ET

053^8

}<ffy SERIE RESEARCH mEmORRRDR

LOGIT MODELS AND CHAOTIC BEHAVIOUR

P. Ni j kamp

A. Reggiani

Researchmemorandum 1989-54 augustus 1989

VRIJE UNIVERSITEIT

FACULTEIT DER ECONOMISCHE WETENSCHAPPEN

EN ECONOMETRIE

A M S T E R D A M

1=1 ^ 0-

^ « T E R ^ x

/

ABSTRACT

Chaos theory and discrete choice theory have been developed as two

separate analytical tools from various different disciplines.

This paper aims to link chaos theory (emerging mainly from physics)

to discrete choice theory (emerging mainly from geography and economics)

by showing the formal conditions under which a dynamic logit model can

exhibit chaotic behaviour.

Furthermore, the analysis will be illustrated and extended by

developing a time-delayed logit model related to a modal (or route)

choice problem incorporating congestion effects in a dynamic setting. It

will be illustrated by means of simulation experiments, that different

types of behaviour can emerge depending on critical values of the

utility function and initial conditions.

1

1. Introduction

An important scientific event in the past decade has been the dis-

covery of chaos theory and its relevance for the social sciences. The

theory of chaos deals with deterministic, non-linear systems which are

able to produce complex motions of such a nature that they sometimes

seem to be completely random. In particular, small uncertainties may

grow exponentially (though all time paths are bounded), leading to very

different trajectories in the long run, so that precise predictions are

- under certain conditions - almost impossible.

After an interesting series of studies on chaotic phenomena in the

field of physics, chemistry, biology, metereology and ecology (see for a

historical review e.g., Nijkamp and Reggiani, 1988d), chaos theory has

been introduced and investigated also in economics and geography with

the purpose of getting a better understanding of the potential useful-

ness of this new dynamical tooi regarding both theory and application in

the social sciences.

The possibility of chaotic behaviour was mainly identified in growth

models or business cycle models (see, among others, the pioneering

studies by Benhabib and Day, 1982; Day, 1982; Guckenheimer et al., 1977,

Grandmont, 1985, and the special issue of Journal of Economie Theory

edited by Grandmont, 1986), in cobweb models (Chiarella, 1988) and in

the economie analysis of long waves (Nijkamp, 1987a, Sterman, 1985).

Chaotic models have also been explored in regional economics and geog­

raphy, for instance in the field of industrial systems (White, 1985),

urban macro dynamics (Dendrinos 1984), spatial employment dynamics

(Dendrinos, 1985), relative population dynamics (Dendrinos and Sonis,

1987; Sonis and Dendrinos, 1987), migration systems (Reiner et al.,

1986) and urban evolution (Nijkamp and Reggiani, 1988d).

Interesting surveys of chaos theory can be found in various con-

tributions (see, for example Boldrin, 1988, Kelsey, 1988, Lung, 1988,

and the special issue of System Dynamics Review, edited by Andersen

1988), giving some further interesting insights into chaotic analysis

related to economics and geography. However, so far the contributions

concerning chaos theory have been developed mainly at a macro level. In

this paper an attempt will be made at linking micro- and macro-dynamic

behaviour by demonstrating how a dynamic logit model (emerging from

micro-economie theory) can exhibit a macro-chaotic behaviour, under

certain conditions of lts dynamic utility function.

2

It is noteworthy that contributions to dynamic logit models are up

till now also very scarce in the literature, despite some few interest-

ing examples (see Ben-Akiva and De Palma, 1986; De Palma and Lefevre,

1983; Fischer and Nijkamp, 1987; Haag, 1986; Leonardi, 1986). In this

respect the present study aims to show also such new developments in

dynamic logit models in order to fill this gap. The first part of this

paper will be mainly theoretical in nature and is devoted to show the

chaotic properties of dynamic logit models: the field of applications

can be manifold, since it is well known that logit models can be applied

to various fields of research, such as migration analysis, transporta-

tion aanalysis, residential choice analysis, industrial location

analysis, etc. (see the special issue of Regional Science and ürban

Economics, edited by Nijkamp, 1987b, and Golledge and Timmermans, 1987).

