Models describing phase shifts of output signal in economic systems under stochastic resonance conditions

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Models describing phase shifts of output signal in economic systems under stochastic resonance conditions

A.G. Isavnin

Stochastic resonance is a phenomenon in which a weak periodic signal, usually in a bistable system, can be amplified by the addition of external noise. Considered initially in the context of ice ages, it has been observerd and theoretically investigated in systems of various types and nature [1]. The origin of stochastic resonance in all these systems lies in the fact that the periodic driving force modulates the probabilities of fluctuational transitions between the coexisting stable states and hence the populations of the states; in turn this gives rise to a comparatively strong modulation of a coordinate of the system with an amplitude proportional to the distance between the stable positions. Since the transitions themselves arise because of noise and the transitions probabilities increase sharply (exponentially, for Gaussian noise) with the noise intensity, the efficiency of the modulation is also sharply increased. The theoretical consideration of stochastic resonance has been carried out, most commonly, for a discrete two-state theory model or, in the case of continuous system, was based on an approximate or numerical solution of the Fokker-Planck equation for a periodically driven system [2]. An alternative approach to stochastic resonance is based on linear response theory. According to this theory, if a system with a coordinate q is driven by a weak force AcosЩt, there arises a small periodic term in the ensemble-averaged value of the coordinate, д<q(t)>, oscillating at the same frequency Щ and with amplitude a proportional to that of the force:

, , (1)

where , . (2)

The quantity ч(Щ) here is the susceptibility of the system. Equation (1) holds for dissipative and fluctuating systems that do not display persistent periodic oscillations in the absence of the force AcosЩt; it is stochastic resonance in bistable systems of this kind (“conventional stochastic resonance”) that is considered in this paper. The function ч(Щ) contains, basically, all information on the response of the system to a weak driving force. It gives both the amplitude of the signal, a, and its phase lag with respect to the force, ?. In turn, the value of 0.25a² gives the intensity (i.e. the area) of the delta-shaped spike in the spectral density of fluctuations (SDF) S(щ) of the system at the frequency Щ of the driving force,

. (3)

The onset of such a spike follows immediately from (1) with account taken of the principle of decay of correlations:

for . (4)

Following [1], the response of the system, in the context of stochastic resonance, is often characterized by the ratio R of the area of the above spike to the value S⁽⁰⁾(Щ) of the SDF at the giving frequency Щ in the absence of periodic driving, i.e., by the signal-to-noise ratio. It is evident from (1)-(3) that R may be expressed in terms of a susceptibility ч(Щ):

, . (5)

Therefore, the evolution of the susceptibility and of S⁽⁰⁾(Щ) with varing noise intensity D show immediately whether or not stochastic resonance (i.e., an increase and subsequent decrease in R with increasing D) is to be expected at a given frequency.

An important advantage of describing stochastic resonance in terms of the susceptibility is that such a description relates stochastic resonance to standard linear-response phenomena (for example, magnetic susceptibility [3]) investigated in physical kinetics. One more advantage is that quite often the systems investigated are in thermal equilibrium or quasiequilibrium. In this case the susceptibility can be expressed immediately in terms of the SDF S⁽⁰⁾(Щ) in the absence of periodic driving via the fluctuation-dissipation relations:

,

, (6)

Where P implies the Cauchy principal part and T is temperature in energy units. It follows from Eqs. (5) and (6) that the onset of stochastic resonance can be predicted from purely experimental data on the evolution of SDF of a system with temperature without assuming anything at all about the equations that describe its dynamics, i.e., for a system treated as a “black box”.

However, it is not only signal-to noise ratio that is important in the context of the influence of noise on the response of the system to a sinusoidal driving force.

The presence or absence of phase shifts in stochastic resonance was a riddle for many years. The first prediction of a phase shift seems to have been due to Gang and Nicolis [4], who concluded that, for an overdamped system fluctuating in a bistable potential,

, (7)

where W⁽⁰⁾ is the sum of the transition rates out of each of the potential wells; similar results were also obtained by McNamara and Wiesenfeld [1]. On the other hand Gammaitoni [5] claimed that analog simulations as well as numerical computations “had ruled out (the phase shifts) as apparently spurious”. Because the onset of the phase shifts follows automatically from the linear response theory approach to stochastic resonance. Gammaitoni assumed this in itself to be a good reason to doubt the applicability of linear response theory to stochastic resonance.

