W O R K I N G P A P E R
RECENT DEVELOPMENTS IN SYSTEM DYNAMICS SOFTWARE
November
1987 WP-87-110
l n l e r n a l ~ o n a l l n s l l t u l e lor Applied Systems A n a l y s ~ s
NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHOR
November
1987 WP-87-110W o r k i n g P a p e r s are interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have received only limited review. Views or opinions e x p r e s s e d h e r e i n d o not necessarily r e p r e s e n t those of t h e Institute or of i t s National Member Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria
This p a p e r is a s h o r t review of a c o n f e r e n c e held in Sevilla, Spain, in October 1987. Organized by t h e Systems Dynamic Society, i t c o n c e n t r a t e d a r o u n d con- c e p t s in methodology and applications of nonlinear system modelling within t h e framework i n t r o d u c e d by J a y F o r r e s t e r a n d h i s followers.
The a t t i t u d e to t h i s a p p r o a c h is c o n t r o v e r s i a l . F o r example, t h e r e s p e c t i v e methodologies d o n o t involve t h e identification of system p a r a m e t e r s and t h e con- s t r u c t i o n of t h e models from available d a t a d o e s n o t involve and may e v e n c o n t r a d - ict with t h e r i g o r o u s c o n c e p t s and techniques of modern system t h e o r y .
The review given h e r e d o e s not discuss t h e r e l e v a n c e of 'System Dynamics".
I t merely g i v e s some information on t h e t o p i c s p r e s e n t e d at t h e c o n f e r e n c e in Sevilla.
A. Kurzhanski Chairman System a n d Decision S c i e n c e s P r o g r a m
RECENT
DEVELOPMENTS IN
SYSTEM DYNAMICS SOFTWARE Zstvhn V6lyiThis p a p e r i s to r e p o r t r e c e n t r e s u l t s in s y s t e m dynamics s o f t w a r e development re- lying mainly o n t h e m a t e r i a l p r e s e n t e d at t h e 1 9 8 6 I n t e r n a t i o n a l C o n f e r e n c e of t h e System Dynamics S o c i e t y h e l d u n d e r t h e t i t l e 'System Dynamics o n t h e Move' [I].
The c o n f e r e n c e t o o k p l a c e between 22-24 O c t o b e r 1 9 8 6 in S e v i l l a , S p a i n . P a r t i c i - p a n t s c a m e f r o m m o r e t h a n twenty c o u n t r i e s f r o m f o u r c o n t i n e n t s a n d included leading p e r s o n a l i t i e s of t h e s y s t e m dynamics community.
S y s t e m dynamics was d e v e l o p e d at t h e e n d of t h e f i f t i e s by J a y W . F o r r e s t e r to as- s i s t i n t h e s t u d y of complex s y s t e m s a r a t h e r wide r a n g e of u s e r s , with l i t t l e or n o mathematical b a c k g r o u n d . According to D. H . Meadows ([6] p . 31.): dealing with q u e s t i o n s a b o u t t h e dynamic t e n d e n c i e s of complex s y s t e m s , t h a t i s , t h e b e h a v i o r a l p a t t e r n s t h e y g e n e r a t e o v e r time. S y s t e m d y n a m i c i s t s are g e n e r a l l y u n c o n c e r n e d with p r e c i s e n u m e r i c a l v a l u e s of s y s t e m v a r i a b l e s in s p e c i f i c y e a r s . They are much m o r e i n t e r e s t e d in g e n e r a l dynamic t e n d e n c i e s ; w h e t h e r t h e s y s t e m as a whole i s s t a b l e or u n s t a b l e , o s c i l l a t i n g , growing, declining, or i n equilibrium.' In mathemat- ical t e r m s t h i s m e a n s t h a t in s y s t e m dynamics we u s e s y s t e m s of o r d i n a r y n o n l i n e a r (functional) d i f f e r e n t i a l e q u a t i o n s f o r r e p r e s e n t i n g r e a l i t y , a n d q u e s t i o n s a n d a n s w e r s are t y p i c a l l y f o r m u l a t e d i n q u a l i t a t i v e t e r m s . A s i s well known, t h e mathematical t h e o r y t h a t d e a l s with s u c h p r o b l e m s i s r a t h e r difficult.
