CHAOS OUT OF STIFF MODELS
Khalid Saeed and N. L. Bach
Asian Institute of Technology
ABSTRA‘
While nonlinear combinations of multiple modes existing in complex oscillatory systems may
generate chaotic behavior in real systems, the studies of chaos attempted in system dynamics
have often resorted to forcing simplistic models of systems to chaos. This paper illustrates
how chaotic modes have been constructed through the creation of mis-specifications and
anomalies in the model structure and parameters. This process has not only reduced the
models to artifacts with little relevance to problem solving but has also invariably
introduced a stiff structure that is susceptible to considerable building up error as numerical
integration methods are used with long simulation times. The paper concludes that a model
must qualify as an empirically valid system by meeting the requirements of the normal system
dynamics practice if the chaotic modes it generates are to be of practical value.
Key Words: Deterministic Chaos, Social Systems, System Dynamics, Models, Computer
Simulation
INTRODUCTION
The existence of coupled major negative feedback loops together with the nonlinear
relationships often found in real systems may give rise to many complex nonlinear
combinations of multiple oscillations of different periodicities. In cases when these
periodicities are non-converging, some of these combinations can be so complex that their
envelope may never seem to repeat while the relationship between successive cycles within
the envelope appears to have no perceptible order. The behavior so created falls within the
definition of deterministic chaos [Andersen 1988].
Unfortunately, in system dynamics literature chaos has been treated largely as an artifact. In
order to create chaos, Researchers have often used simplistic models with unrealistic and
often stiff stricture which not only violates the normal modelling heuristics of system
dynamics but i is also susceptible to much building up error error when numerical integration
methods are used with long simulation times (Kreyszig 1972). The evidence of chaos in real
world data is both limited and inconclusive. Even when asymmetric modes can be observed,
they may be interpreted both as auto-correlated noise or chaos depending on the way the
underlying processes are modeled [Chen 1988]. Since chaotic modes exhibited by the models
appear only with certain parameter values and exogenous inputs lying within narrowly
specified ranges that are susceptible to integration errors, the relevance to the real-world
systems of the chaotic behavior appearing in the models remains unclear; nor has
experimentation with them to-date evolved any principles for system improvement [De
Greene 1990].
This paper demonstrates how complex combinations of non-linear periodicities may create
complex envelops, and how stiffness may corrupt such behavior. It also reexamines the
experimentation carried out earlier by the authors with chaotic models selected from the
literature to show that these models incorporated both unrealistic structure and stiffness.
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Thus chaotic modes exhibited by the models might then be unrelated to the real world as
well as corrupted by building up error [Saeed and Bach 1990, 1991].
It is suggested that the treatments of chaos as an artifact, though interesting, are irrelevant
to the traditional system dynamics objectives of unification of knowledge and policy design
for system improvement [Forrester 1987] because of the weak integration of the chaotic models
with the real world and their susceptibility to integration error. These deficiencies will
have to be overcome if the research on chaos is to be of practical value.
2, CHAOS AS A COMPLEX MODE OF BEHAVIOR
The real-world systems contain many adjustment paths appearing as coupled major negative
feedback loops, each creating an oscillatory mode of a given periodicity. There may also exist
many nonlinear relationships governing the flows associated with the stocks in each feedback
loop. Thus, there are many possibilities of creating complex nonlinear combinations of
multiple frequencies leading to infinitely long envelopes with unrecognizable reletonipe
between the successive cycles — a mode of behavior referred to as chaos.
Since multiple adjustment paths and nonlinear relationships are quite Pervagive in human
systems, the existence of chaotic modes in real world social phenomena cannot be ruled out.
This can be demonstrated by modulating systematically the behavior of a simple linear
workforce-inventory system generating undamped oscillations of a single period by an
exogenous periodic function. The model is developed and simulated using iTHINK 2.0 with
Runga-Kutta-4.! Figure 1 shows the simple linear model of a workforce inventory system
used in our experimentation. In the basic version of this model, workforce adjustment depends
on the inventory discrepancy while production rate is determined by the workforce.
Shipments in the final version of the model used in a subsequent experiment are also
constrained by a nonlinear function representing the inventory limitation.
The exact solution for inventory and workforce for the basic model is of the following form:
= AcosQot + BsinQ,t
where A, B and Q, are functions of the system parameters. When disturbed by a step change
in shipments, this system will generate sustained oscillations.
A forcing function is now applied to this system in the form of an oscillatory disturbance in the
parameter representing Productivity. The forcing function is of the following form:
Productivity*(1+CcosQt)
When the Q, and Q are non-integer and non-converging numbers, the complex oscillatory
pattern generated by this system will have an envelope with an infinitely long period, which
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Hanover, NH 03755, U. S. A.
