Scientific cycle model with delay

Jointly published by Akadémiai Kiadó, Budapest Scientometrics,

and Kluwer Academic Publishers, Dordrecht Vol. 52, No. 1 (2001) 83–95

Scientific cycle model with delay

MAREK SZYDàOWSKI,1 ADAM KRAWIEC2

1Faculty of Mathematics and Physics, Jagiellonian University, Kraków (Poland)2Department of Economics, Jagiellonian University, Kraków (Poland)

In this paper we analyse the growth in scientific results of natural sciences in terms of infinitedynamical system theory. We use functional differential equations to model the evolution ofscience in its sociological aspect. Our model includes the time-to-build of fundamental notions inscience (time required to understand them). We show that the delay parameter describing timerequired to learn and to apply past scientific results to new discoveries plays a crucial role ingenerating cyclic behaviour via the Hopf bifurcation scenario. Our model extends the de SollaPrice model by including death of results as well as by incorporating the time-to-build notion. Wealso discuss the concepts of knowledge and its accumulation used in economic growth theory.

Science evolution as a dynamical system with a time delay

When one thinks about the mathematical description of scientific discoveries and theevolution of science one immediately recalls Goffman and Harman’s seminal paper.6

They considered the history of discoveries in symbolic logic. Their analysis was basedon the Church bibliography3,4 which provided a complete list of publications in the fieldof symbolic logic from 1847 to 1932, beginning with the work of Boole and De Morganand ending with Gödel's incompleteness theorem. Church marked with an asteriskpublications which he thought to be of special interest or importance from the point ofview of symbolic logic, and with a double asterisk publications “which mark the firstappearance of a new idea of fundamental importance”.

In our opinion such a distinction seems to be natural and can be adopted in othercases but then Church’s review paper should be replaced by another one suitable forother disciplines. Our main aim is to present some general mechanism which is genericin a typical scientific discipline like theoretical or experimental physics.

Following Goffman and Harman, scientific discovery may be modelled by acomplete, ordered, and finite set of permutable elements of information formed within“the universe of scientific discourse”.6 The act of a new discovery can be treated as a

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M. SZYDàOWSKI, A. KRAWIEC: Scientific cycle model with delay

successful culmination of a series of efforts to “acquire the necessary informationelements and to find an appropriate set-defining and set-ordering criterion”.

We present the alternative approach to the problem of scientific cycles. Ourapproach differs from Harman and Goffman’s in two points. First, we do notdifferentiate the type of scientists; second, we are interested in papers of similar quality.Therefore, the papers as well as scientists are homogeneous. This is our simplifyingassumption in the analysis.

All scientists constitute a homogeneous group capable of creating important papers(with both theoretical and experimental results). We analyse only such papers. Everyscientist writes important papers which are considered important as long as they arecited. It does not matter if this scientist studies another subject or is alive.

For experimental study we suggest to choose as our homogeneous group of papers,the papers which were of fundamental importance in the Harman and Goffmanclassification. It is a possible way of creating the data set. It could be those papers insome field of science (not necessary symbolic logic) that are important according to theHarman and Goffman classification (often cited papers). In general we consider naturalsciences where the communication is directed through the scientific journals.Operational sense of state variable x is a number of papers which have been cited in agiven year often then a threshold value. Given an access to the appropriate data, the nextstep would be the estimation of the time delay from the observations.21

Alternatively, we can take all the papers published in the field of science but we feelthat their number is more influenced by exogenous factors (e.g., there are less papersduring wars). Of course the periods of war are anomalous. It is an exogenous factorwhich negatively modifies the path of exponential growth. In economics this type ofevents are treated as stochastic phenomena and modelled in stochastic processes. Ourapproach is strictly deterministic.

Apart from being simple, our model is quite general and cannot be restricted to thespecific field of science or mathematics.

We make some general assumptions referring to the model. First of all our system isdescribed in terms of dynamical systems, i.e. the state variable of the system is a welldefined variable. In the considered case it is the number of important papers in a givenperiod of time. Moreover, we assume that the time variable is continuous and the rate ofchange of a number of scientific results is a smooth function of the state variable. Timeevolution of the system is represented by the trajectory in the space of all possible states.This space is called the phase space.