G. P. R i c h a r d s o n a n d A. L. P u g h 111, [77, i d e n t i f y s e v e n s t e p s in a p p r o a c h i n g a p r o b l e m f r o m a s y s t e m dynamics p e r s p e c t i v e :
problem identification and definition system conceptualization
model formulation
a n a l y s i s of model b e h a v i o r model evaluation
policy a n a l y s i s
model u s e or implementation.
This p r o c e d u r e usually involves passing t h r o u g h t h r e e s t a g e s of model r e p r e s e n t a - tion: causal-loop d i a g r a m s , flow diagrams a n d equations. The f i r s t s t a g e in developing t h e f o r m a l model i s to d r a w causal-loop diagrams. Causal-loop d i a g r a m s a r e d i r e c t e d g r a p h s w h e r e nodes r e p r e s e n t elements of t h e system (i. e . v a r i a b l e s of t h e model) a n d e d g e s t h e c a u s a l r e l a t i o n s connecting them. Edges are m a r k e d with positive or n e g a t i v e s i g n s to indicate w h e t h e r i n c r e a s e c a u s e s i n c r e a s e or de- c r e a s e . The s e c o n d , i n t e r m e d i a t e s t e p i s t a k e n by drawing t h e flow diagram, t h a t i s to c l a s s i f y v a r i a b l e s in t h e c a u s a l loop diagram a c c o r d i n g to t h e i r r o l e s in t h e model (like levels, r a t e s , c o n s t a n t s contain a l l t h e information needed to write t h e equations, t h a t i s t h e final, a n a l y t i c a l form.
Despite f r e q u e n t c r i t i c i s m s t h e t e c h n i q u e h a s b e e n widely a p p l i e d , and nowadays i s c o n s i d e r e d as a p r o f e s s i o n . I t h a s p r o v e d to b e s p e c i a l l y useful in education a n d combined with gaming techniques. Argurner~ts w e r e focussed a r o u n d t h e following points:
problems r e l a t e d to q u a l i t a t i v e validation, s e n s i t i v i t y and p a r a m e t e r tuning ('lack of e m p i r i c a l b a s e ' , Oerlemans e t a l . , 151)
u n w a r r a n t e d u s e of mathematical o b j e c t s t h a t are sometimes v e r y difficult to handle ( f o r a l t e r n a t i v e s s e e e. g . J. Talavage, 141)
In s t a n d a r d s y s t e m dynamics, t h e solutions of t h e system of d i f f e r e n t i a l e q u a t i o n s a r e computed b y way of a p p r o x i m a t i n g t h e i n t e g r a l s b y E u l e r ' s method with a given i n t e r v a l length, i. e. considering t h e r i g h t h a n d s i d e c o n s t a n t in e a c h i n t e r v a l . This p r o c e d u r e i s r e a l i z e d in DYNAMO,
[?I,
a c o m p u t e r simulation language developed a t MI'I', t h a t h a s b e e n a s s o c i a t e d with t h e method f r o m t h e v e r y beginning. Even t h e a n a l y t i c form of t h e models, i . e. t h e d i f f e r e n t i a l equations are normally formulat- e d in t h i s language, r a t h e r t h a n in mathematical notations. S i n c e t h e mid s i x t i e s DYNAMO i s b a s e d o n FORTRAN. I t s l a t e s t v e r s i o n s like DYNAMO I11 a n d IV became r a t h e r f l e x i b l e as t h e y a r e c a p a b l e to handle s u b s c r i p t e d v a r i a b l e s a n d DO loops,a c c e p t u s e r d e f i n e d f u n c t i o n s , or o f f e r a possibility of s e l e c t i n g b e t w e e n v a r i o u s i n t e g r a t i o n methods. Mini-DYNAMO i s a simplified v e r s i o n t h a t r u n s o n small com- p u t e r s with 20K core memory while Micro-DYNAMO a n d P r o f e s s i o n a l DYNAMO a r e i n t e n d e d f o r p r o f e s s i o n a l p e r s o n a l c o m p u t e r s l i k e t h e IBM PC. DYSMAP, d e v e l o p e d in t h e s e v e n t i e s at B r a d f o