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might often be the case in reality. A phase plot obtained from this system appears at Figure
2(a). Care has been taken to assure that no stock in the defined system assumes negative
values over the course of the simulation even though the inventory limitation creating a first
order control on the shipments has not yet been applied. The pattern shown in Figure 2(a) is,
however, not strictly chaotic since a systematic relation appears to exist between the various
cycles within the envelope, leading to a discernible order in the pattern generated.
Figure 1: Flow diagram of a simple workforce inventory system forced with an exogenous
cyclical function
amplitude
productivity
inventory limitation
desired inventory
adj coefficient
Figure 2(a):| Phase plot of workforce and inventory showing the complex pattern created by
the model.
age,
When this pattern is further modulated by introducing a gradually sloping nonlinear first
order control function representing the inventory limitation on the shipments, which also
creates some stiffness in the system by introducing a variable time constant in the stock of
inventory, the pattern appears to turn chaotic, as shown in Figure 2(b). The inventory
limiting function applied is also shown in Figure 2(b). The behavior in Figure 2(b), although
chaotic at first, tends to converge to a limit cycle. However, when the inventory limiting
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function is made steeper, which also increases the degree of stiffness of the model by creating
sudden changes in the time constant of the stock of inventory, the converging characteristic
disappears and the system appears to display sustained chaotic behavior, as shown in Figure
2c).
Figures 2(b): Modified pattern when a gradual inventory limitation is introduced
(og
8
: 10.02. om -—
21.16.
56.60 0.25 63.81
Figure 2(c): Modified pattern when a steep inventory limitation is introduced
1: inventory v. workforce
WN
21.38
19.08.
workforce
18.584
Two issues are to be examined here. First, which of the nonlinear functions applied is a
reasonable approximation of the reality? Second, what contributes to the chaos, building up
error resulting from numerical integration of a stiff model or the creation of a non-converging
envelope? ,
As for the first issue, both functions can be defended, depending on the level of aggregation
used in describing the stock representing inventory. If this stock contains many types of
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widgets in sum, a slight shortage in the aggregate could mean a stock out for certain types of
widgets, which would result in turning away a significant number of customers. Thus, a
gradually sloping function would make sense. On the other hand, if inventory represents only
one type of widget, only a high level of shortage would turn away a significant number of
customers, and the steeper second type of function would be more appropriate.
The issue of the source of chaos cannot, however, be settled easily. While there is a
possibility that the nonlinear combination of the two frequencies has generated a non-
converging and complex envelope associated with chaos, there is also ample reason to surmise
that the steep non-linear functions creating sudden changes in time constants have interfered.
with the numerical integration process used. It is widely recognized that a high degree of
stiffness resulting from the presence of steep nonlinear functions in a model can create
significant building up error even when sophisticated integration methods such as Runga-
Kutta-4 as used, which could be responsible for the observed erratic behavior identified as
chaos. Firstly, the computer round off error contributes to the building up error since the
integration interval must remain quite small during the long simulation times used in the
study of chaos. Secondly, since the derivatives change fast for the steep functions, the local
approximation error becomes significant and this leads to selection of inappropriate
integration intervals, which amplifies integration error. The two types of errors create
considerably large building up error [Kreyszig 1972, Pugh-Roberts Associates 1986]
It is quite difficult to say which source is dominating the chaotic patterns shown in Figures
2(b) and 2(c); an infinitely complex envelope or the building up error created over the course of
numerical integration. It can be said, however, that the contribution of integration error is
higher in Figure 2(c) than in Figure 2(b), since the former incorporates.a steeper nonlinear
function. The reader should be reminded here that the former case also displays sustained
chaos.
THE TREATMENT OF CHAOS IN SYSTEM DYNAMICS LITERATURE
The authorszhave carried out extended experimentation with selected system dynamics
models of social phenomena recently used in the studies on deterministic chaos through
computer simulation. The results of this experimentation were reported in the proceedings of
the 1990 International System Dynamics Conference [Saeed and Bach 1990]. All experiments
discussed in Saeed and Bach (1990) were performed using Professional DYNAMO Plus
program with Runga Kutta-4.2 Five well-known models were selected for experimentation.
These included the Waycross and Weidlich models of migratory dynamics discussed in
Rasmussen and Mosekilde 1988, Mosekilde, et. al (1985), Reiner et. al. (1988) and Richardson
and Sterman (1988); two versions of a model of resource allocation in a firm shown to display
chaotic modes respectively by Mosekilde, et. al.(1988) and Andersen and Sturis (1988), and a
simple model of the economic long wave originally developed by Sterman (1985) and shown to
display chaos in Rasmussen, et. al. (1985).
