1.J+(DT) (MRP31.JK-MRP12 JK)
P1=P1I
P2.K=P2.J+(DT) (MRP12.JK-MRP23.JK)
P2=P2I
P3.K=3000-P1.K-P2.K
IMP12.K=A(P2.K-P1.K)+B(I2.K-11.K)
MRP12 .KL=IMP12.K*(B12.K*DP1.K+(1-B12.K)*DP2.K) /AMD
IMP23.K=A(P3,.K-P2.K)+B(13.K-12.K)
MRP23.KL=IMP23 .K* (B23.K*DP2.K+(1-B23.K)*DP3.K) /AMD
IMP31.K=A(P1.K-P3.K)+B(I1.K-I3.K)
MRP31.KL=IMP31.K*(B31.K*DP3.K+(1-B31.K)*DP1.K) /AMD
DP1,K=1-EXP(-P1.K/PO)
DP2 .K=1-EXP(-P2.K/PO)
DP3 .K=1-EXP(—P3.K/PO)
B12 .K=TABHL (SHIFT, IMP12.K,-15,15,5)
B23 .K=TABHL (SHIFT, IMP23.K,-15,15,5)
B31.K=TABHL (SHIFT, IMP31.K ,-15,15,5)
A=1
B=2.0
AMD=50 months
SHIFT=0/.05/.15/.5/.85/.95/1
PO=400 families *
NOTE
NOTE ITALIAN POPULATON
NOTE
QAO> >> rrr yr whew Pe eos
11, K=I1.J+(DT) (MRI31.JK-MRI12.JK)
I1=I1l
I2.K=12.d+(DT) (MRI12.JK-MRI23. JK)
I2=I21
1I3.K=3000-I1.K-I2.K
IMI12 .K=C (I2.K-I1.K)+D(P2.K-P1.K)
MRI12 .KL=IMI12 .K*(A12.K*DI1.K+(1-A12.K)*DI2.K) /AMD
IMI23 .K=C (13 .K-I2.K)+D(P3.K-P2.K)
MRI23 .KL=IMI23.K*(A23.K*DI2.K+(1-A23.K)*DI3.K)/AMD
IMI31,K=C (11,K-I3.K)+D(P1.K-P3.K)
MRI31.KL=IMI31.K* (A31.K*D13.K+(1-A31.K)*DI1.K) /AMD
DI1,.K=1-EXP(-I1.K/PO)
DI2.K=1~EXP(-12.K/PO)
DI3.K=1-EXP(-13.K/PO)
A12 .K=TABHL (SHIFT, IMI12.K,-15,15,5)
A23 .K=TABHL (SHIFT, IMI23.K,-15,15,5)
A31,K=TABHL (SHIFT, IMI31.K,-15,15,5)
C=1
D=-4.5
~582-
SIMULATION RESULTS
The simulation results to be presented in this section were obtained with
constant values of the parameters A, C and D, and only the parameter B has
been gradually reduced from run to run. For most simulations we show a plot
of the Puertorican population in Richmond as function of time together with a
phase-plot in which the Puertorican population in Jonesboro is depicted vs.
the Puertorican population in Richmond.
Figure 3 shows the results of the base case run in which B=2. For reasons of
PUERTORICAN FAMILIES IN RICHYOND
3000 4
2000 +
1000 ~
PUERTORICAN FAMILIES IN JONESBORO
3000
PUERTORICAN FAMILIES IN RICHMOND
Figure 3. Time- and phase-plots obtained with the migration model for
B=2 (base case). The figure shows the approach of a trajectory to the limit
eycle attractor.
~583-
symmetry, our model has an equilibrium point in which both the Puertorican
and the Italian populations are equally distributed between the three di-
stricts. As in the linear case, this equilibrium point is unstable. Even if
the model is initiated in equilibrium, the slightest disturbance triggers an
expanding oscillation. Due to the migration limitation factors, however, the
amplification now stops at a certain wave amplitude - before any of the
populations become negative. The non-linear restrictions thus cause the model
to have a self-sustained oscillation with finite amplitude, a limit cycle.
The phase-plot in figure 3 shows how the limit cycle is approached by a
trajectory starting close to the equilibrium point. The rate at which this
approach occurs is a measure of the dissipation (loss) in the system. In the
present case, the approach is rather slow, and because our problem has 4
state variables, the projection into a two-dimensional state space looks a
little complicated.
