Organizacija, letnik 39 Razprave številka 10, december 2006

The Periodicity of the Anticipative Discrete 
Demand-Supply Model 

Andrej Škraba1, Davorin Kofjač1, Črtomir Rozman2 

1Univerza v Mariboru, Fakulteta za organizacijske vede, Kidričeva cesta 55a, SI-4000 Kranj 

2Univerza v Mariboru, Fakulteta za kmetijstvo, Vrbanska 30, SI-2000 Maribor 

{andrej.skraba, davorin.kofjac}@fov.uni-mb.si, crt.rozman@uni-mb.si


This paper presents an analysis of the periodicity solutions to the discrete anticipative cobweb model. Dubois’ anticipative principle was 
applied in the modification of Kaldor’s cobweb model. Characteristic solutions are gained through the application of a simulation, which 
determines the cyclical behaviour of supply and demand. Z-transform was applied in the determination of the solutions. The interconnection 
between the anticipative definition of the cobweb model and Hicks model is addressed.

Keywords: cobweb, hyperincursivity, system dynamics, anticipative system, nonlinear system, Farey tree, chaos

Obravnava periodičnosti rešitev anticipativnega diskretnega modela ponudbe in povpraševanja 

V prispevku je izvedena analiza periodičnosti v diskretnem anticipativnem cobweb modelu. Pri modifikaciji Kaldorjevega cobweb modela 
je upoštevan Duboisov anticipativni princip. S pomočjo simulacije so prikazane karakteristične rešitve, ki opredeljujejo ciklična nihanja med 
ponudbo in povpraševanjem. Pri določitvi rešitev je bila uporabljena z-transformacija. V prispevku je naslovljena povezava med anticipativno 
definicijo cobweb modela in Hicksovim modelom.

Ključne besede: cobweb, hiperinkurzivnost, sistemska dinamika, anticipativni sistem, nelinearni sistem, Fareyevo drevo, kaos

1 Introduction

The cobweb model presents the adjustment to market supply 
and demand. It is typically viewed as the model of the 
agricultural pricing mechanism. The story behind the model 
might be briefly explained as follows: “The quantity offered 
for sale this year depends on what was planted at the start 
of the growing season, which in turn depends on last year’s 
price. Consumers look at the current prices, though, when 
deciding to buy. The cobweb model also assumes that the 
market is perfectly competitive and that supply and demand 
are both linear schedules.” For a clear and extensive introduction 
to the topic, see Rosser, 2003. The fact, that the 
cobweb model is in the field of discrete dynamics is rather 
an advantage, since the systems of difference equations are 
often easier to grasp. For example, in his enduringly valuable 
scholarly work on the studies of Dynamic Systems, 
Luenberger (1979) firstly addresses difference and differential 
equations later on. The model in question has all the 
characteristics of classical System Dynamics (SD) models: 
equilibrium, competitiveness, human perception, delay and 
adjustment, but somehow it avoids being included settled 
in the common SD model bank of each SD modeller. The 
main reason for the elusiveness of the cobweb model lies in 
its original form, which is not suitable for direct transformation 
to the common elements such as Level (L) and Rate 
(R). The functions of supply Qs(k) and demand Qd(k) can be 
specified in the form:

 
(1)

 
(2)

where a, b, c and d are parameters specific to the individual 
markets. P(k) and Qs(k) should be restricted to the positive 
values. In the cobweb model it is assumed that producers 
supply a given amount in any one time period (determined 
by the previous time period’s price) and then the 
price adjusts so that all the products supplied are bought by 
customers. If we write this in the form of an equation, then 
Qd(k) = Qs(k), which enables us to state that the price is:

 (3)

 Eqs. 1, 2 and 3 are not quite in the proper form for 
performing the transformation to the SD model. One of the 
things is the time argument (k-1). The other is the missing 
Rate (R) elements and the corresponding .t. One should 
expect that the transformation will provide the known equations 
in the familiar form for the structure shown in Fig. 1. 
The model developed should enable us to examine the prop


 (1) 
(2) 
(3) 
RpRqPSupply~DemandSystems’ Adaptivenessf(, )PQQ 
(4) 
(5) 
(6) 
(1) 
(2) 
(3) 
RpRqPSupply~DemandSystems’ Adaptivenessf(, )PQQ 
(4) 
(5) 
(6) 
(1) 
(2) 
(3) 
RpRqPSupply~DemandSystems’ Adaptivenessf(, )PQQ 
669



erties of the cobweb model and also to consider its structural 
and incursive perspective.