r d U n i v e r s i t y , UK, [3], a n d NDTRAN at N o t r e Dame U n i v e r s i t y , USA are widely u s e d as well. T h e s e are e x t e n s i o n s of DYNAMO a n d o f f e r b e t t e r g r a p h i c a l o u t p u t , a n d m o r e s o p h i s t i c a t e d o p t i o n s f o r model c h e c k i n g a n d a n a l y s i s . A s i g n i f i c a n t d e v e l o p m e n t i s STELLA, [ 8 ] , a p r o g r a m r u n n i n g o n t h e Macintosh, fully e x p l o i t i n g i t s e x c e l l e n t g r a p h i c s f a c i l i t i e s . STELLA starts with d e - fining t h e model in t h e f o r m of flow d i a g r a m o n t h e s c r e e n , a n d d e v e l o p s t h e a n a l y t i c a l f o r m a u t o m a t i c a l l y . A n o t h e r i n t e r e s t i n g s y s t e m i s DYSYS, [Z], d e v e l o p e d at t h e U n i v e r s i t y of K a s s e l , FRG, t h a t i s a simple simulation l a n g u a g e w r i t t e n in BASIC with minimal h a r d w a r e r e q u i r e m e n t s , v e r y well s u i t e d f a r e d u c a t i o n a l p u r - p o s e s .
T h e r e s u l t s p r e s e n t e d a t t h e c o n f e r e n c e r e f l e c t t h e r e c o g n i t i o n of t h e p r o b l e m s b r o u g h t u p b y t h e c r i t i c s , as well as t h e e f f a r t s to ease f u r t h e r t h e modelling p r o - c e d u r e .
This r e p o r t i s divided i n t a two p a r t s , o n e d e a l i n g with t r a d i t i o n a l t y p e s o f t w a r e , a n d t h e o t h e r with e x p e r t s y s t e m s t e c h n i q u e s .
2.
NEW
SYSTEM DYNAMICS SOFTWAREW e start t h i s s e c t i o n by a new implementation of a t h e simulation l a n g u a g e DYSMAP.
T h e new v e r s i o n c a l l e d DYSMAPZ, d e v e l o p e d by U n i v e r s i t y of S a l f o r d C o m p u t e r S e r v i c e s S e c t i o n , UK, i s a n e x t e n s i o n of DYSMAP, i . e. i t a c c e p t s a n y DYSMAP p r o - g r a m with s o m e minor modifications. DYSMAPZ i s i n t e n d e d f o r u s e o n microcomput- ers a n d i s b a s e d o n i n t e r p r e t a t i o n at r u n t i m e r a t h e r t h a n t h e u s u a l t e c h n i q u e of t r a n s l a t i o n i n t o FORTRAN. This r e s u l t s in:
t h e possibility of handling r u n t i m e (in a d d i t i o n to s y n t a x ) e r r o r s , in cases m o r e t h a n t e n t i m e s f a s t e r e x e c u t i o n a n d
i m p r o v e d i n t e r a c t i v e p o s s i b i l i t i e s .
An i m p o r t a n t p o s s i b i l i t y f o r c h e c k i n g t h e c o r r e c t n e s s of t h e model i s t h r o u g h t h e dimensional a n a l y z e r . The i n t e r a c t i v e s e t u p means t h a t a n i n t e r a c t i v e s e s s i o n starts with a b a s i c r u n of a model a n d t h e n t h e u s e r i s allowed to s e l e c t a n y form of o u t p u t (including c o l o u r p l o t s of a n y v a r i a b l e s , n u m e r i c a l v a l u e s o n t h e s c r e e n , o r in f i l e s in ASCII f o r m a t ) , t h e n p a r a m e t e r v a l u e s a n d e v e n e q u a t i o n s c a n b e modified a n d t h e model r e r u n . D a t a communication t h r o u g h f i l e s i s p o s s i b l e in two d i r e c - t i o n s . T h e r e f o r e t h e i n t e r a c t i v e e n v i r o n m e n t m a k e s i t e a s y to a p p l y optimization f o r p a r a m e t e r tuning or s e n s i t i v i t y a n a l y s i s . I t i s in f a c t p l a n n e d t o i n c l u d e t h i s f e a t u r e i n t o t h e p a c k a g e . In a d d i t i o n to c o m p u t e d r e s u l t s , o t h e r s u p p l e m e n t a r y in- f o r m a t i o n i s a v a i l a b l e t h r o u g h t h e HELP o p t i o n .
