THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986. 1.049
MAPS : AN EXPERT ADVISOR FOR THE QUALITATIVE
ANALYSIS OF DYNAMICAL SYSTEMS:
M. Andretta (+), M, Lugaresi(++), P. Mazzanti(+++), R. Serra (+)
(+) TEMA, Viale A. Moro 38, Bologna (Italy)
(++) Dept. of Physics, Bologna University
(+++) Dept. of Phisics, Bologna University,
INFN, Sezione di Bologna (Italy)
Abstract. The analysis of the evolution of non linear dynamical
systems is a complex task. The cases where: i) the model equa-
tions can be regarded as a careful and reliable representation of
the real system and, therefore, need no revision or modification;
ii) the parameter values are precisely known; iii) the initial
conditions are precisely known, are rather rare.
At least one of the previous conditions is not fulfilled in most
of the systems of interest for System Dynamicists. Therefore
qualitative analysis of dynamical systems, i.e. the study and
Classification of their asymptotic behaviours, is of extreme
importance, at least in long term models.
The methodologies of knowledge representation recently developed
in the field of expert systems can be applied to this problem. We
therefore developed MAPS, an expert advisor for the qualitative
analysis of dynamical systems. MAPS takes the system equations
as input, classifies them according to their features and
performs the necessary calculations at each stage, sending
appropriate messages to the modelist.
At present MAPS deals with autonomous second order systems of
ordinary differential equations.Further developments are foreseen
concerning the study of higher order sytems and the design of an
“equations database " for comparison with previously analyzed
equations.
INTRODUCTION
When dealing with models of complex systems, we are faced with
many uncertainties (Serra et al. (1984a, 1986b), Sedehi et al.
(1986)). It often happens that the basic model equations are
essentially empirical, lacking a firm theoretical basis. This is
almost always the case, with the following two major exceptions +
*) those physical systems which allow a direct application of
"natural laws ” (e.g., planetary motion);
*) those socio-economical systems which are completely artificial
(e.g., accounting) and can therefore be perfectly known, at least
in principle (Bartezzaghi and Mariotti(1983)).
4.050 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
Besides ‘this very fundamental kind of uncertainty, which involves
the model structure, there are also well known problems relating
to the parameter estimation and, in the case of dynamical models,
to the determination of the initial conditions.
It is therefore necessary not to consider a single dynamical
system and its time evolution, but rather a family. of dynamical
systems, which may differ with respect to model structure,
parameters’ values or initial conditions, or any combination of
the previous sources of uncertainty.
These remarks are of course familiar to every system dynamist ;
the usual recipe to deal with such a situation is to resort to
several computer simulations, evaluating the robustness of the
model to different modifications. It is a very difficult and
complicated task, although it may prove a way to learn . some
interesting features of the system under study. We already
pointed out the opportunity to use reduced models in order to
compress the dimensionality of the phase space in such analysis
(Sedehi et al. (1984,1986)).
The use of reduced models seems interesting when we are
interested in the long term behaviour of the system, neglecting
the fast dynamical variables which are "slaved" by the slower
variables, also called order parameters (Haken (1977), Serra et
al. (1986a)). The elimination of the fast variables amounts to a
kind of projection of the original model equations on the
subspace of the order parameters. However, a price needs to be
payed in order to achieve this reduction of complexity : noise is
introduced, leading to a stochastic reduced model. This, in turn,
implies that the system description must be given in terms of
partial differential equations of the Fokker-Planck type.
However, there exist several cases where the level of noise is
low enough to allow the use of deterministic reduced models,
without introducing appreciable errors.
A simple example is the following (Haken (1977), Serra et al.
(1986a) ¢ consider a stochastic differential equation of the
following type :
M (1)
x, = fx) + ANd)
where A(t) is gaussian white noise. We also suppose for
simplicity that the system is of the gradient type, i.e. :
OV(x)
fx) =— (2)
Ox:
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986. 1.051
The "solution" of the Langevin equation (Eq.1) involves the
determination of the time evolution of the probability density of
the order parameters, p(x,t). There exist standard recipes in
order to determine the evolution equation for p(x,t), which is a
partial differential equation named after Fokker and Planck. Its
asymptotic solution can be given in analytical terms in the case
of gradient systems (of course, that’s why we have chosen this
example) +
P_(x) = Qe Vv (3)
X= fbd (4)
where @ is a normalization constant and D a measure of the
noise intensity. We can directly verify on &q.(3) that the
extrema of the asymptotic probability density are extrema also of
the potential function V(x), and are therefore stationary
solutions of the deterministic dynamical system (Eq.4). In
particular, stable equilibrium points of the deterministic system
(i.e., minima of V(x)) correspond to maxima of the asymptotic
probability density, and their determination allows a semi-
quantitative analysis of the stochastic system (Eq.1). Indeed, if
the noise level is low, the stochastic system will be found
almost always near the extrema of the probability density.