Furthermore, it will also be shown in this paper how the introduction of

time lags in a dynamic logit model may exhibit a more complex behaviour

under specific conditions of the utility function. Finally the stability

of a dynamic logit model in case of a constrained optimization process

will be examined.

It should be noticed that in the sequel we assume some elementary

knowledge on the mathematics of chaos and non-linearity, so that exten-

sive mathematical definitions can be dropped. For further details we

recommend some Standard references: Collet and Eckmann (1980), Devaney

(1986), Guckenheimer and Holmes (1983), loos (1979) and Poston and

Stewart (1986).

2. Existence of Chaos in Dynamic Logit Models

In this section it will be shown that a discrete choice model of the

logit type can exhibit chaotic behaviour under some conditions of the

utility function.

Let us consider the following dynamic logit model:

P j t = exp(ujjt)/S exp(uJjt) (2.1)

where P. represents the share of the population choosing a given dis-

tinct alternative j (j=l,...,J) at time t and u. the utility of

choosing j at time t.

Model (2.1) is the temporal version of a static logit model (see

Heekman, 1981). It is noteworthy that it can also emerge as a solution

3

of an optimal control spatial interaction model, whose objective func-

tion is a cumulative entropy function (see Nijkamp and Reggiani, 1988a;

1988b) .

Let us first consider the rate of change of P. with respect to time

(i.e., d P ./dt): J ' u

^ d F ^ " *j,t " Ht I ^ ^ j . t ^ exP<uJ,t^ (2-2)

Expression (2.2) leads after some simple computational exercises (see

also Nijkamp and Reggiani, 1988c) to:

P. = u. _ P. _(1-P. _) (2.3) j,t J,t j,t' jft'

where u. = du. /dt represents the time rate of change of u. Now it J i J ) - J > *-

is interesting to explore in more detail possible trajectories of u.

By assuming, for instance, that the utility of choosing alternative j is

increasing linearly with time, through a fixed a., we get:

u. _ = cc. (2.4)

so that the final expression (2.3) becomes:

P. = a. P. ,_ (1-P. J (2.5) J.t J J,t j,t'

Expression (2.5) is the well-known continuous time logistic equation

depicting logistic growth (based on the Verhulst equation).

Let us now approximate equation (2.5) in discrete time by considering a

unit time period:

P. «_,,-P. - a. P. _ (1-P. J (2.6)

j,t+l j,t j j,t j.t'

which can be re-arranged as follows:

a. p- ^i - (a.+l) P. _ (1 - TT P- J ' (2-7) j,t+l J j,t a.+l j,t/

It is now interesting to compare eq. (2.7) with the Standard discrete

logistic growth model for a biological population X (X <1):

4

X t + 1 - N Xt (1-Xt) (2.8)

where N is a parameter reflecting the growth rate (0<N<4).

The equivalent of N in our equation (2.7) is a.+l, so we can rewrite

(2.7) as follows:

P. f . - N P. f (1 • p. ) (2.9) j,t+l j,t N j,t'

N-l It is evident that for an increasing value of N we have -rr— -*• 1, and

consequently (2.9) coincides with (2.8). The model (2.8) has been

thoroughly investigated by May (1976). This author showed that if:

K N O (2.10)

the fixed point (1-^ , 1~ ) i-s a n attractor and the system will settle

at the stable point . For N=3 the system bifurcates to give a cycle of

period 2. Next for:

3<N<4 (2.11)

successive bifurcations give rise to a cascade of period doublings in

the range 3<N<3.8... In particular for the value N = 3.8... a fixed

point of period three appears giving rise to a chaotic regime, according

to the famous statement of Li and Yorke (1975) "Period three implies

chaos". The sequence of events of this process is shown in Fig. 1;

details can be found in May (1976), Collet and Eckmann (1980) and

others.

It is now interesting to show that also the dynamic (discrete) logit

model as specified in (2.9) belongs to the same family of May-models.

In fact, if we map in the plane (P. N) eq. (2.9), we get exactly the J >

same type of bifurcations as in the May model leading to a fixed period

of period three (and hence to a chaotic behaviour), the only difference

being the upper limit greater than 1 (see Fig. 2). However, this diagram

shows that for N — 3 the system bifurcates and gives rise to unfeasible

For the definition of a fixed point, an attractor and other terms re-lated to non-linear analysis see also Nijkamp and Reggiani (1988d).

5

Fig. 1 Bifurcation Diagrams of a May-Logistic Map

(2.95<N<4) Y-axis = X ; X-axis = N

Fig. 2 Bifurcation Diagrams of a Logit Map

(2.95<N<4). Y-axis = P. ; X-axis = N

6

values of P. (in particular P.>1). In this case one would have to switch

to values of P.<1, thus causing sudden jumps in the system's trajectory.

The straightforward conclusion is that when -3<N<4, the discrete

logit map (2.9) enters an unstable regime: in other words when the slope

a. of the utility function u. with respect to time (related to the

dynamic logit model (2.2)) is less than » 2 (note: a. = N-l), we have

stable or periodic solutions in choosing alternative j; when 2<Q.<3

chaotic behaviour emerges, characterized by unpredictable movements

which are hardly foreseeable by modellers and planners.

3. Chaotic Dynamics in Delayed Logit Models

In this section the dynamic behaviour of a delayed logit model will

be examined. In other words, we will model the growth of the population

share P. (choosing option j) whose ability to grow in any given time

span is govemed by the population in the previous time span; i.e., we

transform equation (2.9) in the following way:

p- -j.1 = N P . f l - ^ P . .) (3.1) j,t+l j,t N j.t-l'

Clearly, in population biology terms, the new discrete logit model

(3.1) contains a non-linear term regulating the population size with a

time delay of one generation.

Equation (3.1) can be transformed into a two-dimensional system by

introducing the auxiliary variable:

Q- • . - * • - i (3.2) J.t J,t-1

so that eq. (3.1) becomes:

^ t + l - ^ j . t ^ ^ Q j . t ) (3.3)

Qj,t+1 = Pj,t

System (3.3) is a special case of more general classes of well-known

prey-predator relationships. Q. may here be interp reted as an expected

(cumulative) lack of availability of alternative j (e.g., lack of road

capacity in travel behaviour) (predator) whose influence will be the

reduction of population P. (prey) through the parameter N-l.

7

The stability analysis shows a fixed point at (0,0), which is stable

for 0<N<1, but which becomes unstable for N>1.

The second fixed point (1,1) is unstable for 0<N<1; it is stable for

1<N<2 but it is subject to a Hopf bifurcation for N=2 (see Annex A).

These results are indeed most interesting if they are compared with

those emerging from the Standard delayed logistic equation for a

biological population P (see Maynard Smith, 1968):

pt + i = N pt u-*t-i> (3.4)

th where P represents the population density in the t generation and N is

a parameter reflecting the growth rate. Model (3.4) has been inves-

tigated by several authors (see for example Aronson et al., 1982;

Lauwerier, 1986; Pounder and Rogers, 1980).

In particular it has been shown that the fixed point ( N-l N-l N ' N '

related to (3.4) is stable for 1<N<2. As N passes through the value 2,

this fixed point loses stability via a Hopf bifurcation, giving rise to

a chaotic regime. Consequently the parameter values showing a chaotic

attractor are N>2 for both the delayed logit model (3.1) and the delayed

logistic model (3.4). the only difference being the value of the non-

trivial fixed point. It is also interesting to note that equation (3.4)

spawns for N = 2.27 a point of homoclinic tangency (see Pounder and

Rogers, 1980), leaving a strange attractor whose shape is quite similar

to the pictures obtained by Henon (1976) (See Fig. 3).

Fig. 3 The Strange Attractor for the Delayed Logistic

Map (3.4)

Source: Aronson et al., (1982, p. 309)

Yaxis = P t+1 Xaxis

8

If we observe now the bifurcation diagrams (see Fig. 4 and F

emerging from both cases, we may conclude that also in the delaye

model (3.1) a chaotic regime appears for 2<N<2.27.

Fig. 4 Bifurcation Diagrams for a Delayed Logistic

Map (1.9 < N < 2.27) Y-axis - P ; X-axis = N

Fig. 5 Bifurcation Diagrams for a Delayed Logit Map

(1.9 < N < 2.27) Y-axis = P ; X-axis = N

9

It should also be noted that - in our specific case of a delayed

logit model - the chaotic regime means that negative (or greater than 1)

values of P. appear thus leading to unfeasible system's behaviour (i.e.,

extinction) (see Fig. 6).