A r e l a t e d p a p e r b y R. K e l o h a r j u (Helsinki S c h o o l f o Economics, Finland) a n d E . F.
Wolstenholme ( U n i v e r s i t y of B r a d f o r d Management C e n t r e , UK) p r e s e n t e d a new v e r s i o n of DYSMOD (Dynamic Simulation Model Optimiser a n d D e v e l o p e r ) t h a t w a s d e v e l o p e d in t h e l a t e s e v e n t i e s as a s u p p l e m e n t to DYSMAP. The s o f t w a r e i s c u r r e n t l y u n d e r f u r t h e r d e v e l o p m e n t by B r a d f o r d a n d S a l f o r d U n i v e r s i t i e s , UK, in o r d e r t o b e combined with DYSMAPZ. DYSMOD u s e s a hill climbing r o u t i n e f o r op- timization. The DYSMAP simulation ruri i s u s e d in t h e c o n t e x t of t h e optimization r o u t i n e t o c o m p u t e t h e v a l u e of t h e g o a l function. According t o p r a c t i c a l e x p e r i - e n c e , a r a t h e r l a r g e n u m b e r of i t e r a t i o n s (100 o r m o r e ) i s n e e d e d to a c h i e v e t h e optimal v a l u e s , b u t t h e individual i t e r a t i o n s a r e n o t v e r y e x p e n s i v e .
The a u t h o r s l i s t t h e following u s e s of optimization i n s y s t e m dynamics models:
f i t t i n g t h e model to p a s t d a t a ,
finding d e s i r a b l e p a r a m e t e r (and t a b l e f u n c t i o n ) v a l u e s f o r b e t t e r p e r f o r - mance of t h e s y s t e m ,
finding d e s i r a b l e d e c i s i o n s , s t r u c t u r e , or simplification, b y t a k i n g e. g . con- v e x combinations of e x p r e s s i o n s , t h a t r e p r e s e n t a l t e r n a t i v e p o l i c i e s , at t h e r i g h t h a n d s i d e of e q u a t i o n s , o r f o r c i n g t h e e f f e c t s of c e r t a i n p a r t s of t h e model to z e r o ,
s e n s i t i v i t y a n a l y s i s , by r e l a x i n g t h e found optimal v a l u e a n d c h e c k i n g t h e l a r g e s t p o s s i b l e c h a n g e in k e y p a r a m e t e r s .
Unlike t h e t r a d i t i o n a l ways of c a r r y i n g o u t t h e s e t a s k s , t h e method allows a l l t h i s to b e d o n e simultaneously, a f e a t u r e t h a t c o n s t i t u t e s a m a j o r c o n c e p t u a l a d v a n - t a g e . The s t r u c t u r e of t h e p r o g r a m i s shown in t h e F i g u r e 1.
COMPILATION
OBJECTIVE FUNCTION MAXlMlSE OR MlNlHlSE?
PARAHETER RANGES?
LENGTH OF SIMULATION?
SIZE OF STEP?
/SlMULAT ION
Figure 1.
The p r e s e n t a t i o n by H. Krallmann, B. R i e g e r a n d G . Hentzelt of t h e Technische U n i v e r s i t a e t (West-)Berlin r e p o r t s a b o u t a system t h a t c o n s t i t u t e s a u s e r f r i e n d l y communication a n d c o n t r o l system f o r existing system dynamics models implemented on a mainframe c o m p u t e r . I t should b e pointed o u t h e r e t h a t t h e system i s univer- sal in t h e s e n s e t h a t i t could b e e a s i l y extended to h a n d l e o t h e r t y p e s of models as well. I t s major aim i s given by t h e a u t h o r s to a s s i s t in:
s e l e c t i n g p r e f o r m u l a t e d models
g e n e r a t i n g a l t e r n a t i v e s by changing model p a r a m e t e r s
controlling simulations ( s t o r a g e a n d handling of model r e s u l t s ) g r a p h i c p r o c e s s i n g of r e s u l t s
providing model documentations.