This conclusion applies also to a wider class of systems. We
therefore conclude that, in the case of sufficiently low noise
level, a deterministic analysis is sufficient to get a picture of
the system’s destiny, precise enough for most practical purposes,
provided that we remember that only stable equilibrium points
should be considered, and that transitions among different
locally stable states are allowed (although their rate may be
also very low).
We also remark that, qualitatively, the basic content of the
well-known central manifold theorem in dynamical systems theory
(Guckenheimer and Holmes (1983)) is that the asymptotic behaviour
of a deterministic system of ordinary differential equations can
be inferred from that of a lower dimensional one.
We are thus lead to the conclusion .that the analysis of
deterministic reduced models may be very informative. In
particular, we are interested in a discussion of the different
asymptotic behaviours and their stability. properties
("qualitative” analysis of dynamical systems). A fortiori, we are
interested in such analysis when low dimensional deterministic
systems stem directly from our modelling hypotheses, without
being the result of the application of a projection operator to a
more detailed model.
1.052 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
THE GOALS OF MAPS
Some analytical results are available, concerning the analysis we
are interested in. The usefulness of the analytical approach in
dealing with uncertain systems cannot be overemphasized, since it
allows us to avoid several computer trials whose interpretation
may be very cumbersome. Such analytical results are particularily
rich in the case of low dimensional systems (Guckenheimer and
Holmes (1983), Serra et al. (1986a, 1986b)). The two dimensional
case is particularily simple, since chaotic behaviour is
excluded, while it may be observed in higher dimensional systems.
Usually, the “qualitative” analysis of dynamical systems requires
an interplay between analytical and numerical computation. While
several routines can be found in the numeric domain, the
analytical calculations are usually carried on "by hand".
The main idea behind the system MAPS (Mathematical Advising
Production System) is that a meaningful subset of these analyti-
cal calculations can be carried on in an authomatic way, due to
progresses in computer algebra systems and expert systems.
There exist presently several systems which are capable of analy-
tical calculations. Among the general purpose systems the most
famous is doubtless Macsyma (Buchberger et al. (1982)). It has
many symbolic computing capabilities both to perform basic
algebraic and analytical calculations (e.g. expressions
semplifications and differentiation, computation of limits,
definite and indefinite integrals, functions expansion in Taylor
or Laurent series, analytical solutions of algebraic and
differential equations) and to solve problems in some applied
mathematics areas (e.g. general relativity, high energy physics).
Since its first implementation, in 1972, Macsyma has been used in
many areas, to explore problems, for example, in atomic
scattering cross sections, antenna theory; maximum likehood
estimation, economics.
Another powerful system is Scratchpad, originally implemeted by
I.B.M. using an experimental System/360 LISP system at the
begining of the seventies. Scratchpad II (Jenks (1985)) is
presently used only within 1.B.M.’s scientific centers. Despite
its poor diffusion, Scratchpad II deserves to be mentioned
. because of the extensible language approach adopted in its design
that can be considered a sort of mathematical data and operators
abstraction. . The user language, in fact, contains a set of
basic syntactic constructors, described by notations similar to
those in conventional mathematics. These basic constructs may be
extended by the user, so that Scratchpad II may be considered a
very-high-level non procedural language for “mathematical
manipulations.
Besides the general purpose systems, there are also many systems
dedicated to the solution of specific problems . Applications of
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986. 1.053
these systems include celestial mechanics (e.g. CAMAL and
TRIGMAN), quantum electrodynamics (e.g. REDUCE and SCHOONSCHIP),
general relativity (e.g. CAMAL and SHEEP), high energy physics
(e.g. SCHOONSCHIP and ASHMEDAI), optics (e.g CAMAL), chemistry
(e.g. FORMAC and PERTRAN) and electronics (e.g. REDUCE). Some of
these systems have grown out to general purpose ones (van Hulzen
and Calmet (1982), Bordoni and Miola (1985)).