In the first approximation we consider the evolution of the number of papers whichmark the first appearance of a new idea of fundamental importance which corresponds

84 Scientometrics 52 (2001)


to the publication with a double asterisk in the Church bibliography. The state variablex(t) is the number of these papers published in time t. In the general case this functionbecomes the only separated component of the vector state variables and the evolution ofthe states corresponding to a single asterisk in the Church bibliography can also bemonitored. In this terminology the system is considered on the invariant submanifold,because the interaction of other papers is negligible.

Therefore all our assumptions regarding the model can be expressed in the form ofthe following system

� ( ) ( ), , ,....,x t f x i ni i i= = 1 2

where x(t) ∈ 5n and f i ∈ C ∞ and a dot denotes differentiation with respect to t.

In real evolution there are many different ideas having possible influence on thegrowth of science after some time. The delay T can be interpreted as such a time whichis necessary to build the notion, connected with time needed for deeper understanding oftheory content. Here, the delay parameter T means the time needed for writing anessential paper, however this parameter may have different interpretations. Amongphysicists working on a given problem there is a common feeling whether an obtainedresult is important and essential in the present status of science.6 We assume that therate of creation of new results in time t is proportional to the number of results in timet–T. We further assume that the increase of essential papers represents the growth ofscience. These papers are necessary to write a new essential paper because they are builtupon older results. For example, the notion of energy was well known since the Newtontimes, whereas the full understanding of this notion was achieved by E. Noether 200years later. Now we have a similar problem with the notion of energy localization.

We assume that changes in the time evolution of the system – natural sciences – attime t depend on both the state of the system at the current time, x(t), and on the state ina certain past time x(t−T). The feedback g(x(t−T)) between present changes and paststates can be written as

�( ) ( , ( ), ( ( ))x t F t x t g x t T= − (1)

where T > 0 is a time delay. We assume the simplest version of an identity feedback. Insuch a case the dynamical system (1) is a delay differential equation (DDE). Thesolution of (1) is a function of time t, it depends also on some initial function φ(t)defined on the interval [−T,0].

System (1) constitutes the simplest model of growth of natural sciences in which therate of creation of new results x(t) depends on the state of science (measured by anumber of scientific results) in the past t−T.

Scientometrics 52 (2001) 85


In a more realistic case many different time delays can be included in the model. Letthe increase of knowledge given by a number of essential papers in a unit of time be alinear combination of the total number of papers in different past moments Ti with somecoefficients αi.

�( ) ( )x t x t Tii

i= −∑α

where ∑i αi = α. Provided that all delays are the same Ti = T for all i, we have

�( ) ( ) .x t x t T= −α

Our model is the simplest one. It is included only a single constant delay parameterτ. Nevertheless such a toy model allows us to explain observed periodicity in thedynamics of sciences.6,12,15 There is a good evidence of exponential growth ofscientific results in the initial development of a new theory.8,13

Following the de Solla Price model – science is a dynamical process with a positivefeedback. New results in output are put in input, i.e., he considered a system with apositive feedback

�( ) ( ) .x t x t= α (2)

Solving this equation we obtain exponential growth of scientific results with the rateα. The real growth seems to be limited rather than growing to infinity. Therefore weintroduce the Miller mechanism of death of some results, in which new results makesome results obsolete or are the generalization of others

� ( ) ( ) .x x t x t= −α β 2 (3)

Let us note that the notion of creative destruction can be mention in this context.1

This model can also be enriched with the time delay. In general, the new papersdepend on results obtained up to the time t−T. It can be interpreted in two ways. Firstly,there is the time necessary to write and prepare a paper and secondly, there is the timenecessary to understand and incorporate these older results into a new paper.Additionally, the death of results takes place in a time t when new papers are published

� ( ) ( ) .x x t T x t= − −α β 2 (4)

Let us note that after rescaling the time variable according to the rule t → α −1τ

and redefinition of a state variable in such a way that x → x ([(τ)/(α)] − ( )τα α

− T =

x– (τ– − τ) from system (3) we obtain (after dropping bars) the very simple system



� ( ) ( ) .x x x= − −τ γ ττ 2 (5)

where now a dot denotes differentiation with respect to the new parameter τ and thephysical parameter is defined as γ = β / α.