To accentuate the form of the stable attractor rather than of the initial
transient, in the following simulations we have started the plotting routine
only after the transient has died out. Figure 4 thus shows a phase-plot of
the limit cycle attractor for B=2.
PUERTORICAN FAMILIES IN JONESBORO
3000
2000
1000 4
° T T 71
° 1000 2000 3000
PUERTORICAN FAMILIES IN RICHMOND
Figure 4. Phase-plot of the limit cycle attractor for B=2 (1-cycle).
Figure 5 shows the simulation results obtained for B=1.8. In the time-plot we
now observe alternating high and low maxima for the Puertorican population in
Richmond, and the period of the stable attractor has now doubled relative to
the l-cycle in figure 4. In the phase~plot we see how the attractor has
folded itself, and it now closes only after two revolutions.
If B is reduced to 1.765 we obtain the simulation results of figure 6. There
are now 4 different maxima in the time plot, two high maxima and two low
maxima. The difference between the two low maxima is approximately 17%, while
~584-
the difference between the two high maxima only amouns to about 3%. Compared
with a difference of about 45% between the high and low maxima generated by
the first period doubling, this agrees with the theoretical prediction that
the split between the loops must decrease from period doubling to period
doubling. The universal constant a which describes this reduction in loop
split only applies after the first few period doublings have removed those
aspects of the process which depend on the particular problem. From the
phase-plot of figure 6 we see how the attractor now has folded twice, a 4-
cycle.
PUERTORICAN FAMILIES IN RICHMOND
2000 +
T T T
1000 1100 1200 1300 1400
1
1500
Te
_PUERTORICAN FAMILIES IN JONESBORO
2000 -
1000 +
° T T 1
° 1000 2000 3000
PUERTORICAN FAMILIES IN RICHMOND
Figure 5. Time- and phase-plots for the stable 2-cycle attractor obtained
with B=1.8. ,
-585—
As B is further reduced, the bifurcation process continues, although it
becomes harder and harder to follow with the tools that we are applying in
this study. For B=1.760 we obtain the 8-cycle shown in figure 7. At B=1.7 the
threshold to chaos is exeeded, and from then on the system in general behaves
in an aperiodic and random manner. This is illustrated in figures 8 and 9 for
B=1.3 and B=1.1, respectively. Now, the phase-plots for the stable attractor
no longer close, and the variation of the Puertorican population with time
appears completely stochastic with random succession of high and low, broad
and narrow maxima. When looking at figure 9 it should be recalled that the
curves describe a stationary behaviour for a simple, purely deterministic
systen.
PUERTORICAN FAMILIES IN RICHMOND
3008.7
2000 ~
8
8
8
8
PUERTORICAN FAMILIES IN. JONESBORO
2000 ~
1000 ~
T T 7
0 1000 2000 3000
PUERTORICAN FAMILIES IN RICHMOND.
Figure 6. Time- and phase-plots for the stable 4-cycle attractor obtained
with B=1.765.
-586-
PUERTORICAN FAMILIES IN JONESBORO
1
3000
PUERTORICAN FAMILIES IN RICHYOND
Figure 7. Phase-plot for the stable 8-cycle attractor obtained with B=1.760.
PUERTORICAN FAMILIES IN RICHWOND
2000 4 |
1000 4
° T T 71
1000 1500 2000 2500
Figure 8. Time-plot for the chaotic attractor obtained with B=1.3.
-587-
PUERTORICAN FAMILIES IN RICHYOND
00
n\
2000 +
1000 4
|
“i000 1880 2000 2500
TE
PUERTORICAN FAMILIES IN JONESBORO
3000
2000
N\
IN AN)
SROPVAM\
Oo 1000 2000 3000
PUERTORICAN FAMILIES IN RICHMOND
Figure 9, Time- and phase-plots for the chaotic attractor obtained with
B=1,1. Note that the deterministic process now appears completely random.
DISCUSSION
From a stricter point of view, simulations of time-behaviour can of course
not be taken as proof for chaos. The system in figure 9 may still be perio-
dic, just with a period of more than 1500 months. During the last decade, a
number of methods have therefore been developed (or adopted) which can give
more convincing evidence for chaotic behaviour. On such method is to deter-
mine the largest Lyapunov exponent Ly (Young, 1983 and Froeyland, 1983). If
this exponent is positive, the attactor is chaotic. (For a limit cycle L,=0,
and in the case of a stable equilibrium point L,<0).