The MAPS system has been implemented with the aid of another
computer algebra system, muMath, developed by D.Stoutmeyer and
co-workers (Stoutmeyer (1985)). This choice was due to the fact
that the muMath package was designed for personal computers,
while retaining several powerful features. Mumath is written in
muSimp, a high level Structured IMPlementation language,
especially designed for artificial intelligence applications
on personal computers. It was designed at the end of the
seventies by Stoutmeyer and Rich, in order to give the user the
power of a functional language like Lisp but with a much ‘easier
syntax.
On the other hand, several progresses have been achieved in
expert systems (Harmon and King (1985)). They can be seen as
systems performing in a way similar to a human expert, in narrow
domains. A major concept relating to this field is that of
heuristic knowledge (Nilsson (1982), Winston (1984)), namely that
kind of knowledge which is embedded in rules of thumb, mental
habits, etc., rather than in books and formal teaching. The major
reason for the use of such heuristic knowledge is the need to
reduce large search spaces, "trying first" those solutions which
can be regarded as most probable, although by no means sure, on
the basis of previous experience. A large body of techniques
aiming at formalizing these rules through direct interaction with
experts is known as "knowledge engineering”.
The qualitative analysis of dynamical systems usually involves a
large part of formal mathematical knowledge, plus. some
heuristics: here too we have a large search space, where “rules
of thumb" can be applied in order to keep the problem manageable.
We can so summarize the goals of MAPS : to assist the modelist in
the qualitative analysis of deterministic dynamical systems,
performing a parametric analysis of the nature and stability of
the asymptotic states. The analysis cannot be made completely
authomatic, but it needs an interaction between the system and
the modelist. The system should incorporate not only some mathe-
matical knowledge, but also some “knowledge of the
mathematician",i.e. heuristics. MAPS is a mixed system, involving
both analytical and numerical computations.
1.054 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
THE MAPS PROTOTYPE
There exists presently a first prototype of MAPS for PC IBM
compatibles, which is already a useful tool in the qualitative
analysis of dynamical systems and which has also proved useful in
order to analyze the requirements for a more complete system. In
its present version, the system supports the qualitative analysis
of two-dimensianal autonomous systems of the following kind :
x= f(x, ¥, PB, €) (5)
| Yr aby. e.é)
Here x and y are the two dynamical variables, c denotes a set of
constants (literal and/or numerical) and p denotes a scalar
parameter : the asymptotic analysis is performed as a function of
the values of this parameter. The analysis is performed within a
domain of values of x;y, and p which is asked by - the system.
Further specifications include the sign of the literal constants.
The system performs a diagnosis on the proposed system, verifying
whether it belongs to some peculiar class (e.g., linear,
hamiltonian or gradient) in order to simplify the analysis.
This diagnosis is performed basing upon the (symbolic) Jacobian
matrix of the system, which will be useful also in the following,
when stability of equilibria will be checked. The next step is
the search for equilibrium points, i.e. solutions of the
algebraic system :
f a=0
(x, ¥, B,C) (6)
glx, y,p, @)=0
These solutions are searched in order of increasing difficulty.
First the existence of solutions like (0,0),(x,;0),(Osy) is
checked. Such solutions can eventually be used in order to
reduce the complexity of the algebraic system through
factorization, trying to reduce the order of, say, f by
expressing it as
fix, yh = (x = xu) (y = yol? f(x, y) (7)
where a or b may eventually vanish. Moreover, it is possible to
look for solutions by direct substitution.
For every equlibrium point, a linear stability analysis is
performed, as a function of the parameter p. This involves the
determination of the nature (real or complex) of the eigenvalues
'HE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986. 1.055
of the Jacobian matrix, evaluated at the equlibrium point, and of
the sign of the real eigenvalues and of the real part of the
complex eigenvalues. The nature of the equilibrium points can be
determined according to the information summarized in Table 1.
EIGENVALUES CONFIGURATION
Real distinct positiv Unstable node
Real distinct negative Stable node
Real distinct opposite signs Saddle
Real coincid. negative Stable "star" node
Real coincid. positive Unstable "star" node
Compl. conjug. zero real part Center
Compl. conjug. neg- real. part Stable focus
Compl. conjug. pos. real part Unstable focus
1 Eigenvalue = 0 Degenerate point
TABLE 1. Classification of the equilibrium points.
There is also the possibility to perform numerical simulations,
in order to complement the analytical results. In fact, MAPS
includes an interface with a set of numerical integration
routines ; presently, the definition of the parameter values to
test is entirely left to the user.