It can be easily shown that the exact solution of (5) in the case of a vanishing delayis given by

x Ce C( ) ( ), .τ γ τ= − = <− −1 1 1const

Equation (5) with zero delay describes the simplest model of populationgrowth which was proposed by Verhulst in the XIX century. For example the growthof biomass, described by the number of cells N, is governed by the equationdN/dt = µN − νN2 where the first term corresponds to the birth of cells whereas thesecond one is related to the death of cells. Verhulst assumed that probability of deathdepends on the size of population, i.e. the probability is greater for a larger population.Asymptotically as t → ∞ the population remains constant (N∞ = µ / ν).

It is evident that critical points of system (5) have the same form as thecorresponding system with the zero delay τ. It is a simple consequence of the fact thatcritical points represent stationary states of the system, i.e. the system is invariant withrespect to time translation t → t + const. Therefore the critical points of equation (5) are

x01 0= (6)

x02

11=

�

��

��=

−−β

αγ . (7)

Critical point (6) represents the trivial state of the system in which there are no newgreat ideas and discoveries in a given field. Critical point (7) represents the state of thesystem which can be interpreted as the constant rate of creation of new results. For γ = 1new ideas coincide with the dying ones. For γ < 1 a new idea explains more and replacean old explanation. At last, for γ > 1 new ideas not only replace the old ones but alsoexplore new areas in the field.

From the formal point of view the difference between (3) and (5) is that if we putα = 1 then parameter β is expressed in the units of α. The existence of correspondingsymmetry of the system (3) means that parameter γ is a physical parameter of the model.

The delay in DDEs provides a natural method by which constant coefficientequations can be solved, even when these equations are nonlinear as in our example.However, this method requires tedious computations and often yields cumbersomesolutions.



An analytically simpler method of describing solutions to DDEs, which is wellknown from the theory of ordinary differential equations, is the analysis of thecharacteristic equation. In our case it is the equation for linearised equation (3), i.e

�( ) ( *)( ) * ( *)( ) .x t x x t T x x x t= − − − −α β2 (8)

After centring the fixed point at the origin (8) we obtain x−x* = y, the linear equationfor the deviation from the fixed point

dy

dty t T y= − −α β( ) (9)

where β = 2β x* and constants α, β >0. Note that a similar equation was also obtained

by Kalecki in his business cycle model.10

It is easy to check that the real eigenvalue at the critical point is

λ α β α= − = − <2 0x* ,

i.e., this point is a stable node.For the local analysis of stability of the delay equation it is sufficient to consider the

characteristic equation for the linearised system (9). We assume that there is a solutiony∝eλt and the characteristic equation for system (8) has the form

λ α βλ= −−e Τ . (10)

Our idea is to search for cyclic behaviour in system (8). This behaviour is analogousto the economic phenomenon known as business cycle. Therefore it can be described asa scientific cycle.

The creation of cyclic behaviour may be understood in terms of the bifurcationtheory. There are many papers on the Hopf bifurcation theorem used in infinitedimensional systems and there are also several books applying it to various problems inscience.7,9,14 The Hopf bifurcation takes place if the pair of imaginary eigenvaluescrosses transversally the imaginary axis on the Gauss plane.14 In order to prove this factwe can check that

1) there is a pair of conjugated complex solution of (10) in the formλ = σ ± iω and there is only one for which the real part of eigenvalueReλ = 0;

2) the transversality condition is fulfilled, i.e. d

dt Re λ(t) ≠ 0.

After decomposition of equation (10) into real and imaginary parts, we obtain

σ α ω βσ= −−e TΤ cos( ) (11)



− = −ω α ωσe TΤ sin( ) . (12)

Because of reflection symmetry of this equation ω → − ω we can assume that ω > 0. Thecyclic behaviour can appear if σ = 0, and from (11)-(12)

tan arctan( ) ,ω ωβ

ω ωβ

πbi bi bi bibiT T j= − ⇒ = + j∈Z ,

β ω α ω α β2 2 2 2 2+ = ⇒ = −bi .

This means that if α2>β2 there always is ωbi and, consequently, a periodic solutionwith the approximately constant period

P ≅ =−

2 22 2

πω

π

α βbi.

The dependence of bifurcation value of T on system parameters α and β is shown on thediagrams. Figure 1 presents relationship between Tbi and α with β = 0.1. On this figurewe can see the saturation effect for higher values of parameter α. Otherwise if α = 1 isfixed then the relation between Tbi and β is demonstrated on Figure 2. Here we can seethe weak dependence of bifurcation value on constant β.