1 ? 2 0

, b

i 4 K

1 ! 21

ï 1 '

11 -0 471

Jfl. Ü i I' I !

i

l!

I

lil! Il' f U M

ij ! i j l i! I! i' i' !l !l' I I i il I t l ! I

1 h

I II

I i 1

1!

Ii ff

L (f !! iJ ff

! l

/ l i l

ii

t'

u.u 25.00 » • i • ' i

50.00 75.00 100.00

Tirn>?

Fig. 6 Delayed Logit Model for N = 2

In conclusion for the parameter values 0<N<1, system (3.3) embodies

stability in its fixed point (0,0); while for 1<N<2 system (3.3) repre-

sents stability in its fixed point (1.1). Consequently, in a delayed

logit model more restrictive conditions for the parameter N emerge (in

comparison to the non-delayed logit model (2.9)) in order to have a

stable behaviour.

Once again (see also Nijkamp and Reggiani, 1988c) the introduction

of time lags in dynamic equations shows a richer spectrum of complex

behaviour which has to be taken into account, mostly in planning and

forecasting problems.

4. An optimal Gontrol Approach for a Dynamic Logit Model

In this section we will present an optimal control approach in order

to evaluate the stability of a dynamic logit model in case of a con-

strained optimization process. In particular we will use here as the

objective function a cumulative entropy function whose use in dynamic

10

spatial interaction analysis has been suggested elsewhere (see Nijkamp

and Reggiani, 1988a and Sonis, 1986).

If we assume - in line with the previous sections - a spatial inter­

action system (e.g., a transportation system of a residential choice

system) , we may assume the following objective function:

max U = I [-0 E P. (inP.-l) - j8« S o. (ina.-l)] dt (4.1) JQ ^ - J J J l -j J J

where the first term in brackets represents the cumulative entropy of

the actors concerned (i.e., the maximum probability of choosing the

alternative j for a given time horizon T ) , while the second term in

brackets represents the maximum interaction between the marginal

Utilities of the individuals (see also, e.g., Wilson, 1981). The coeffi-

cients /L and /3„ represent the relative weights attached to the two

utility components.

The state variable is P. while we assume as control variable the J

marginal utility a. (depending on the attributes of the Utilities). Then

we get the following optimal control problem

U = [ [-B, S P. (inP.-l) - 0 O E a. (ina.-l)] J 0

P l j J J 2 j J J

s. t.

P. = cc. (l-P.)P. J J J J

EP. = 1 j J

(4.2)

Clearly, the evolution of the state variable P. in (4.2) should obey

assumption (2.4) defining a..

Owing to the static constraint in (4.2), we have to use the follow­

ing Lagrangean L as a first-order condition for a maximum:

L = -p. S P. (in P.-l) - /30 S a. (in a.-l) + A. a. P. (1-P.) F l j J J ^2 j j 2 J J J J

+M (S P.-l) (4.3) j J

where A. is the costate variable related to the P. constraint and u is J J

the Lagrangean multiplier related to the static constraint.

11

The necessary first-order conditions for a global maximum are:

3L 8a.

3 u

8L SP.

J =

• -A.

J

3L 3A. =

• P.

J

Vi

Vi

Vi

(4.4)

The conditions (4.4) are also sufficiënt for optimality since the objec-

tive function is a concave function. Next by using the condition:

f^- - -/90 in a. + X. P. (1-P.) - 0 (4.5) 5«j H2 j 3 3 3

we can derive:

A. in a. = -^— P. (1-P.) (4.6)

From equation (4.6) we can see that the shadow (dynamic) price A. is

positive, when a.>l and negative when a.<l. The expression for the op-

timal value a. is therefore the following (see also Fig. 7):

a. - exp [X.?. (1-P )//92] (4.7)

Aj Pj d-Pj)/^

Fig. 7 Relationship between marginal Utilities

and costate variables

12

Front expression (4.7) and Fig. 7 it is clear that a.=l is an equilibrium

point for the values P.=0 and P.=l. It is also evident that a.=l is a J J J

bifurcation point, from which two paths emerge.