 As the Wiener’s cybernetics principle (Wiener, 1961) 
stands firm in the system’s theory, the cobweb model principle 
stands as the basic linearized principle construct for the 
systems interaction dependence and will probably remain 
the basic starting tool for the quantitative analysis of complex 
systems.

2 Transformation to the SD Form

Fig. 1 shows, that the price P and quantity Q of the goods 
should be stated as the Level elements depending on the 
Rates that determine the change in price and quantity supplied. 
The theory behind the cobweb model states that the 
quantity supplied at the present depends on the price in the 
past. Therefore the price and the quantity supplied should 
be dependant variables as illustrated in Fig. 1. Restating the 
above Equations, while eliminating the time argument k-1, 
gives us the following set of equations:

(4)

 
(5)

 
(6)

 Eqs. 4, 5 and 6 will enable the determination of the 
Rates elements. Let us determine the Rate element for the 
change of Price Rp(k). As the equations are in the different 
form the Rate will be determined as R(k)=L(k+1)-L(k):

(7)

 A little more work will be needed for the RQ(k) since 
special time considerations had to be taken. We will apply 
the time arguments of k+1 and k+2 in order to loose the k-1 
argument that is present in Eq. 2:

 

(8)

 Since the time k* argument with the consideration of 
Eq. 1 and 2 actually represents the past, i.e. the k-1 argument, 
we should state the equations for P(k-1) and Qs(k-1). 
Eq. 3 will enable us to state P(k-1): 

 
(9)

 (10)

 Eq. 10 is set by the fact that Qd(k) = Qs(k) and Eq. 4. 
The consideration of the k-1 time argument is necessary in 
order to perform calculations in the model. At each time 
step the previous values are needed in order to perform the 
calculation. By inserting

in Eq. 10 we get:

 
(11)

 By inserting Eq. 11 and 9 into Eq. 8 we get a simplified 
form of the rate equation:

 
(12)

 As the result of the above algebraic manipulation, the 
cobweb model could be stated in the standard SD form: 

 
(13)

 
(14)

 
(15)

 
(16)

with the starting conditions and Qs(0)=x 
where x represents the starting perturbation of the model. In 
the above set of equations the .t is introduced which is not 
present in the formulation of the classical cobweb model. If 
.t=1 then the model is equivalent to the classical cobweb.

 RpRqPSupply~)PQQ 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
Organizacija, letnik 39 Razprave številka 10, december 2006

RpRqPSupply~DemandSystems’ Adaptivenessf(, )PQQ 
(1) 
(2) 
(3) 
RpRqPSupply~DemandSystems’ Adaptivenessf(, )PQQ 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(1) 
(2) 
(3) 
RpRqPSupply~DemandSystems’ Adaptivenessf(, )PQQ 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(1) 
(2) 
(3) 
RpRqPSupply~DemandSystems’ Adaptivenessf(, )PQQ 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(1) 
(2) 
(3) 
RpRqPSupply~DemandSystems’ Adaptivenessf(, )PQQ 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(1) 
(2) 
(3) 
RpRqPSupply~DemandSystems’ Adaptivenessf(, )PQQ 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(1) 
(2) 
(3) 
RpRqPSupply~DemandSystems’ Adaptivenessf(, )PQQ 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
Figure 1. The main elements of the proposed cobweb SD structure

 (11) 
(12) 
(13) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(11) 
(12) 
(13) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(11) 
(12) 
(13) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(11) 
(12) 
(13) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(11) 
(12) 
(13) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(11) 
(12) 
(13) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
670



Fig. 2 shows the SD structure of the cobweb model corresponding 
to Eqs. 13, 14, 15 and 16. There are two levels 
represented, P and Qs, and two rate elements, RP and RQs. 
The model’s behaviour is determined by the input parameters 
a, b, c and d as well as by the perturbation parameter 
p. The element P0 represents the initial value of the level 
element P. The initial value of the level element Qs is equal 
to the arbitrary value of perturbation p.