Figure 1. Dependence of Tbi on α with β = 0.1



Figure 2. Dependence of Tbi on β with α = 1

Note that if the delay is close to the bifurcation value, say T ≅ Tbi then from(11)-(12) we can obtain an infinite set of ωk, k = 0,1,... which determine one main cycle(k = 0) and cycles with shorter periods (k = 1,2,...).

Now we must check the transversality condition

Re Re ( ) Re

Re ( ) ( )

∂λ∂

∂∂

α β αλ

λ λ β σ ω σ

λ λΤ Τ

Τ Τ= − = − =

= − + = − − −

− −e e

2 2

and

Re .∂λ∂

∂σ∂

ωσΤ ΤT T= =

= = >bif 0

2 0

Finally we can conclude our discussion. The generalized de Solla Price model with adelay parameter admits endogenous cyclic behaviour which is represented by a limitcycle in the phase space for α > β (i.e. always if β = 0). This cyclic behaviour isgenerated by the Hopf mechanism of bifurcation to a periodic orbit.



The problem of stability of the cyclic solution is not the subject of the present paperand will be considered in our further research. We are going to extend our analysis todetermine the real evolution of a number of publication in physics by listing the materialcovered by the Science Citation Index.

Economic concept of knowledge and cyclic development of science

There is a new approach to social sciences using complex systems theory. It seemsto be very attractive from the methodological point of view as well as the explanation ofthe Kondratiev cycles or chaos and rationality in economics.11

Knowledge and learning are the key elements of the modern theories of economicgrowth. The classical Solow model of growth introduced exogenously definedexogenous variable called effectiveness of labour. Then the changes of the physicalcapital per worker and effectiveness of labour are sources of growth. The effectivenessof labour can be treated here as an abstract knowledge or technology. The other possibleinterpretation is the education and skills of the labour force.19

Both the physical capital per worker and effectiveness of labour cannot account fordifferences in growth in a country's past and present as well as differences in growthamong countries. Therefore, the new growth theory assumes the effectiveness of labourto has endogenous character and economic agents make rational decisions about seekingand using knowledge. These analysis could be extended in two directions.

First, there is the accumulation of human capital, being a worker's abilities, skillsand knowledge.

The second way is more interesting in our analysis. There is introduced a newresearch and development (R&D) sector where new technologies are produced. Now theeffectiveness of labour is the knowledge produced by R&D. However there are manytypes of knowledge beginning from basic scientific knowledge to knowledge applied forspecific aims (e.g. producing a microprocessor, hard disk and other componentsnecessary to make a computer). Basic scientific research is of interest here.

In the course of the development of natural sciences and mathematics we observethat results of research have been made available relatively freely. The motivation ofresearch is not the desire to make profit, and therefore it is supported by governmentsand different organizations. The economics behind it is relatively easy because it isgiven away freely and is used in production creating positive externalities. Therefore, itis valuable to society and should be subsidized.17,18,20



Alternative theories stress that important innovations and progress of knowledge isnot the work of a special kind of economic sector but is the work of extraordinarytalented individuals.2,16 Our point of view is close to this conception in this sense thatwe also stress out the role of individuals. In conclusion the analysis of knowledgeaccumulation and development of science is important to sociology and economics.Basing on the Solow model we assume the exogenous technical progress with constantgrowing rate g

� ( ) ( ) ,A t gA t=

and knowledge grows exponentially. This assumption seems to be compatible with deSolla Price's proposition. Further we can consider the economic growth model withknowledge accumulation in the following way � ( ) ( ) ,A t gA t T= − where the effect ofdeath of older results is neglected. The equations of the model have the form

�( )( )

( )( )k t sk t n g

A t T

A tk t= − + −�

��

α

� ( ) ( ) ,A t gA t T= −

where s is the saving rate, k = K/AL, and n is the rate of population growth.

Final remarks

We considered a simple model of growth of scientific ideas what allow us to obtainsome new results.

1. We formulate de Solla Price's approach in terms of functional differentialequations. In the special case of zero delay and without dying of old results we obtainthe de Solla Price law.

2. We discuss the effects of delay connected with the time in which a new idearipens. We have found that there are periods of high and low science activity without theexternal perturbations such wars, famine, political perturbations. This cyclic behaviouris proved by using the Hopf bifurcation theory.