SYSTEM STRUCTURE
The system is written in muSIMP, i.e. the host language of
muMath. This choice obviously simplifies the interface between
MAPS and muMath. Moreover, a functional language like muSimp is
particularily well suited for the kind of problems we are dealing
with. As far as hardware is concerned, an IBM PC compatible with
at least 512 kbytes of RAM memory is required.
MAPS is organized as a production system (Nilsson (1982),Winston
(1984), Harmon and King (1985)), i.e. a system composed by a
global database, a set of "if...then..." production rules, a
control system and a knowledge acquisition module. It presently
incorporates about 80 rules. The antecedents of the rules have
the following form:
1.056 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
if < Left Hand Side > = < Right Hand Side >
the consequent being always a muSIMP function. Each rule is
triggered by a pattern matching mechanism between the two sides
of its antecedents and the database. A rule may have two kinds
of. antecedents: askable and derivable. When an askable
antecedent is met, and there are no informations corresponding to
its LHS in the database, the control system of MAPS asks the user
for additional” information. On the contrary, the control system
automatically computes the missing information corresponding to
the LHS of a derivable antecedent. When all the antecedents of a
rule are satisfied, the muSIMP function of the consequent may be
applied.
Qne of the major problems in building a production system is the
so. called conflict set resolution, that is the problem of
singling the proper rule to be applied out of those which might
be activated at each step of the resolution process. We have
tackled this problem organizing the rules into a hierarchical
structure of contexts ("contesti"). A context is a subset of
rules, which are involved in the solution of some specific
subproblem; in every context the rules are ordered according to
their priority. There is only one active context at a time, and
the control system of MAPS analyzes the rules in order of decrea-
sing priority. A special context, called directing context, con-
trols the activation of the other contexts, depending on the
status of the database. Figure 1 sketches the hierarchical
structure of the contexts.
DIRECTING CONTEXT
(rot vot,
Equilibria Locat Linear fiumerieai
calculation stability solutions cure
context context context
—}—}-—, es
« Tr
Factorwation Analysis
context context
Fig. 1. The hierarchical structure of the contexts.
‘THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986, 1.057
AN APPLICATION OF MAPS
Let us now describe an application of MAPS, in order to show the
way how it works : we will perform the qualitative analysis of
the equilibrium points of the following system, which can be
regarded as"a modified version of the well known Lotka-Volterra
equations :
ax? + pxy
xe
8
| y=by-pexy 8)
The corresponding output is shown in Appendix I. At present, the
output of MAPS is written in Italian; anyway, a new English
version is going to be developed. At first, the system asks the
user the equations to be analyzed, the allowed domains of values
of x, y and p and the signs of the constants. Then it computes
the symbolic Jacobian matrix of the system and automatically
analyzes the equations, verifying if they belong to some
Particular class (in this example the equations are polinomials,
not linear ....). If the system is linear, the program
algorithmically calculates the equilibrium point. If, as in this
case, the system is not linear, MAPS searches for the equilibrium
points using the above mentioned heuristics strategy. The program
now displays the solutions it has found with this method and asks
the user if he wants to perform the factorization of the equa-
tions. In the example we report MAPS factorizes the equations and
it finds another equilibrium point. Now, the first part of the
analysis is completed and a swap of the working memory is
required. This is necessary because of the limited amount of RAM
(320 kilobytes) muSIMP can address. When the loading of the
second part of MAPS is done, the program performs a linear
stability analysis for every equilibrium point, as a function of
the parameter p. The final results of the analysis are displayed
and the program asks if the user wants to activate the numeric
context.
The reasons why we have introduced a numerical interface in the
system are the following: 1) impossibiliy of getting exact
results due to intrinsic complexity of the differential system.
This complexity may reflect itself into an excessive computer
overhead (é@xcessive computing time or memory occupation). 2) The
possibility of getting numerical results for various inital
conditions and different values of the various parameters is an
useful feature, also in the case when the main goal of the MAPS
system succeeds. The numerical interface consists of a number of
programs written in Basic which perform the following operations:
1.058 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
a) translation into Basic of the representation, in terms of
lists, of the system of differential equations. Constant names
and the parameter p are stored together with their ranges. The
correspondence between the matematical functions in muSIMP and
those in Basic is exact.
b) Explicit construction of the subprogram (in Basic) containinig
the system to be integrated.
