Published in
Rep.
Prog.
Phys.
61, 889–997 (1998).
Analogue studies of nonlinear systems
D G Luchinsky†§,PVEMcClintock† and M I Dykman‡
† Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
‡ Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
Received 23 February 1998
Abstract
The design of analogue electronic experiments to investigate phenomena in nonlinear
dynamics, especially stochastic phenomena, is described in practical terms. The advantages
and disadvantages of this approach, in comparison to more conventional digital methods,
are discussed. It is pointed out that analogue simulation provides a simple, inexpensive,
technique that is easily applied in any laboratory to facilitate the design and implementation
of complicated and expensive experimental projects; and that there are some important
problems for which analogue methods have so far provided the only experimental approach.
Applications to several topical problems are reviewed. Large rare fluctuations are studied
through measurements of the prehistory probability distribution, thereby testing for the first
time some fundamental tenets of fluctuation theory. It has thus been shown for example that,
whereas the fluctuations of equilibrium systems obey time-reversal symmetry, those under
non-equilibrium conditions are temporally asymmetric. Stochastic resonance, in which the
signal-to-noise ratio for a weak periodic signal in a nonlinear system can be enhanced
by added noise, has been widely studied by analogue methods, and the main results
are reviewed; the closely related phenomena of noise-enhanced heterodyning and noise-
induced linearization are also described. Selected examples of the use of analogue methods
for the study of transient phenomena in time-evolving systems are reviewed. Analogue
experiments with quasimonochromatic noise, whose power spectral density is peaked at
some characteristic frequency, have led to the discovery of a range of interesting and often
counter-intuitive effects. These are reviewed and related to large fluctuation phenomena.
Analogue studies of two examples of deterministic nonlinear effects, modulation-induced
negative differential resistance (MINDR) and zero-dispersion nonlinear resonance (ZDNR)
are described. Finally, some speculative remarks about possible future directions and
applications of analogue experiments are discussed.
§ Permanent address: Russian Research Institute for Metrological Service, Ozernaya 46, 119361 Moscow, Russia.
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Contents
Page
1. Introduction 891
2. Technical details 893
2.1. Circuit design 893
2.2. Noise generators 898
2.3. Signal acquisition and analysis 900
3. Large rare fluctuations 901
3.1. The concept of large fluctuations 901
3.2. Prehistory probability distribution 904
3.3. Time asymmetry of classical fluctuations 915
3.4. Status of the experiments on large fluctuations 928
4. Stochastic resonance 929
4.1. What is stochastic resonance? 929
4.2. Stochastic resonance as a linear response phenomenon 931
4.3. Non-conventional forms of stochastic resonance 936
4.4. Noise-enhanced heterodyning 954
4.5. Noise-induced linearization 958
5. Transient effects 961
5.1. Swept-parameter systems 961
5.2. Decay of unstable states and transient multimodality 965
6. Phenomena induced by quasimonochromatic noise 972
6.1. Quasimonochromatic noise and its generation 972
6.2. Transitions in a bistable potential 974
6.3. The stationary distribution 976
6.4. Large fluctuations and observation of a switching point 979
7. Deterministic nonlinear phenomena 981
7.1. Modulation-induced negative differential resistance 982
7.2. Zero-dispersion nonlinear resonance 984
8. Future directions 989
Acknowledgments 990
References 990
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1. Introduction
Analogue simulation provides a method of solving complicated dynamical equations. One
builds an electronic model of the system under study and then investigates its behaviour,
usually under the influence of external forces of some kind, using a digital data processor
to analyse the response. This approach has been found especially useful in relation to
stochastic problems, i.e. those where the system of interest is subject to random fluctuations
(noise), either of external or internal origin. Contrary to a still widespread misapprehension,
analogue simulation does not require large or costly apparatus: an ‘analogue computer’ as
such is no longer needed. