Lecture presented at the workshop "Complex Systems in Natural and Social Sciences" (CSNSS’99), 14-17 September 1999, Kazimierz Dolny, Poland

Science as dynamical system with a time delay

We assume that changes in the time evolution of the system -- natural sciences -- at time t depend on both the state of the system at the current time, x(t), and on the state in a certain past time $x(t-\tau)$. The feedback $g(x(t-\tau))$ between present changes and past states can be written as

$$\dot{x}(t) = F(t,x(t),g(x(t-\tau))$$ (1)

where $\tau > 0$ is time delay. It is assumed the simplest version of the identity feedback. Then dynamical system (1) is called a delay differential equation (DDE).

The system (1) constitutes the simplest model of growth of natural sciences in which the rate of creation of new results x(t) depends on the state of science (measured by a number of scientific results) in the past $t-\tau$.

In the real evolution there is many different ideas having influence on the growth of science after delay time. The delay $\tau$ can be interpreted as a notion of time to build, connecting with time needed for deeper understanding of theory content. Here the delay parameter $\tau$ means the time needed for writing an essential paper, however this parameter may have different interpretations. Among physicists working on a given problem there is a common feeling whether obtained result is important and essential in the present status of science [1]. We assume that the speed of creation of new results in time t is proportional to a number of results in time t-T. We consider that increase of essential papers represents the growth of science. These papers are necessary to write a new essential paper because it builds up on older results. Increase of knowledge given by a number of essential papers in a unit of time is a function of a total number of papers in different past moments T_i with some coefficients $\alpha_i$.

$$\dot{x}(t) = \sum_{i} \alpha_i x(t-T_i)$$

where $\sum_{i} \alpha_i = \alpha$. Provided that all delay are the same T_i = T for all i then

$$\dot{x}(t) = \alpha x(t-T)$$

Our model is the simplest in which there is included only single constant delay parameter $\tau$. This toy model we can explain observed periodicity in dynamics of growth sciences [1,4,6]. There is a good evidence of exponential growth of scientific results the initial developing of a new theory [2,5].

Following the de Sola Price model -- science is a dynamical process with a positive feedback. New results in output are given in input i.e.

$$\dot{x}(t) = \alpha x(t).$$ (2)

This model can be enriched with both time delay and the Miller mechanism of dying of some results

$$\dot{x} = \alpha x(t-T) - x^2(t).$$ (3)

In general, there is some delay and new results in output appear in input with delay.

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

An analytically simpler method of describing solutions to DDEs , which is well known from the theory of ODE ,is the analysis of the characteristic equation. In our case it is the equation for linearized equation (3), i.e

$$\dot{x}(t)= \alpha (x - x^{*} (t-T) - 2 x\vert _{x=x^{*}}(x-x^{*}) .$$ (4)

After centering the fixed point at the origin (4) we obtain x-x^*=y, the linear equation for the perturbation around a fixed point

$$\frac{dy}{dt} = \alpha y(t-T) - \beta y$$ (5)

where $\beta = 2 x^{*}$ and constants $\alpha, \beta > 0$. Let us note that similar equation was obtained by Kalecki in his business cycle model [3].

We assume that there is a solution $y \propto e^{\lambda t}$ and the characteristic equation for system (4) has the form

$$\lambda = \alpha e^{- \lambda T} - \beta .$$ (6)

Our idea is to search for cyclic behaviour in the system (4). This behaviour is analogous to economic phenomenon known as business cycle. Therefore it can be described as scientific cycle.

The creation of cyclic behaviour is understood in terms of bifurcation theory. The Hopf bifurcation takes place if the pair of imaginary eigenvalues crossing transversally an imaginary axis on the Gauss plane. To prove this fact we can check that:

1) there is the pair of conjugated complex solution of (6) in the form $\lambda = \sigma \pm i \omega$ and there is only one for which real part of eigenvalue ${\rm Re} \lambda = 0$;

2) the transversality condition is fulfilled, i.e. $\frac{d \ }{dt} {\rm Re} \lambda(t) \ne 0$.

After the decomposition the equation (6) on real and imaginary part we obtain

$$\sigma \alpha e^{- \sigma T} \cos(\omega T) - \beta$$ = (7)

$$- \omega \alpha e^{- \sigma T} \sin(\omega T)$$ = (8)

Because of a reflection symmetry of this equation $\omega \to - \omega$ we can assume that $\omega > 0$. The cyclic behaviour can appear if $\sigma =0$, and from (7-8)

$$\tan \omega_{\rm bi} T_{\rm bi} = - \frac{\omega}{\beta} \Ri... ...(\frac{\omega_{\rm bi}}{\beta} + 2j\pi), \quad j \in {\bf C},$$

$$\beta^2 + \omega^2 = \alpha^2 \Rightarrow \omega_{\rm bi} = \sqrt{\alpha^2 - \beta^2}$$

This means that if $\alpha^2 > \beta^2$ there always exist $\omega$ and consequently a periodic solution with the approximately constant period

$$T \cong \frac{2 \pi}{\omega} = \frac{2 \pi}{\sqrt{\alpha^2 - \beta^2}}.$$

Now we must check the transversality condition

$$\begin{eqnarray*} {\rm Re} \frac{\partial \lambda}{\partial T} &=& {\rm Re} \fr... ...- \lambda^2 - \lambda \beta] = -(\sigma^2 - \omega^2) - \sigma \end{eqnarray*}$$

and

$${\rm Re} \left. \frac{\partial \lambda}{\partial T} \right\... ... \sigma}{\partial T} \right\vert _{\sigma =0} = \omega^2 > 0.$$