 The response of the classic cobweb model developed 
by SD methodology is shown in Fig. 3 and Fig. 4. The 
parameter values applied and the description of the system’s 
response are shown in Table 1. 

a 

b 

c 

d 

p 

Response 

400 

-20

-50

15

20

Stable

200

-8.1

-43

12

90

Unstable

160

-2

-20

2

55

Dyn. Stable 





 There are three possibilities: a) a Stable system, where 
the supply and demand converge, b) an Unstable system 
where the supply and demand diverge and c) the Dynamically 
stable system shown in Fig. 4, where the price and 
demand neither converge nor diverge. 

 A dynamically stable response indicates the periodical 
solution that will be of interest in further examination of 
the model. In general a solution yn is periodic if yn+m = yn for 
some fixed integer of m and all of n. The smallest integer for 
m is called the period of the solution. In our case the solution 
in Fig. 4 is a two-cycle solution and in general the following 
definition will be applied (Shone, 1997): 

 Definition 1. If a sequence {yt} has, for example, two 
repeating values y1 and y2, then y1 and y2 are called period 
points and the set { y1, y2 } is called a periodic orbit.

 This periodical response of the system is important 
because real agricultural systems depend on the cyclic 
behaviour and could be controlled only by regarding the 
periodicity of such systems. Examples from real cases could 
easily be found in crops as well as in the stock.

1 Separation of the Structural Elements

The structure of the model in Fig. 2 shows that the Price and 
Quantity are related. However the structure can be repre


Organizacija, letnik 39 Razprave številka 10, december 2006

PRPRQsQsP0a
........
bcdp
Figure 2. System Dynamics structure of the cobweb model

60-300-1000100300500600-60-200204020050100150QQPP200250681012141618a) Stableb) Unstable
......
......
......
......
Figure 3. Response of the SD cobweb model: a) Stable, b) Unstable

Table 1. Parameter values

671



sented in a different way. By transforming the cobweb model 
into an SD form, the model could become non-autonomous 
depending on the variable .t. The following two equations 
represent the different formulation of the cobweb model:

(17)

 

(18)

 This reformulation represents Qs and P as the non-
related quantities. The only boundary that exists are the 
coefficients. The rate elements should be determined in 
order to formulate the complete SD model:

 

(19)

(20)

 In order to meet the initial conditions of the model, 
the QS(k-1) should be determined:

 
(21)

 Equations for P and QS in standard SD form are as 
follows: 
(22)

 
(23)

 
(24)

 
(25)

 Eqs. 22, 23, 24 and 25 represent the cobweb model in 
a separated SD form as shown in Fig. 5. Note that the terms 
for P and QS are related only to the coefficients a, b, c, d and 
p. P(k+1) is dependent only on the value of P(k) and the 
coefficients a, b, c, d and p, but not on QS. Respectively for 
the QS(k+1). 

2 Anticipative Formulation

Comparison of the structures shown in Fig. 2 and Fig. 5 
indicates that P and QS depend only on the parameter values 
of a, b, c, d and p, i.e. on the initial conditions. Eqs. 22, 23, 
24 and 25 enable the determination of the entire anticipative 
(future event) chain, while the equation:

 
(26)

and Eq. 21 enable the determination of the feedback (past 
event) chain. The representation of the Feedback ~ Anticipative 
chain is shown in Fig. 6. The dynamics of interest 
are therefore the chain’s dynamics that are dependant on 
the parameters a, b, c, d and p. Both chains are actually 
dependant on strategy dynamics which could be formulated 
as f(a, b, c, d, p, t).

 (17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
(25) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
(25) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
(25) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
(25) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
(25) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
(25) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
(25) 
Organizacija, letnik 39 Razprave številka 10, december 2006

050257512510015001020304050607080T0QPDynamically Stable
......
...... 
(11) 
(12) 
(13) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
(25) 
(11) 
(12) 
(13) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
(25) 
Figure 4. Response of the SD cobweb model: Dynamically 
stable

P11RPRQsQsP0a
........
bcdp
Figure 5. A cobweb model in SD form ~ separated elements

 (26) 
(27) 
672



Application of the hyperincursive algorithm and 
inspection of the equations gained with Dubois’ (Dubois 
and Resconi, 1992) formulation of logistic growth and 
previous research (Kljajić, 1998; Kljajić, 2001, Kljajić et al., 
2005), yields the following set of equations for the hyperincursive 
cobweb model:

 
(27)

 
(28)

with the initial conditions:

(29)

 
(30)

 
(31)

 (32)

 The coefficients A and B in Eq. 27 could be replaced 
by the terms P(k+1) or P(k) while the coefficients C and D 
in Eq. 28 can be replaced by QS(k+1) or QS(k). This yields 
16 different combinations of system defined by Eqs. 27 and 
28 that should be studied. The appropriate explanation of 
the modified system structure should follow the techniques 
of graphical solutions for homogenous difference equations 
(Puu, 1963; Azariadis, 1993).

The system combination further examined will have the following 
terms: A=P(k+1), B=P(k) C= QS(k+1) and D= QS(k). 
This yields the following set of equations:

 
(33)

 
(34)

Organizacija, letnik 39 Razprave številka 10, december 2006

 (26) 
(27) 
(28) 
(29) 
(30) 
(31) 
(32) 
(33) 
(34) 
P(k+2)(k+2)
(k+1)(k+1)
(k+1)(k+1)
(k+1)(k+1)(k)
(k)
(k)
(k)
PRPRPRPRQsRQsRQsQsQsP0(k)
Q0sQ0s(k)
P0abcdpINITINITINITINIT 
(35) 
(36) 
..................................
Feedback chainAnticipative chainInitial Conditions
......
.s....s......s.....Qs......s......s......s.....
a(t)
....
b(t)c(t)d(t)p(t) 
(26) 
(27) 
(28) 
(29) 
(30) 
(31) 
(32) 
(33) 
(34) 
P(k+2)(k+2)
(k+1)(k+1)
(k+1)(k+1)
(k+1)(k+1)(k)
(k)
(k)
(k)
PRPRPRPRQsRQsRQsQsQsP0(k)
Q0sQ0s(k)
P0abcdpINITINITINITINIT 
(35) 
(36) 
(26) 
(27) 
(28) 
(29) 
(30) 
(31) 
(32) 
(33) 
(34) 
P(k+2)(k+2)
(k+1)(k+1)
(k+1)(k+1)
(k+1)(k+1)(k)
(k)
(k)
(k)
PRPRPRPRQsRQsRQsQsQsP0(k)
Q0sQ0s(k)
P0abcdpINITINITINITINIT 
(35) 
(36) 
Figure 6. Feedback ~ Anticipative chain

 (26) 
(27) 
(28) 
(29) 
(30) 
(31) 
(32) 
(33) 
(34) 
P(k+2)(k+2)
(k+1)(k+1)
(k+1)(k+1)
(k+1)(k+1)(k)
(k)
(k)
(k)
PRPRPRPRQsRQsRQsQsQsP0(k)
Q0sQ0s(k)
P0abcdpINITINITINITINIT 
(35) 
(36) 
(26) 
(27) 
(28) 
(29) 
(30) 
(31) 
(32) 
(33) 
(34) 
P(k+2)(k+2)
(k+1)(k+1)
(k+1)(k+1)
(k+1)(k+1)(k)
(k)
(k)
(k)
PRPRPRPRQsRQsRQsQsQsP0(k)
Q0sQ0s(k)
P0abcdpINITINITINITINIT 
(35) 
(36) 
(26) 
(27) 
(28) 
(29) 
(30) 
(31) 
(32) 
(33) 
(34) 
2)(k+2)
1)(k+1)
1)(k+1)
(k+1)(k+1)(k)
(k)
(k)
(k)
PRPRPRPRQsRQsRQsQsQsP0(k)
Q0sQ0s(k)
P0abcdpINITINITINITINIT 
(35) 
(36) 
(26) 
(27) 
(28) 
(29) 
(30) 
(31) 
(32) 
(33) 
(34) 
P(2)(k+2)
k+1)(k+1)
k+1)(k+1)
(k+1)(k+1)(k)
(k)
(k)
(k)
PRPRPRPRQsRQsRQsQsQsP0(k)
Q0sQ0s(k)
P0abcdpINITINITINITINIT 
(35) 
(36) 
(26) 
(27) 
(28) 
(29) 
(30) 
(31) 
(32) 
(33) 
(34) 
2)(k+2)
1)(k+1)
1)(k+1)
(k+1)(k+1)(k)
(k)
(k)
(k)
PRPRPRPRQsRQsRQsQsQsP0(k)
Q0sQ0s(k)
P0abcdpINITINITINITINIT 
(35) 
(36) 
P(k+2)(k+2)
(k+1)(k+1)
(k+1)(k+1)
(k+1)(k+1)(k)
(k)
(k)
(k)
PRPRPRPRQsRQsRQsQsQsP0(k)
Q0sQ0s(k)
P0abcdpINITINITINITINIT
Figure 7. Structure of the hyperincursive Cobweb model; Euler integration, .t=1

673



Eqs. 33 and 34 with the initial conditions stated by Eqs. 
29 ~ 32 could be modelled as shown in Fig. 7. The structure 
represents the cobweb model in a hyperincursive form 
modelled using classic SD elements. The Euler integration 
method is applied with the time-step .t=1.

 Eqs. 33 and 34 could be reformulated in order to show 
the dependency of the future-present-past events as follows:


 (35)

 
(36)

 Eqs. 35 and 36 state that the value of the present is 
dependent on the past as well as on the future. This paradoxical 
statement is realizable since the formulation of a 
feedback ~ anticipative chain could be stated. Fig. 7 has 
two delay chains, one for P and one for QS. One might note 
that the level and rate elements are dependant only on the 
coefficients and initialization values.

3 The Periodicity of the System

The z-transform is the basis of an effective method for the 
solution of linear constant-coefficient difference equations. 
It essentially automates the process of determining the coefficients 
of the various geometric sequences that comprise 
a solution (Luenberger, 1979). The application of the z-
transform on Eqs. 33 and 34, with initial conditions stated 
by Eqs. 29 ~ 32, gives:

 
(37)

An inverse z-transform yields the following solution:

 

(38)

 The following equation should be solved in order to 
acquire the conditions for the periodic response of the system:


 
(39)

 Let us compute a numerical example of a periodic 
solution by applying the z-transform. The period examined 
will be the period of 9 i.e. n=9. One should insert the condition 
n=9 into Eq. 39. The following possible solution for the 
initial condition is worth examining:

 
(40)

 The term (let us denote the term as z*) 
could be expressed in the following way using three different 
imaginary values in polar form:

 
(41)

 
(42)

 
(43)

 By inserting Eqs. 41, 42 and 43 into Eq. 40 and performing 
a trigonometric reduction, one gets the following 
solutions:

 (44)

 By inspecting Eq. 40 and considering the equation for 
the roots of complex numbers (Kreyszig, 1997):

 
(45)

the general form of the solution for the parameter d could 
be assumed:

 (46)

where n is the period and m = 1, 2, 3, …, n-1. A similar procedure 
could be performed for the arbitrary period of n. More 
general solutions, which also regard the parameter, b that 
was fixed for the purpose of determining the solutions, is:

 
(47)

 In some cases the solutions could be expressed in 
an alternative algebraic or trigonometric form. Table 2 
below shows the solutions for the parameter d up to the 
period of n=12. Alternative solutions could be expressed 
as the roots of the polynomial. The table incorporates the 
Shape symbols, which are important in the study of the 
response of dynamical systems. This is especially the case in 
the examination of complex nonlinear dynamical systems 
(Sonis, 1999; Matsumoto, 1997; Hommes, 1998). Mapping 
of the system and the visualization of the periodic solution 
is therefore important for the analysis of periodic or chaotic 
solutions of differential and discrete difference equations. 
The description of the shape is taken from the vocabulary 
of proper shapes although the response of the system is primarily 
not proper. The numerical values of the solutions for 
parameter d are important since these values also confirm 
the findings of Sonis (1999) on the domain of attraction for 
2D dynamics by n-dimensional linear bifurcation analysis. 
One of the important conditions gained by the proposed 
inspection is the value of the period n=10, which is in close 
relation to the period n=5. The value of the parameter d is

 with the numerical value being d=1.61803… 

This solution represents the “Golden Ratio” (.). Some of 
the different representations of the solution for the value of 
parameter d with period of n=10 are:

 (34) 
P(2)(k+2)
(k+1)(k+1)
(k+1)(k+1)
(k+1)(k+1)(k)
(k)
(k)
(k)
PRPRPRPRQsRQsRQsQsQsP0(k)
Q0sQ0s(k)
P0abcdpINITINITINITINIT 
(35) 
(36) 
(34) 
P(k+2)(k+2)
(k+1)(k+1)
(k+1)(k+1)
(k+1)(k+1)(k)
(k)
(k)
(k)
PRPRPRPRQsRQsRQsQsQsP0(k)
Q0sQ0s(k)
P0abcdpINITINITINITINIT 
(35) 
(36) 
31)31(i+. 
(39) 
(40) 
31)31(i+. 
(41) 
(42) 
(43) 
(44) 
(45) 
(46) 
(39) 
(40) 
31)31(i+. 
(41) 
(42) 
(43) 
(44) 
(45) 
(46) 
(39) 
(40) 
31)31(i+. 
(41) 
(42) 
(43) 
(44) 
(45) 
(46) 
(38) 
(39) 
(40) 
31)31(i+. 
(41) 
(42) 
(43) 
(44) 
(45) 
(46) 
(38) 
(39) 
(40) 
31)31(i+. 
(41) 
(42) 
(43) 
(44) 
(45) 
(46) 
(38) 
(39) 
(40) 
31)31(i+. 
(41) 
(42) 
(43) 
(44) 
(45) 
(46) 
Organizacija, letnik 39 Razprave številka 10, december 2006

 (47) 
215.
=d(48) 
(48) 
(49) 
215.
=d
...
.
...
.
.
..
.
.
.
.
<
.
<.
.
=
+
+
++
1111111111kkkkkkkkkkkkPPPifPPPifPPPifPPPR (50) 
(37) 
(38) 
(39) 
(40) 
(41) 
(42) 
(43) 
(44) 
(37) 
(38) 
(39) 
(40) 
31)31(i+. 
(41) 
(42) 
(43) 
(44) 
(37) 
(38) 
(39) 
(40) 
(41) 
(42) 
(37) 
(38) 
(39) 
(40) 
215.
=d(48) 
674



(48)

 The first solution of the parameter d with the period 
of n=10 connects the discrete system considered with the 
Fibonacci numbers given by the infinite series:

 
(49)

 The fact that the periodicity conditions of the discrete 
system examined incorporates the golden ratio number ., 
could be observed in other studies (Brock and Hommes, 
1997) of complex nonlinear expansions of the basic cob-web 
systems, e.g. in Brock and Hommes “Almost Homoclinic 
Tangency Lemma”. One should expect that the symmetrical 
response in n-mapping should follow the pattern with a 
synchronization match, e.g. in a certain point of the solution. 
The source of the condition mentioned is presented 
using the above procedure. (The value of parameter d for the 

period of n=5 is ). The periodicity conditions 

are similar to the parameter values gained for the domain of 
2-d dynamic attraction by Sonis (1999). This set of parameters 
is augmented with two values for the periods 8 and 12, 
which are not stated in (Sonis, 1999). 

4 Nonlinear Setup and Results

The system’s responses were gained according to the parameter 
values gathered in Table 2. The changes were made to 
parameter d, which yielded the synchronization patterns 
as shown in the shape column. The parameter values were 
gained from the simulation, where the range of parameter 
d was set at [-40, 40] with .d=0.001. The condition for the 
determination of the parameter values was set by the rule of 
acceptable error between steps of the simulation and definition 
1 of the synchronization. 

Table 2. Parameter values at sync.

desc.

a

b

c

d

p

Triangle

400

-20

-50

20.0000

160

Quadrangle

400

-20

-50

-0.0010

160

Pentagon

400

-20

-50

-12.3671

160

Pentagram

400

-20

-50

32.3620

160

Hexagon

400

-20

-50

-20.0000

160

Nonagram

400

-20

-50

-6.9450

160

Hexagram

400

-20

-50

15.3070

160





 Let us consider the following expansion of the model 
(Škraba et al., 2005, Škraba et al., 2006) and let us define the 
adaptive nonlinear rule R as:

(50)

 48) 
(48) 
(49) 
...
.
...
.
.
..
.
.
.
.
<
.
<.
.
=
+
+
++
1111111111kkkkkkkkkkkkPPPifPPPifPPPifPPPR (50) 
(52) 
215.
=d(48) 
(48) 
(49) 
215.
=d
...
.
...
.
.
..
.
.
.
.
<
.
<.
.
=
+
+
++
1111111111kkkkkkkkkkkkPPPifPPPifPPPifPPPR (50) 
(52) 
Organizacija, letnik 39 Razprave številka 10, december 2006