The problem of phase shifts in stochastic resonance is also interesting from the viewpoint of relating stochastic resonance to standard resonance phenomena [6]. It is well known in physics that when the frequency Щ of an external driving force is swept through the resonant frequency of a system the phase lag ? of the signal in the system decreases monotonically from nearly zero for small Щ to nearly -180⁰ for large Щ, passing through approximately -90⁰ at the resonant value of Щ. Conventional stochastic resonance in bistable systems arrises because, with increasing noise intensity, the probabilities of the fluctuational transitions between the stable states become of the same order of magnitude as or larger than the frequency Щ of the driving force, thereby switching on the mechanism of strong response associated with the transitions. So, the physics is different from that in a standard resonance, and the dependence of the phase lag on the noise intensity would not necessarily be expected to be the same as ?(Щ) in a resonating system. Last but not least, the investigation of the phase shifts under stochastic resonance can give an important extra argument in relation to whether or not stochastic resonance can properly be treated as a linear response phenomenon.

It was shown, that phase shifts do indeed accompany stochastic resonance. However, in continuous systems, they take a form completely different from that predicted for two-state systems. One usually treats the simplest nontrivial system: an overdamped Brownian particle moving in a symmetric bistable potential and, in addition, driven periodically,

, , (8)

where ж(t) is zero-mean Gaussian noise of intensity D,

. (9)

In the absence of periodic forcing, the system (8), (9), irrespective of the particular form of the potential U(q), is quasithermal: its distribution over energy U(q) (an overdamped system has potential energy only) is Gibbsian, with temperature

T=D. (10)

Therefore, the fluctuation-dissipation relations (6) hold and, for weak periodic driving force, i.e. for small A in (8) , the susceptibility ч(Щ) can be expressed in terms of spectral density of fluctuations S⁽⁰⁾(щ) for A=0. Explicit expressions for S⁽⁰⁾(щ), ч(Щ) can be obtained analytically for small noise intensities (low temperatures) D<<ДU where, for general double-well potential, ДU is the depth of the shallower potential well, and ДU=1/4 for model (8).

Many economic processes and effects can be treated with the same approaches as in physics [7]. So, it is possible to use the models mentioned above to describe responses of multistable economic systems to a weak periodic influence, for example periodic advertisements, published experts views, weekly or monthly waves in time series of prices, etc. Existing models enable one to get quite good forecasts of the periodic response at appropriate noise level of economic systems.

economic expert periodic price

References

1. McNamara B., Wiesenfeld K. The theory of stochastic resonance // Phys.Rev.A. - 1989 .- V.39. - № 9. - p.4854-4869.

2. Jung P., Hanggi P. Stochastic nonlinear dynamics modulated by external periodic forces // Europhysics Letters. - 1989. - V.8. - № 6. - p.505-510.

3. Sadykov E.K., Isavnin A.G. Intensification of an alternating magnetic field in a system of small magnetic particles // Physics of the Solid State. - 1994. - V. 36. - № 11. - p.1843-1844.

4. Gang H., Nicolis G., Nicolis C. Periodically forced Fokker-Planck equation and stochastic resonance // Phys.Rev.A. - 1990. - V.42. - № 4. - p.2030-2041.

5. Gammaitoni L., Marchesoni F., Menichella-Saetta E., Santucci S. Stochastic resonance in bistable system // Phys.Rev.Lett. - 1989. - V.62. - № 4. - p.349-352.

6. Isavnin A.G. Stochastic resonance in finely dispersed magnetics: a comparison of discrete and continuous models // Russian Physics Journal. - 2002. - V.45. - №11. - p.1110-1114.

7. Romanovsky M.Yu., Romanovsky Yu.M. Introduction to econophysics. Statistical and dynamical models. - 2007. - 280 p.

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