In that case a planner - who wants to avoid chaotic behaviour in the

state variable - may manipulate the attributes of the utility function

or try to identify the optimal value of X. from the second condition in

(4.4) and to check whether this value is negative (and hence low mar-

ginal Utilities) in order to have a value of a.<l for stable behaviour.

A further remark is that from (4.6) the following relationship can also

be derived:

a. £n a. = X. P./fl0 (4.8)

In other words, the entropie form of a. is directly related to the

dynamic evolution of a logit model through the shadow price A..

Consequently, we may expect that if we also want to control a

delayed logit model no drastic change will occur. In particular when the

expected mobility Q. (see Annex B) is equal to zero, the marginal

utility Q. is decreasing or increasing if the costate variable A. is

negative or positive.

In conclusion, it is clear from our optimal control analysis, that

stable behaviour of the actors (following dynamic logit model) emerges

when their marginal Utilities take low values, and in particular for a.

less than 1.

5. Conclusions

In this paper the possibility of chaotic behaviour for both a

dynamic logit and a delayed logit model has been shown.

In this respect some further remarks are in order. Firstly since

chaos appears in both these models for high values of the marginal

utility a., we have to examine the economie or physical conditions

producing high a.'s and hence impossible predictions on the actor's

behaviour.

Secondly, the chaotic possibility emerging from dynamic logit models

evokes the need to raise the question whether other types of discrete

choice models (such as probit, nested logit, etc.) can exhibit, under

13

certain condltions, chaotic behaviour; in other words if chaos can ap-

pear not only for actors whose behaviour can be represented by

multinomial logit models suffering from the well-known IIA assumption.

Thirdly, an optimal control entropy model applied to a dynamic logit

model shows once more the importance of the marginal utility a., from

which stable behaviour can be controlled. Thus this parameter appears to

be of critical importance for the emergence of chaos.

Acknowledgements

The second author would like to thank the computer assistance by

Andrea Gerali and Luca Schionato (University of Bergamo). Also the

grants C.N.R. n. 88.00409.11 and M.P.I. 60% 1988 are gratefully acknow-

ledged.

14

Annex A Stability Analysis for the Delayed Logit Model

In this Annex, we will analyse the stability properties of the

delayed logit model. For the sake of complicity we rewrite system (3.3)

as follows:

V i - ^ J . t ^ - T V (A.1)

Qj , t+1 Pj , t

It is easy to see that the fixed points (i.e. the values that do not

change when the mapping is iterated) are Ff.(0,0) and F, (1,1).

The local behaviour of the map (A.1) at a fixed point F is governed

by its local linearization for which the Jacobian J taken at F can be

calculated from the corresponding matrix. Since:

J =

N - (N-l) Q (N-l) P. J

(A.2)

we get:

J(0,0)

N

1

0

0 (A.3)

the characteristic equation of the matrix (A.3) is:

K - NK = 0 (A.4)

with eigenvalues K, = 0 , K„ = N. It is evident that |K1|<1 and [K„|<1

for N<1, leading to the stability of the fixed point F~.

15

Then if we consider:

J(1,D 1 1-N

(A.5)

we get the following characteristic equation:

K - K + (N-l) = 0 (A.6)

with eigenvalues:

K± 2 - (1 ± V 5 - 4N/2) (A.7)

The characteristic equation (A.6) is the same as the one emerging

from the linearization of the logistic delayed model (3.1) around its N-l N-l f ixed point (—TJ— , —zr~) , which was thoroughly investigated by Pounder

and Rogers (1980).

Consequently we have a stable solution for 1<N<2. For N=2, we enter

in a Hopf bifurcation.

As a synthesis, following Pounder and Rogers (1980), we can sum-

marize the results in a compact final form (see table 1):

0<N<1

1<N<1.25

1.25<N<2

2<N

stable node

unstable node

unstable node

unstable node

unstable node

stable node

stable focus

unstable focus

Table 1. Stability Solutions for the Delayed

Logit Model (3.1)

16

Annex B. An Optimal Control Approach for a Dynamic Delayed

Logit Model

In this Annex we will analyze an optimal control problem whose ob-

jective function is a cumulative entropy function as defined in (4.1),

subject to a dynamic delayed logit model, as defined in (3.3).