 (47) 
215.
=d(48) 
(48) 
(49) 
215.
=d
...
.
...
.
.
..
.
.
.
.
<
.
<.
.
=
+
+
++
1111111111kkkkkkkkkkkkPPPifPPPifPPPifPPPR (50) 
(52) 
(47) 
215.
=d(48) 
(48) 
(49) 
215.
=d
...
.
...
.
.
..
.
.
.
.
<
.
<.
.
=
+
+
++
1111111111kkkkkkkkkkkkPPPifPPPifPPPifPPPR (50) 
(52) 
1.35

1.3

1.25

P(kp1)

1.2

1.15

1.1

1.05

0.198

0.196

0.197

0.2

d

0.199

0.201

0.202

0.203

Figure 8. Bifurcation diagram in the range d . [0.196, 0.204]

675



Since the forced nonlinear rule has been applied, we 
have arrived at the characteristic nonlinear bifurcation diagram. 
Let us observe the response of the system at the period 
of p=6 as one of the polygon rules, which should provide the 
periodicity of the system considered. The beginning of the 
bifurcation corresponds to the value of the parameter d=1, 
which has been indicated in the analysis of the initial 2-d discrete 
map. Period six is followed by p=7 and p=8. However, 
the analytical proof of the periodicity would be hard since 
the underlying Farey sequence defines the adapted nonlinear 
2-d discrete map. Such evidence is also found in other 
works on nonlinear system analysis, for example (Brock and 
Hommes, 1997; Gallas and Nusse, 1996) or in the recent 
works of Puu (Puu and Shushko, 2004; Puu, 2005).

 Consider another generic alteration of the initial 
anticipative cobweb model:

(51)

 Slight modification of initial Hicks’ model (Puu, 1963) 
gives this interesting response. The system can be represented 
in three dimensions, which reveals the periodicity of 
the system for which the previously determined conditions 
of the Farey tree generally still hold. Fig. 9 shows the 3d 
bifurcation diagram for the altered model. You can see the 
four attractors, which are simultaneous and represent the 
four possible equilibrium states for the trade dynamics. This 
4-cycle characteristic is preserved during the alteration of 
the parameter d, which can be observed in the Fig. 10. The 
four dots on the centre-right side of the figure represent the 
four-cycle characteristic of the response. The larger orbits 
indicate the steep change in the modus of the system.

 In order to analyze the preservation of the periodic 
solutions, the most significant periodic solution, i.e. the 
period of 6, has been applied to the system in Eq. 50, which 
is restated in the following form:

 (52)

 This proposed model, with certain limitations, yields 
the periodicity solutions that are related to the system attractor. 
For example, for period 10 the initial values are: PKP1=-
1, PKP1=-1.61803, PZ=1.61803, d=1.61803 and v=1. Fig. 11 
represents the response of the system for period 6, where 
the parameter d=1. The starting points of the attractor are 
from the interval (-2, 2). The simulation for the determination 
of the attractor was performed using 30,000 random 
starting points. Fig. 12 shows the period 6 attractor with 
6 attractive regions, which are doubled, actually making 
12 beams of periodicity. The centre of the attractor reveals 
the distinguishing 6-sided polygon shape. Fig. 12 shows a 
magnification of the centre of the period 6 attractor. 

5 Conclusion

The story revealed from the hyperincursive model developed 
here raises the following questions: a) Does a change in the 
strategy change the structure or does it only change the relations 
between the elements of that structure? b) Does a change 
in the strategy change the future as well as the past?

 A change in the strategy would mean a new and differ


 ...
.
...
.
.
..
.
.
.
.
<
.
<.
.
=
+
+
++
1111111111kkkkkkkkkkkkPPPifPPPifPPPifPPPR (50) 
(52) 
(51) 
.4
.202402468
.4
.2024PkPzPkp1 
(51) 
.4
.202402468
.4
.2024PkPzPkp1
Organizacija, letnik 39 Razprave številka 10, december 2006

Figure 9. The emergence of four synchronous attractors in the nonlinear situation where d=0.26131278 and b=0.33

676



ent future and should also mean a different past if the strategy 
change occurred earlier. The hyperincursive cobweb 
model consideration enables us to change the future as well 
as the past chain of events. However, a different examination 
of the system dynamics is proposed where change in the 
key parameters is performed while observing the change in 
a complete future and past chain rather than observing the 
classical time response of the system. 