©) Numerical calculation by using suitable integration routines.
The values chosen for the initial conditions, the literal
constants and the parameter p are carefully controlled . against
the previously defined associated ranges. .It is also possible to
graphically represent the results obtained on the screen (high or
low resolution) or to print them or both.
CONCLUSIONS
We have presented the first prototype of an expert advisor for
the qualitative analysis of dynamical systems. This system
integrates analytical, numerical and knowledge representation
environments to study and classify the equilibrium points of a
second order system of ordinary differential equations. The first
benchmarks are positive. The three enviroments appear to be well
integrated and the equilibrium points analysis is feasible also
for relatively complex systems.
Some limitations arise from the characteristics of the functional
programming environment and of the computer algebra system. In
particular, the muSIMP impossibility to address more than 320
kbytes compelled us both to adopt a simple (not memory expensive)
resolution startegy, and to segment the program into two parts.
The Algol-like output’ of muMATH and the absence of general
functions for the simplification of mathematical expressions,
make the output from MAPS sometimes difficult to read. Anyway»
these problems will be at least partially overcome with the new,
already announced version of muSIMP muMATH.
Among the most important extensions of the system. which we
envisage, provided that enough memory is available (either on PCs
or more powerful computers), we mention the diagnosis of possible
limit cycles, the analysis of higher order systems and the
development of an “equations database”, with an efficient search
strategy, for the comparison between already studied cases and
new ones.
‘HE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986. 1.059
ACKNOWLEDGEMENTS
We gratefully acknowledge many helpful discussions with L.Gardini
and G.Zanarini and the financial support of Enidata (grant
N. O21ED86)
REFERENCES
Bartezzaghi,E. and Mariotti,S. (1983). Modelli per le decisioni
strategiche aziendali. Angeli (Milano).
Bordoni,L. and Miola,A. (1985). Applicazione della manipolazione
algebrica e simbolica. Rivista di informatica, 15, pp.143-166.
Buchberger,B., Collins,6.£., Loos,R. eds. (1982). Computer
algebra; symbolic and algebraic computation. Springer, Wien.
Guckenheimer,J. and Holmes,P. (1983). Nonlinear oscillations,
dynamical ‘systems and bifurcations of vector fields. Springer,
Berlin.
Haken,H. (1977). Synergetics. Springer, Berlin.
HarmonsP. and King,D (1985). Expert systems, Wiley, New York.
Jenks,R.D. (1985). A brief introduction to Scratchpad II. The
Scratchpad II Newsletter, 1, p.1
Nilsson,N.J. (1982). Principles of artificial intelligence.
Springer, Berlin.
Sedehi,H.A., Serra,R., Vassallo,S. (1984). Theoretical approach
to long term company behaviour . Proc. Intl. Conf. System
Dynamics, Oslo, p.351
Sedehi,H.A., Serra,R., Vassallo,S. (1986) . Adiabatic elimination
of fast relaxing variables in socio-economic dynamical models.
Eur. J. Oper. Res.» 25, p.l
Serra,R.» Zanarini,G.» Andretta,M., Compiani,M™., (1986a).
Introduction to the physics of complex systems. Pergamon, Oxford.
Serra,R. and Zanarini,G. (1986b). Tra ordine e caos. CLUEB,
Bologna.
1,060 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
Stoutmeyer,D.R. (1985). A preview of the next IBM-PC version of
muMath. In B.Buchberger ed., Eurocal 85, Springer, Berlin, pp.33-
4a
van Hulzen,J.A. and Calmet,J. (1982). Computer algebra systems.
In B.Buchberger et al. eds., Computer algebra,symbolic and
algebraic computation, Springer, Wien, pp.221-244
Winston,P.H. (1984). Artificial intelligence. Addison-Wesley,
Reading.