For the sake of simplicity we will approximate system (3.3) in con-

tinuous terms (by supposing a unit time period) as follows:

P. = P. ,_,, - P. „ = (N-l) P. _ - (N-l) Q. ^ P. _ j j , t+1 j , t v J . t v XJ , t j , t

(B.l)

Q. = Q. ., - Q. = P . - Q. vj vj,t+1 vj , t j , t vj , t

By recalling the condition a. — N-l , system (B.l) results:

P. = Q. P. (1-Q.) J J J J

Q. - P. - Q. J J J

(B.2)

System (B.2) is a special type of prey-predator relationship. P.,

representing the prey, is the share of the poptilation choosing alterna-

tive j (e.g., the form or mode of travel in a transport system) and Q.

(predator) can be integrated as the expected mobility or the lack of

capacity in the mode of transport j (see also Nijkamp and Reggiani,

1988c).

The optimal control problem is therefore the following:

17

rT

max U = [-/3, S P. (inP.-l) - p0 S a. (ina.-l)] dt Jg i ^ ^ i ^ ^

s.t.

P. = a. P. (1-Q.) J J J J

Q.= P. - Q. 3 J J

(B.3)

E P. = 1 j J

S Q. = 1

j J

where a. represents the control variable, and P. and Q. are the state J J J

variables.

The first-order condition for a maximum can be expressed by intro -

ducing the following Lagrangean:

L = -P1 S P. (in P.-l) - 02 E a. (in a.-l) +

+ A. a. P. (1-Q.) + V- (P- - Q-) + M (S P.-l) + J J J J J J 1 j J

+ f (S Q.-D (B.4) j J

where A. and il), ave the costate variables and u, and £ the static shadow J J

prices. Then the optimal value a. results from:

§£-- -/?2 in Q j + Aj Pj (1-Qj) = 0 (B.5)

From (B.5) we can derive the condition:

in a = X P (1-Q ) / (B.6)

which shows that the costate variable A. can be positive or negative,

depending on whether or not a.>l.

18

Analogously to section 4, we can derive the optimal value for a.

a. = exp [AjPj (1-Qj)/^] (B.7)

From (B.7) it is clear that when P.=0 or Q.=l, we get the equi-

librium value a.=l (see also Fig. 8).

»j Pj d-Qj)/-32

Fig. 8 Relationships between optimal control

values and shadow prices related to a

dynamic delayed logit model

It is clear that a. =1 is a bifurcation point. Depending on the sign

of A., the optimal value can be decreasing or increasing and hence lead-

ing to stable or unstable behaviour.

19

References

Andersen, D.F. (ed.) , Chaos in System Dynamic Models: Special Issue System Dynamics Review, vol. 4, no. 1/2, 1988.

Aronson, D.G., M.A. Chory, G.R. Hall and R.P. McGehee, Bifurcation from an Invariant Circle for Two-Parameter Families of Maps of the Plane: A Computer-Assisted Study, Communications in Mathematical Physics. vol. 83, 1982, pp. 303-354.

Ben-Akiva, M., and A. De Palma, Analysis of a Dynamic Residential Location Choice Model with Transaction Costs, Journal of Regional Science, vol. 26, n.2, 1986, pp. 321-341.

Benhabib, J. and R.H. Day, A Characterization of Erratic Dynamics in the Overlapping Generations Model, Journal of Economie Dynamics and Control. vol. 4, 1982, pp. 37-55.

Boldrin, M., Persistent Oscillations and Chaos in Dynamic Economie Models: Notes for a Survey, SFI Studies in the Science of Complexitv, Addison-Wesley Publ. Co., 1988, pp. 49-75.

Ch iarella, C., The Cobweb Model: lts Instability and the Onset of Ghaos, Economie Modelling. vol. 5, no. 4, 1988, pp. 377-384.

Collet, P. and J.P. Eckmann, Iterated Maps on the Interval as Dynamical Systems. Birkhauser, Boston, 1980.