-588-
The Lyaponov exponents L, are limiting values for time approaching infinity
of certain functions 4 te) which are derived from the eigenvalues of the
locally linearized systen as one moves along a trajectory. The number of
Lyapunov exponents and the number of functins ag (t) equal the number of state
variables in the system. In our problem there are thus 4 Lyapunov exponents.
It is a relatively simple matter (Rasmussen, 1985) to construct a DYNAMO-
program which will calculate the function 4,(t) associated with the largest
Luapunov exponent as well as the sum=A,(t) over the four A-functions. A,(t)
and 2a, (t) control the fundamental stability properties of the system.
Unfortunately, the Euler integration procedure of DYNAMO II is not good
enough for this purpose. (It gives L,>0 for a limit cycle). We have therefore
performed our simulations by means of COLTS using a Gear Predictor Corrector
integration procedure. COLTS (Behrens,1980) is a simulation language which
accepts DYNAMO statements, but has a number of facilities beyond those of
DYNAMO II.
Figure 10 shows the Puertorican population in Richmond as a function of time
plotted together with a curve representing t°A,(t). As time approaches
infinity, the (average) slope of this last curve determines the value of the
largest Lyaponov exponent. It is seen that L,>0. This agrees with the requi-
rements for a chaotic attractor. A more detailed exposition of the use of
Lyapunov exponents and other non-local stability measures will be presented
in a forthcoming paper.
Pamilies/100
PUERTORICAN FAMILIES IN RICHMOND
Figure 10. The Puertorican population in Richmond as a function of time
plotted together with the function tray (t). In this simlation B=1.3, and the
attractor is chaotic.
-589-
ACKNOWLEDGMENTS
We are indebted to Prof. W. Weidlich, Institute of Theoretical Physics,
University of Stuttgart, West Germany, who during a meeting in Oslo in the
fall of 1984 suggested that chaos could occur in a simple migration systen.
We have corresponded with Prof. Weidlich during the development of our work.
Considering, however, our differences in methodology as well as in detailed
model structure, Prof. Weidlich has suggested that we publish our results
independently.
REFERENCES
Behrens J. Chr. and P.E. Soerensen: ‘COLTS (COntinuous Long-Term Simula-
tion)’, TIMS Studies in the Management Sciences 14, p. 161-177 (1980).
Eckmann, J.-P.: ‘Roads to Turbulence in Dissipative Dynamical Systems’,
Review of Modern Physics 53, p. 643-654 (1981).
Feigenbaum, J. Mitchell: ‘Universal Behavior in Nonliner Systems’, Los Alamos
Science, Summer 1980, p. 4-27.
Froeyland, Jan: ‘Lyapunov Exponents for Multidimensional Orbits’, Physics
Letters 974, p. 8-10 (1983).
Helleman, Robert H.G.: ‘Self-Generated Chaotic Behaviour in Nonlinear Mecha~
nics’, Fundamental problems in Statistical Mechanics 5, ed. E.G.D. Cohen,
North Holland Publ., Amsterdam (1980), p. 165-233.
Homer, Jack B.: ‘Worker Burnout: A Dynamical Model with Implications for
Prevention and Control’, Proc. 1984 International System Dynamics Conference,
Oslo, Norway, p. 454-473.
Mass, J. Nathaniel: ‘Economic Cycles: An Analysis of Underlying Causes’,
Wright-Allen press (1975).
Meadows, Dennis L.: Dynamics of Commodity Production Cycles, Wright-Allen
press (1970).
Rasmussen, S., E. Mosekilde and J.D. Sterman: ‘Bifurcations and Chaotic
Behaviour in a Simple Model of the Economic Long Wave’, Journal of the System
Dynamics Society, summer 1985.
Sterman J.D.: ‘A Simple Model of the Economic Long Wave’, Proc. 1983 Inter-
national System Dynamics Conference, Vol. 1, Pine Manor College, Mass. p. 75-
110,
Young, L-S.: ‘Entropy, Lyapunov Exponents, and Hausdorff Dimension in Diffe-
rentiable Systems’, IEEE Transactions on Circuits and Systems, CAS 30, Pp.
599-607 (1983),
Zemon S., J. Zucker, J.H. Wasko, E.M. Conwell, and A.K. Ganguly: ‘Parametric
Amplification of Ultrasonic Waves in CdS’, Applied Physics Letters 12, p.
378-380 (1968).