‘THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986. 1.067
APPENDIX I. AN ANALYSIS WITH MAPS: THE LOTKA-VOLTERRA SYSTEM
2 MAPS O8
--- Mathemetical Advising Production System (MAPS). Rel. 1.1 ---
INSERISCI LA PRIMA EQUAZIONE DEL SISTEMA
EG1 2 DX/DT = < espr >_ ‘
17 EOL 2 DX/DT == AX - PF XY
INSERISCI LA SECONDA EQUAZIONE DEL SISTEMA
E02 DY/DT ©= < espr >_
1? E02 2 DY/DT e= -B YD + OR YK _
INSERISCI IL PRIMO ESTREMG DEL RANGE DELLA VARIAEILE X ¢
ESi : < estrinf >_
1? ES1:6_
INSERISC] IL SECONDO ESTREMD DEL RANGE DELLA VARIAEILE Xx :
< estrsup >_
INSERISCI 1L FRIMO ESTREMO DEL RANGE DELL@ VARIAEILE Y =
ESi : < estrinf >
1? ES1:0_
INSERISCI IL SECONDO ESTREMO DEL RANSE DELLA VARIABILE Y ¢
ES2 2 < estrsup >_
INSERISCI IL FRING ESTRENG DEL RANGE DEL FARAMETRO F :
Es1
S estrant >
17 EETrO_
1.062 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
INSERISCI IL SECONDD ESTREMO DEL RANGE DEL PARAMETRO P ¢
ES2 : < estrsup >_
1?
SFECIFICA IL SEGNO DELLE COSTANTI
>@% or €@% or If (-Indeterminato-)
STO CALCOLANDO LO JACOBIANO DEL SISTEMA
STO VERIFICANDO SE L’ORIGINE E’ UN PUNTO DI EQUILIBRIO
RIDUZIONE DEL SISTEMA ALL’ASSE x
RIDUZIONE DEL SISTEMA ALL'ASSE Y
I FUNTI DI EQUILIBRIO FINORA CALCOLATI SONO:
CO, 6)
VUOI FROVARE A FATTORIZZARE LE EQUAZIONI
DX /DT == AX - YF X
DY /DT =
RY? + CY FX
S/N 5D
Ss
STO PROVANDO A FATTORIZZARE LE EQUAZIONI
STO CALCOLANDO I FUNTI DI EQUILIBRIO DEL SISTEMA LINEARE
FUNTI DI EQUILIBRIO CALCOLATI SONO :
60,03
A BMC ROD) , A/F
FUNTI D1 EQUILIBFIO INTERWI AI RANGES FREFISSATI SONU:
CARO RD), B/E)
Ce. o>
LQ FASE DT DPTFRMINA7IONF DET RINT DT FONT TRRIN
IME 1900 INIEMIATUINNAL GUNFEKENGE UF IME SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986. 1.063
E’ TERMINATA
SELEZIONA IL DRIVE PER IL SALVATAGGIO MOMENTANEO
DELLA BASE DEI FATTI (A/B/C 7D) C
VIENE ESEGUITA LA FASE DI SALVATAGGIO
AL TERMINE BATTERE + CONTINUE ( < drive > )&
? CONTINUE (C)&
@: FALSE
2@: BASEFATTI
xy
STO CALCOLANDD GLI AUTOVALORI DEL SISTEMA RELATIVI AL PUNTO DI EQUILIBRIO :
CA B/(C FOZ) , A/F)
GLI AUTOVALORI SONO:
“A B/(2 PY + @ BOC1/2) CB ~ 4 PIO /20/(2 FD
TA B/(2 PF) ~ @ BSC1/2) (CB 4 PIO /20/(2 FD
SE Et VERA LA RELAZIONE
(B= 4 P/EO2 > @
IL SISTEMA AMMETTE DUE AUTOVALORI REALI DISTINTI +
TA E/(2 F) + A BOC/2) (B= 4 FS 1/20/02 FD
TA B/(2 FP) ~ A BOC/2) (CB - 4 PISL/2 (2 FD
ESEI SONO ENTRAMEI DI SEGNO < ©.
FERTANTO, IL FUNTO DI EQUILIBRIO +
CA B/(C PS2) 4 A/F)
—* UN NODO STABILE
STO CALCOLANDO GLI AUTOVALOR] DEL SISTEMA RELATIVI AL PUNTO DI ECUILIBRIO ¢
(6,6)
GLI AUTOVALORI SONO:
4
o
1.064 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
PER OGNI VALORE DI F INTERNO AL RANGE PREFISSATO
(© FINF)
IL SISTEMA AMMETTE DUE AUTOVALORI REALI DISTINTI
a
@
UNO DEI DUE AUTOVALORI £? UGUALE A ZERO, MENTRE L’ALTRO E’ > @
PERTANTO, IL PUNTO DI EQUILIBRIO : (@ ©)
—* UN PUNTO DEGENERE DI TIPO INSTABILE
VUDI PASSARE AL CONTESTO NUMERICO 7 (S}/Ni)
VYUOI CONTINUARE 7 (S/N) N
@:, TRUE