C u r r e n t l y t h e s y s t e m c o n s i s t s of a l i b r a r y of DYNAMO models, a (commercial) sta- t i s t i c a l analysis p a c k a g e c a l l e d SAS 83 to p e r f o r m t h e g r a p h i c a l r e p r e s e n t a t i o n of r e s u l t s and t h e i r own c o n t r o l p r o g r a m . The o p e r a t i o n of t h e p a c k a g e i s done t h r o u g h menus a n d i s s u p p o r t e d by t h e HELP f e a t u r e . I t s s t r u c t u r e shown in F i g u r e 2.
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Finally w e mention t h e p i e c e of s o f t w a r e c a l l e d SDSE (System Dynamics S o f w a r e in E d u c a t i o n ) b y A. Toval, A. R e q u e n a , S. Martinez a n d J . Monreal of t h e U n i v e r s i d a d d e M u r c i a , S p a i n . A s t h e name i n d i c a t e s t h e s y s t e m w a s d e s i g n e d to facilitaLe t h e t e a c h i n g a n d u s e of S y s t e m Dynamics in e d u c a t i o n , namely i n s c h o o l s a n d h i g h s c h o o l s . The s y s t e m was w r i t t e n i n BASIC a n d was implemented f o r IBM PC (compati- b l e s ) u n d e r M S DOS.
3. EXPERT SYSTEMS FOR
SYSTEM DYNAMICS
T h e e x p e r t s y s t e m t y p e s o f t w a r e to b e i n t r o d u c e d h e r e h a v e t h e g e n e r a l aim to c o v e r , in a d d i t i o n to t h e n u m e r i c a l a n d adminisLrative t y p e o p e r a t i o n s , as much of t h e s y s t e m a n a l y s i s p r o c e s s as p o s s i b l e .
This means, in m o r e d e t a i l , in t h e s y s t e m d y r ~ a m i c s c o n t e x t to i n c l u d e :
t h e s p e c i f i c a t i o n of v a r i a b l e s a n d implications c o n n e c t i n g t h e m to e s t a b l i s h t h e s y s t e m b o u n d a r i e s a n d t h e c a u s a l d i a g r a m ,
c l a s s i f i c a t i o n of t h e v a r i a b l e s t o c o n s t r u c t t h e flow d i a g r a m , writing t h e model e q u a t i o n s ,
a n d (of c o u r s e ) simulation, o u t p u t a n d d o c u m e n t a t i o n .
L e t u s start with EASDM, or a n E x p e r t Aid f o r S y s t e m Dynamics Modelling t h a t was p r e s e n t e d b y J.C. Gonzalez a n d G. F e r n a n d e z f r o m t h e L a b o r a t o r y f C y b e r n e t i c s a n d S y s t e m s T h e o r y of t h e P o l y t e c h n i c a l U n i v e r s i t y of Madrid. The m a j o r novelty of t h e s y s t e m i s t h a t i t i s a b l e to c a r r y o u t t h e t h i r d s t e p of t h e p r e v i o u s l i s t , t h a t i s t h e c o n s t r u c t i o n of t h e flow d i a g r a m i n a s e m i a u t o m a t i c way. EASDM was p r o - grammed i n PASCAL a n d PROLOG f o r p e r s o n a l c o m p u t e r s with M S DOS o p e r a t i n g s y s t e m . (PROLOG c a n nowadays b e r e g a r d e d as t h e s t a n d a r d l a n g u a g e f o r
'knowledge r e p r e s e n t a t i o n ' t h a t i s f o r l o g i c a l p r o g r a m m i n g . )
EASDM h a s a m o d u l a r s t r u c t u r e c o n s i s t i n g of t h e following units:
definition module, to i n t r o d u c e v a r i a b l e s a n d l o g i c a l c o n n e c t i o n s , c l a s s i f i c a t i o n module, to c o n v e r t t h e a b o v e i n t o a flow d i a g r a m , e q u a t i o n module, t o w r i t e t h e f o r m a l model,
p r o c e s s i n g module, to t r a n s l a t e t h e f o r m a l model i n t o a PASCAL, p r o g r a m , simulation module, to p e r f o r m c o m p u t a t i o n s , a n d c o n v e r t t h e r e s u l t s i n t o t h e d e s i r e d n u m e r i c a l or g r a p h i c f o r m .