3. Our considerations give us the foundation for modelling the knowledgeaccumulation in economic growth theory. The standard assumption in economics is thatknowledge increases with the constant rate.



Our model is fully endogenous without any exogenous influences. We do not knowto what extent the exogenous factors can dominate. However, we focus on endogenousmechanism of generating cycles in science growth. We analyse the problem of timeneeded to elaborate new ideas from the old ones. We claim that it is a possiblemechanism of cyclic behaviour in our model. Of course, there is another problem ofestimation its influence on the real growth of science.

In our opinion all models which explain observed cyclic behaviour in the number ofnew scientific results can be divided into two different groups. First, endogenous modelswhich explain scientific cycles in terms of some internal mechanisms. Second,exogenous models where cyclic behaviour develops from external factors. While we donot deny the importance of the latter mechanism, in this paper we focus our attention oninternal causes of cyclic behaviour.

It would be helpful here to concentrate on the status of obtained results. Our aim wasto bring a relatively simple model which can explain cyclic behaviour observed byGoffman and Harman. They observe that twenty-five year cycle was maintained forsymbolic logic up to present time. They also observed the period equalled to 12.5 yearwhen a new idea of fundamental importance would be expected to occur. From ourpoint of view this fact suggests the presence of a limit cycle in the corresponding phasespace. Because limit cycles are present only in nonlinear systems we have empiricalevidence of nonlinearity in evolution of the system. From our point of view, Goffmanand Harman found a special solution – an unstable focus – in the phase space.

If we put specific values of parameters α and β taken from experience we obtaindifferent values of the period of oscillations. For example, the cycle with period of 12.5year – the results obtained by Goffman and Harman – corresponds to α2−β2 = (0.16π)2.The sense of parameters α and β can be established for T = 0 then α = x−1(dx/dt)denotes the constant rate of creation of new results and β describes old results becomingworthless from the point of view of new discoveries. If β ≅ 0 then α ≅ 0.16π ≅ 0.5.

If we assume for simplicity that all papers are important and should be included inthe model then we can use the de Solla Price analysis of period τ when results double tocalculate relation between period T and parameter τ.

ατ

= ln 2

and on the other hand

α π β= +4 2

22

T.



From the above two equations we obtain

τπ β β

=+

=>>T

T

T2

2 2 2 24 2

1 1 44

( ) ln

..

If we put β = 0.4 then τ = 9 years which approximately corresponds to τ for growth ofAmerican science.5

*

It is the extended version of the paper which was presented at the 11th International Congress of Logic,Methodology and Philosophy of Science, August 20-26, 1999, Cracow, Poland. The paper was supported byKBN grant no. 1 H02B 009 15. The authors are grateful to dr K. Ma�lanka for useful comments.

References

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2. W. BAUMOL, Entrepreneurship: productive, unproductive, and destructive. Journal of PoliticalEconomy, 98 (1990) 893–921.

3. A. J. CHURCH, Symbolic Logic, 1 (1936) 121.4. A. J. CHURCH, Symbolic Logic, 3 (1938) 178.5. D. J. DE SOLLA PRICE, Science since Babylon. Yale University Press, New Haven, 1961.6. W. GOFFMAN, G. HARMAN, Mathematical approach to prediction of scientific discovery. Nature,

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17. W. D. NORDHAUS, The optimal rate and direction of thechnical change. In: Essays on the Theory ofOptimal Economic Growth, KARL SHELL, (Ed.) pp. 53-66, MIT Press, Cambridge, 1967.

18. E. S. PHELPS, Models of technical progress and the golden rule of research. Review of Economic Studies,33 (1966) 133–146.

19. D. ROMER, Advanced Macroeconomics. Mc-Graw-Hill, New York, 1996.20. K. SHELL, A model of inventive activity and capital accumulation. In: Essays on the Theory of Optimal

Economic Growth, KARL SHELL, (Ed.) pp. 67-85, MIT Press, Cambridge, 1967.21. H. VOSS, J. KURTHS, Reconstruction of non-linear time delay models from data by use of optimal

transformations. Physics Letters A 234 (1997) 336–344.

Received April 26, 2001.

Address for correspondence:MAREK SZYDàOWSKI

Faculty of Mathematics and Physics, Jagiellonian University,Orla 171, 30-244 Kraków, PolandE-mail: [email protected]


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Scientific cycle model with delay