The Waycross and Weidlich models incorporate the same causal structure, representing the
2Professional DYNAMO Plus is a trade mark of Pugh-Roberts Associates, Five Lee Street,
Cambridge, MA 02139, U.S. A.
migratory dynamics of multi-ethnic communities resulting from the imbalances between the
two population groups in three adjacent neighborhoods. However, the former model has a
very steep first-order control on the outflows from the population stocks in the three
neighborhoods, while the latter incorporates a very steep function representing the effect of
the population imbalance between pairs of neighborhoods on migratory flows. Both
formulations are not only unnecessarily complex and unrealistic, they also create considerable
stiffness in the model. Minor corrections creating reasonable measures of imbalance and first-
order control eliminate the chaotic modes in these models, replacing them with limit cycles,
Anderson and Sturis (1988) and Rasmussen and Mosekilde (1988) use the same model of
resource allocation in a firm to produce chaotic modes, although they refer to the model
differently and use slightly differing parameters and slopes of non-linear functions. The
model deals with resource allocation between production and sales activities in a firm whose
total resources are fixed. Product availability is treated in the model as a function of the
product inventory only, rather than of the inventory coverage that takes into account both
supply and demand. Customer loyalty is then modelled as a very steep non-linear function of
availability. This unrealistic and stiff structure creates chaos which disappears either when
availability is reformulated as inventory coverage, or when the nonlifear function
representing customer loyalty is made less steep. Both these changes make the model realistic
as well as less stiff.
Sterman's model of long wave contains a simple and generally robust structure representing an
aggregate production sector that orders capital from itself according to required production
capacity. For normal parameter values, the model exhibits a characteristic limit cycle. It
produces chaotic behavior when one of its behavioral functions is made extremely steep and
an unrealistically high cyclical exogenous disturbance is applied [Rasmussen et. al. 1985].
This chaotic mode disappears when the amplitude of the exogenous disturbance is decreased,
or when the slope of the questionable behavioral function is reduced to a realistic value, both
measures also reducing stiffness.
We have traced the occurrence of chaos in the experimented models to the four types of
modelling errors summarized in Table 1. These are: non-robust rate equations; an unrealistic
decision information basis; an unrealistic order of magnitude of response to information; and
excessive exogenous disturbance. We have now further discovered that each modelling error
also made the models excessively stiff. Minor changes in the experimented models, which
improve their correspondence to reality while simultaneously reducing stiffness, eliminate
chaotic modes.
CHAOTIC MODES AND SYSTEM DYNAMICS MODELLING HEURISTICS
Our experimentation suggests that chaotic behavior appears in the models used in the
literature because of mis-specifications and errors in the model formulation and, possibly, also
because of stiffness. Both problems can be easily avoided by following normal system
dynamics modelling heuristics. Figure 3 illustrates the widely practiced, although
informally implemented, modelling heuristics recommended for system dynamics modelling
work. Empirical evidence is the driving force both for delineating the micro-structure of the
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model and for verifying its behavior, although the information on the behavior may reside in
the historical data and that concerning the micro-structure in the experience of people
[Forrester 1980].
Table1: — Pattern of modelling errors in experimented models
Sources of Chaos
Non-Robust Unrealistic Unrealistic Excessive
Models Rate Equation Information Response Exogenous
Formulation Basis to Information _ Disturbance
Weidlich
2. Business
Policy: *
Rassmussen/
Andersen
3. Macro- * *
Economics:
Sterman,
Long Wave
Source: Saeed and Bach(1990)
The first requirement of the method is to organize historical information into what is known
in the jargon as;"reference mode." The reference mode leads to the formulation of a "dynamic
hypothesis” expressed in terms of the important feedback loops existing between the decision
elements in the system that create the particular time variant patterns contained in the
reference.‘mode. The dynamic hypothesis must incorporate causal relations based on
information on the decision rules used by the actors of the system, and not on correlations
between variables observed in the historical data.
A formal model is then constructed incorporating the dynamic hypothesis along with the
other structural detail of the system relating to the problem being addressed. To have
credibility, the model structure must be "robust" to extreme conditions and be "identifiable" in
the "real world," where the real world consists both of theoretical expositions and
experiential information. A model might undergo several iterations in order to achieve an
acceptable structure.
Once a satisfactory correspondence between the model and the real-world structure has been
reached, the model is subjected to behavior tests. Computer simulation is used to deduce time
paths of the variables of the model, which are reconciled with the reference mode. If a
discrepancy is observed between the model behavior and reference mode, the model structure
is re-examined and, if necessary, modified. In rare cases, such testing might also unearth
missing detail concerning the reference mode, leading to a restatement of the reference mode,
although for most cases, the reference mode delineated at the start of the modelling exercise
must be regarded as sacred. When a close correspondence is achieved simultaneously between
structure of the model (including its parameters) and the theoretical and experiential
information on the system, and also between the behavior of the model and the empirical
evidence about the behavior of the system, the model is accepted as a valid representation of
the system [Bell & Senge 1980, Forrester & Senge 1980, Richardson & Pugh 1981].