T h e functioning of t h e c l a s s i f i c a t i o n module i s b a s e d o n t h e o b s e r v a t i o n , a n d t h e r e l a t e d t h e o r y , t h a t a c o r r e c t l y d r a w n c a u s a l l o o p d i a g r a m a l r e a d y c o n t a i n s all t h e n e c e s s a r y s t r u c t u r a l i n f o r m a t i o n f o r drawing t h e flow d i a g r a m . C o r r e c t n e s s i s a n e s s e n t i a l f e a t u r e h e r e , t h a t i s e n s u r e d by t h e d i a g n o s t i c c a p a b i l i t e s included i n t o t h e module.
A n o t h e r s y s t e m with good s u p p o r t i n g c a p a b i l i t i e s in t h e d e s i g n of t h e model i s LSC, a Continuous Simulation S o f t w a r e d e v e l o p e d in t h e I n s t r u m e n t a t i o n a n d C o n t r o l La- b o r a t o r y , A l g e r i a p r e s e n t e d by M . Gaci a n d A. B a b a a m e r . The s y s t e m i s p a r t l y im- p l e m e n t e d o n a m a i n f r a m e c o m p u t e r a n d i s w r i t t e n in PASCAL a n d FORTRAN.
The s o f t w a r e l e a d s t h e u s e r t h r o u g h t h e s t a g e s of modeling as i n d i c a t e d in F i g u r e
S y s t e m d r f i n l t i o n
IC
s e t s d e s c r i p t i o n
R e l a t i o n s e t d e s c r i p t i o n
S y s t e m r e e l d a t a
I
a n a l y s i s
o d e 1 v a l i d i t y
a I Y e s
+
-F i g u r e 3.
M o d e l s i m u l a t i o n
-
Y e s
The system works i n t e r a c t i v e l y , using a high level language t h a t with t h e following set of 'environments':
s t a r t i n g environment: model name and simulation t y p e
f o r m a l d e c l a r a t i o n environment: r e g r o u p i n g commands, defining v a r i a b l e t y p e s
initialization environment: defining initial values, c o n s t a n t s etc.
r e l a t i o n s environment: defining e q u a t i o n s
edition environment: editing simulation r e s u l t s a n d o u t p u t c o n t r o l environment: checking t h e c o h e r e n c y of t h e model
Model definition i s c a r r i e d o u t by passing p e r h a p s s e v e r a l times t h r o u g h t h e s e en- vironments.
The s o f t w a r e u s e s i n t e g r a t i o n by G e a r ' s algorithm f o r stiff systems. F o r nonstiff systems t h e modified Hamming a n d Runge Kutta Gill a l g o r i t h m s or simply E u l e r ' s method c a n b e used. A s p e c i a l o p t i o n i s a v a i l a b l e f o r d i s c r e t e systems. I t i s planned t h a t a s e n s i t i v i t y analysis module b e included, computing a sensitivity c o e f f i c i e n t a c c o r d i n g to p . M . F r a n k .
F u r t h e r modules of t h e system are (not fully implemented yet):
compilation module: lexical a n d s y n t a c t i c analysis, and c o d e g e n e r a t i o n f o r m a l d e r i v a t i o n module
d a t a s t r u c t u r e module l i b r a r y module
g r a p h i c s module.
Finally w e c o n s i d e r a c o n t r i b u t i o n t h a t we r e g a r d as t h e most s u b s t a n t i a l among t h o s e r e p o r t e d i n t h i s review. The s y s t e m i s called MAPS (Mathematical Advising P r o d u c t i o n System), a n d is a n e x p e r t a d v i s o r f o r t h e q u a l i t a t i v e a n a l y s i s of dynam- i c a l systems. In i t s p r e s e n t f o r m MAPS d e a l s with s e c o n d o r d e r autonomous systems of o r d i n a r y d i f f e r e n t i a l equations. I t i s implemented i n a p r o t o t y p e v e r s i o n o n a n IBM PC. Minimal RAM r e q u i r e d i s 512 Kb. The s y s t e m i s w r i t t e n in muSIMP, i. e. t h e h o s t language of muMATH and u s e s t h e l a t t e r f o r c a r r y i n g o u t symbolic calcula- tions.