Day, R.H., Irregular Growth Cycles, American Economie Review, vol. 72, 1982, pp. 406-414.

Dendrinos, D.S., Turbolence and Fundamental Urban/Regional Dynamics, Paper presented at the American Association of Geographers, Washington D.C., April 1984.

Dendrinos, D.S., On the Incongruous Spatial Employment Dynamics, Technoloeical Change. Employment and Spatial Dynamics. (P. Nijkamp, ed.), Springer-Verlag, Berlin, 1986, pp. 321-339.

Dendrinos, D.S. and M. Sonis, The Onset of Turbolence in Discrete Relative Multiple Spatial Dynamics, Applied Mathematics and Computation, vol. 22, 1987, pp. 25-44.

De Palma, A., and C. Lefevre, Individual Decision-Making in Dynamic Collective Systems, Journal of Mathematical Sociology. vol. 9, 1983, pp. 103-124.

Devaney, R.L., Chaotic Dynamical Systems. Benjamin Cummings pPubl. Co., Menlo Park, CA, 1986.

Fischer, M.M. and P. Nijkamp, From Static Towards Dynamic Discrete Choice Modelling: A State of the Art Review, Regional Science and Urban Economics. vol. 17, no.1, 1987, pp. 3-27.

Golledge, R. and H.J.P. Timmermans (eds.), Behavioural Modelling in Geographv and Planning. Croom Helm, London, 1988.

20

Grandmont, J.M., On Endogenous Competitive Business Cycles, Econometrica. vol. 53, 1985, pp. 995-1046.

Grandmont, J.M. (ed.), Special Issue Journal of Economie Theory. vol. 40, 1986.

Guckenheimer, J. and P. Holmes, Non-Llnear Oscillations. Dynamical System and Bifurcation of Vector Fields. Springer Verlag, Berlin, 1983.

Guckenheimer, J., G. Oster and A. Ipatchi, The Dynamics of the Density Dependent Population Models, Journal of Mathematical Biology. vol. 4, 1977, pp. 101-147.

Haag, G., A Stochastic Theory for Residential and Labour Mobility in-cluding Travel Networks, Technological Change. Emplovment and Spatial Dynamics (P. Nijkamp ed.), Springer-Verlag, Berlin, 1986, pp. 340-357.

Heekman, J.J. Statistical Models for Discrete Panel Data, Structural Analysis of Discrete Data with Econometrie Applications (C.F. Manski and D. McFadden, eds.), MIT Press, Cambridge, Mass., 1981. pp. 114-178.

Hénon, M., A Two-dimensional Mapping with a Strange Attractor, Communications in Mathematical Physics, vol. 50, 1976, pp. 69-70.

Iooss, G., Bifurcations of Maps and Applications. North-Holland, New York, 1989.

Kelsey, D., The Economics of Chaos or the Chaos of Economics, Oxford Economie Papers. 40, 1988, pp. 1-31.

Lauwerier, H.A., Two-Dimensional Iterative Maps, Chaos (A.V. Holden, ed.), Manchester University Press, Manchester, 1986, pp. 58-95.

Leonardi, G., An Optimal Control Representation of a Stochastic Multistage-Multiactor Choice Process, Evolving Geographical Structures (D. Griffith and A. Lea, eds.), Martinus Nijhoff, The Hague, 1983, pp. 62-72.

Li, T.Y. and J.A. Yorke, Period Three Implies Chaos, American Mathematical Monthly. vol. 82, no. 10, 1975, pp. 985-992.

Lung, Y., Complexity and Spatial Dynamics Modelling: From Catastrophic Theory to Self-Organizing Process: a Review of Literature, The Annals of Regional Science, vol. 22, no. 2, 1988, pp. 81-111.

May, R., Simple Mathematical Models with Very Complicated Dynamics, Nature. vol. 261, 1976, pp. 459-467.

Maynard Smith, J., Mathematical Ideas in Biology. Cambridge University Press, Cambridge, 1968.

Nijkamp, P., Long-Term Economie Fluctuations: A Spatial View, Socio-Economic Planning Sciences, vol. 21, no. 3, 1987a, pp. 189-197.