Figure 3: System dynamics Modelling Process
EMPIRICAL EVIDENCE
COMPARISON AND STRUCTURE COMPARISON AND
VALIDATING
RECONCILIATION | eens RECONCILIATION
Nn roannanom
DEDUCTION OF
MODEL
BEHAVIOR
ACTERNATIVE OEE CH SIMULATION
When dealing with stiff real systems, the normal heuristics of system dynamics practice
guard against the creation of a stiff model by separating the short-range and longe-range
dynamics and modelling them independently (Saeed 1991). A single model must incorporate
the integration processes with medium-range time constants represented as stocks, those with
relatively small time constants as auxiliaries, those with relatively long time constants as
constant parameters to avoid stiffness. The normal heuristics of system dynamics practice also
require that, to preserve the integrity of a model and maintain the dominance of its internal
trends, outside disturbances should be kept small so that they do not overpower the forces
embodied in the model structure. Models with very steep functions, and parameter sets
creating excessive stiffness, or those driven by powerful exogenous cyclical functions, therefore
violate standard system dynamics modelling practice and should be viewed as artifacts with
no real-world problem-solving relevance.
The study of chaos as an artifact forced out of models that violate the normal modelling
heuristics of system dynamics is quite meaningless. The appearance of chaotic modes in such
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models may often signal the existence of anomalies in the model, calling for a revision of its
structure and parameter specifications to improve its correspondence with reality and also to
minimize any experimental error created in working with it.
WHAT SHOULD RESEARCH ON CHAOS SEEK?
We do not rule out the possibility that chaotic modes exist in real-world social phenomena
when all system relations are assumed to be deterministic. The existence of memory and
nonlinear responses to information in human systems may give rise to new weighting functions
for the repetitive decisions taken, although a pattern might exist in the roles the human
actors play on a long-term basis, which can create chaotic modes. However, notwithstanding
the many learned attempts to create chaotic modes with models of physical, biological and
social phenomena, we are of the view that experimentation with models alone without
reference to reality and without a specific policy focus, is more alchemy than life science. The
study of chaos as an artifact will often force the system to chaotic modes through subjective
adjustments in the model without justifying relevance to reality, which would create mis-
specifications and anomalies in the system structure and parameters, yet without creating any
practical insights.
To be of practical value, research into chaos must concentrate on addressing the issue of the
relevance of chaotic models to real-world phenomena and on policy design for system
improvement. To accomplish this, evidence of chaos must be sought in real-world data and in
realistic models of systems which also have realistic parameter sets. There has been some
progress in that direction in the recent work explaining noisy physical and biological
phenomena (Olson and Schaffer 1990, Mosekilde 1990), although still without a clear policy
focus. As for social systems, Sterman has reported the occurrence of chaos in a model
representing a multi-tier market system embodied in a game, when parameters related to the
behavior of a significant minority of the subjects playing the game (20%) were used. This
minority response to the decision-making information given to them was more aggressive than
for the majority whose parameter set produced stable behavior (Sterman 1988). Although
people acting in a gaming situation may not act realistically, and the models given to them
may also not‘fully embody the real process they abstract, such experimentation may provide
both realism and a policy focus in the treatment of chaos.
CONCLUSION
Chaotic behavior in many of the models discussed in the literature appears to arise from
modelling errors and from the problems of numerical integration methods, giving the
impression that chaos might be an artifact related to models and numerical integration. This
impression is, however, a function of inappropriate research designs that have focussed on
chaos as an artifact, often forcing the models of systems to chaotic modes through the creation
of mis-specifications and anomalies in their structure and parameters, without seeking
practical insights. We are uncertain how such studies of chaos can be related to the objective
of real-world problem-solving, which social science in general and system dynamics in
particular seeks to accomplish.
The existence of coupled major negative feedback loops together with the nonlinear
relationships often found in real systems may give rise to many complex nonlinear
combinations of multiple oscillations of different periodicities. Some of these combinations
can be so complex that their envelope may never seem to repeat, while the relationship
between successive cycles within the envelope appears to have no perceptible order. The
behavior so created falls within the definition of deterministic chaos. Experimentation with
simple relationships appears to confirm this point. Thus chaotic modes may exist in real
systems. However, to be of practical value, the research into chaos must deal with empirical
information and realistic models of real-world systems, with the aim of establishing the
relevance of chaos to real-world phenomena and to policy design for system improvement, and
not on the achievement of chaotic modes as artifacts from unrealistic models.
The authors acknowledge with thanks the communications received from Erik Mosekilde and
John Sterman over an extended period of time (1990-1992) that greatly helped in the
preparation of this paper. Roger Hawkey edited the final draft.
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