Based o n t h e qualitative t h e o r y of d i f f e r e n t i a l equations, MAPS s t u d i e s t h e asymp- t o t i c b e h a v i o u r of dynamical systems, performing a p a r a m e t r i c a n a l y s i s of t h e na- t u r e a n d s t a b i l i t y of t h e asymptotic states. I t i s pointed o u t t h a t u n d e r c e r t a i n con- ditions, l a r g e r complex systems c a n b e r e d u c e d to simpler (deterministic) o n e s in s u c h a way t h a t from t h e qualitative behaviour of t h e l a t t e r conclusions c a n be drawn f o r t h a t of t h e f o r m e r . In t h i s way t h e s o f t w a r e p r o v i d e s us, although in a v e r y limited r a n g e , with a mathematically correct a n s w e r to some of t h e questions usually posed i n system dynamics. The analysis c a n n o t be made completely au- tomatic, a n d it n e e d s a n i n t e r a c t i o n between t h e system a n d t h e modeler. According to t h e a u t h o r s a final system should i n c o r p o r a t e , along with formal mathematical knowledge, some h e u r i s t i c s , as well.
The c u r r e n t l y implemented v e r s i o n s t u d i e s systems of t h e form:
where x a n d y a r e s c a l a r state v a r i a b l e s , F i s a c o n s t a n t v e c t o r a n d p i s t h e s c a l a r p a r a m e t e r along which t h e analysis i s p e r f o r m e d .
The s t e p s of analysis:
verification w h e t h e r t h e system belongs to s o m e s p e c i a l c l a s s (like l i n e a r , hamiltonian e t c . )
s e a r c h f o r equilibrium points
l i n e a r s t a b i l i t y analysis f o r e a c h equilibrium point as a function of t h e param- eter p
.
In o r d e r to complement a n a l y t i c a l r e s u l t s , numerical simulations c a n a l s o b e p e r - formed.
4. REFERENCES
1. ARACIL, J., J. A. D. MACHUCA, M. KARSKY (Eds) S y s t e m D y n a m i c s : o n t h e Move P r o c e e d i n g s of t h e 1986 I n t e r n a t i o n a l C o n f e r e n c e of t h e System Dynam- i c s S o c i e t y , Sevilla, S p a i n , (1986)
2. BOSSEL, H . , U m w e l t d y n a m i k
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30 Programme &er k y b e r n e t i s c h e Umwelter- f a h r u n g e n a u f jedem BASIC-Rechner, TeWi V e r l a g , M u n i c h 1 9 8 5 .3. CAVANA, R . Y., R . G. COYLE, ASUA?' User Manual, U n i v e r s i t y o f B r a d f o r d , ( 1 9 8 2 )
4 . LEGASTO J R , A. A., J . W. F O R R E S T E R , J. M . L Y N E I S ( E d s ) S y s t e m Dynamics, N o r t h H o l l a n d , A m s t e r d a m , N e w Y o r k , O x f o r d ( 1 9 8 0 )
5. OERLEMANS, T., M. TJELLINGS, H . D E V R I E S World Dynamics: Social Feed- back May Give Hopefor t h e h t u r e , N a t u r e 2 3 8 , pp 251-255. ( 1 9 7 4 )
6 . R A N D E R S , J. (Ed) Elements of t h e S y s t e m D y n a m i c s Method, T h e MIT Press, C a m b r i d g e , M a s s a c h u s e t t s , USA, ( 1 9 8 0 )
7. RICHARDSON, G . P . , A. L. P U G H 111, I n t r o d u c t i o n to S y s t e m D y n a m i c s Model- i n g w i t h Dl'A?dMO, T h e MIT P r e s s , C a m b r i d g e , M a s s a c h u s e t t s , USA, ( 1 9 8 1 )
8. RICHMOND, B., A User's G u i d e to