The following procedure proposition emerges, which enables 
the anticipative formulation of the classical dynamic 
system. Since the hyperincursive systems are hard to determine 
(Dubois and Resconi, 1992; Rosen, 1985), the anticipatory 
mechanisms developed should be applied. Therefore, 
the model should a) be transformed into the separated form, 
b) provide the property of the past-future chain and c) apply 
the hyperincursive structure to the model studied. 

Organizacija, letnik 39 Razprave številka 10, december 2006

.40
.2002040
.2000200400600
.30
.20
.100102030PkPzPkp1
.50050
.50050
.50510x 104Pkp1PkPz
.40
.2002040
.2000200400600
.30
.20
.100102030PkPzPkp1
.50050
.50050
.50510x 104Pkp1PkPz
Figure 10. The preservation of four synchronous attractors in the nonlinear situation where d=0.26151152 and b=0.33

Figure 11. The period 6 attractor with the parameters: d=1, b=1 and v=1

677



The model developed shows that, by the statement of 
the general rule of the system, the synchronization of the 
entire feedback-anticipative chain could be gained by setting 
the appropriate strategy in the form of the value of the 
parameters set, which should be time dependant. 

The idea for the simulation proposed in this paper is quite 
different from the common paradigm. The structure of 
the model should yield the entire feedback-anticipative 
chain and observation should be made of the entire system 
response. This provides new and quite challenging responses 
that should initiate further interest and examination of 
the proposed model. 

 One of the interesting responses from the model is the 
helix like shape that is synchronized at certain time steps. 
The entire feedback-anticipative chain, i.e. all the point set, 
is synchronized according to the period of the system. The 
solution of the periodicity conditions for the 2-d discrete 
linear cobweb map provided the means to determine these 
periodicity conditions and an analytical approach using z-
transformation provides the proper way to determine the 
periodic solutions. The emergence of a Farey tree as the 
rational fraction representation yields the organization of 
the periodicity solutions. The model developed shows that, 
by the statement of the general rule of the system, the synchronization 
of the entire feedback-anticipative chain could 
be achieved by setting the appropriate strategy in the form 
of the value of the parameters set, which should be time 
dependant. The bifurcation experiment with the nonlinear 
mapping provided the example of periodicity transposition 
to systems of higher complexities. Period 6 has been 
determined as one of the most stable periodic solutions, as 
has been explicitly shown by the analysis of system’s attractors.


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Figure 12. A magnification of the centre of the period 6 attractor from Fig. 11 with the parameters: d=1, b=1 and v=1

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Acknowledgement 

This research was supported by the Ministry of Higher Education, 
Science and Technology of the Republic of Slovenia 
(Program No. UNI-MB-0586-P5-0018).

Andrej Škraba obtained his Ph.D. in the field of Organizational Sciences 
from the University of Maribor. He works as a researcher, 
teaching assistant and assistant professor at the University of 
Maribor, Faculty of Organizational Sciences in Cybernetics and 
DSS Laboratory. His research interest covers modelling and simulation, 
system dynamics, decision processes and multiple criteria 
decision-making. Recently his interests have been focused on the 
analysis of nonlinear chaotic systems.

Davorin Kofjač received his Associate’s degree in computer science 
at the Faculty of Electrical Engineering and Computer Science 
in Maribor in 1999. He achieved his B.Sc. degree in information 
systems management at the Faculty of Organizational 
Sciences in Kranj in 2002. Currently he is working as a young 
researcher at the Faculty of Organizational Sciences. His research 
interests include modeling and simulation, artificial intelligence 
and operational research.

Črtomir Rozman achieved his PhD. at University of Maribor, Faculty 
of Agriculture. He is active as Assistant Professor for Farm 
management and Agronomic information systems in the Department 
for Agriculture Economics and Rural Development (Faculty 
of Agriculture, University of Maribor). His research includes development 
of DSS for farm management (simulation modelling, multi 
criteria decison analysis) and economics of agricultural production 
with the emphasis on organic agriculture. He is also involved in 
teaching activities as Head of Department and as thesis supervisor 
at post graduate study programmes. He is author or coauthor of 
121 papers including 6 papers in journals with impact factor.

Organizacija, letnik 39 Razprave številka 10, december 2006

679