Nijkamp, P. (ed.), Discrete Spatial Choice Analysis: Special Issue Regional Science and Urban Economics. vol. 17, no.1, 1987b.

Nijkamp, P., and A. Reggiani, Entropy, Spatial Interaction Models and Discrete Choice Analysis: Static and Dynamic Analogies, European Journal of Operational Research, vol. 36, no. 2, 1988a, pp. 186-196.

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Nijkamp, P., and A. Reggiani, Dynamic Spatial Interaction Models: New Directions, Environment and Planning A. vol. 20, 1988b, pp. 1449-1460.

Nijkamp, P. and A. Reggiani, Chaos Theory and Spatial Dynamics, Proceedings IX Annual Conference of Italian Regional Science Association. Torino, vol. 2, 1988c, pp. 583-602.

Nijkamp, P., and A. Reggiani, Theory of Chaos in a Space-Time Perspective, Paper presented at the thirty-fifth North American Meeting of the Regional Science Association, Toronto, Canada, 1988d.

Poston, J.M.T. and H.B. Stewart, Non-Linear Dynamics and Chaos. John Wiley and Sons, Chichester, 1986.

Pounder J.R. and T.D. Rogers, The Geometry of Chaos: Dynamics of a Non-Linear Second-Order Difference Equation, Bulletin of Mathematlcal Biology. vol. 42, 1980, pp. 551-597.

Reiner, R., M. Munz, G. Haag and W. Weidlich, Chaotic Evolution of Migratory Systems, Sistemi Urbani. vol. 2/3, 1986, pp. 285-308.

Sonis, M., A Unified Theory of Innovation Diffusion, Dynamic Choice of Alternatives, Ecological Dynamics and Urban/Regional Growth and Decline, Paper presented at the Conference on Innovative Diffusion, Venice, 1986.

Sonis, M. and D.S. Dendrinos, Period-Doubling in Discrete Relative Spatial Dynamics and the Feigenbaum Sequence, Mathematlcal Modelling: an International Journal, vol. 9, 1987, pp. 539-546.

Sterman, J.D., A Behavioural Model of the Economie Long Wave, Journal of Economie Behaviour and Organization. vol. 5, 1985, pp. 17-53.

Wilson, A.G., Catastrophic Theory and Bifurcation. Croom Helm, London, 1981.

1986-1 Peter Nijkamp New Technology and Regional Development

1986-2 Floor Brouwer Aspect5 and Application of an Integrated Peter Nijkamp Environmental Model with a Satellite Design

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1986-3

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Peter Nijkamp

Peter Nijkamp

Peter Nijkamp Jacques Poot

Henk Folmer Peter Nijkamp

Floor Brouwer Peter Nijkamp

Han Dieperink Peter Nijkamp

Peter Nijkamp Aura Reggiani

E.R.K. Spoor

V. Kouwenhoven A. Twijnstra

F.C. Palm E. Vogel vang

M. Wortel A. Twijnstra

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F.C. Palm C.C.A. Winder

25 Vear* of Regional Science: Retrospect and Prospect

Information Centre Policy in a Spatial Perspeetive

Structural Dynamics in Cities

Dynamics of Generalised Spatial Interaction Models

Methodological Aspects of Impact Analysis of Regional Economie Policy

Mixed Oualitative Calculus as a Tool in Policy Model ing

Spatial Dispersion of Industrial Innova-tion: a Case Study for the Netherlands

A Synthesis between Macro and Micro Models in Spatial Interaction Analysis, with Spe­cial Reference to Dynamics

De fundamenten van LINC

Overheidsbetrekkingen In de strategie en organisatie van ondernemingen

A short run econometrie analysis of the in­ternational coffee market

Flexibele Pensioenering

Causes of Labour Market Imperfections in the Dutch Construction Industry

The Stochastic life cycle consumption mo­del: theoretical results and empirical evidence

1986-17 Guus Holtgrefe DSS for Strategie Planning Purposes: a Future Source of Management Suspicion and Disappointment?

1986-18 H. Visser H.G. Eijgenhuijsen J. Koelewijn

The financing of industry in The Nether­lands

1986-19 T. Wolters Onderhandeling en bemiddeling in het be-roepsgoederenvervoer over de weg

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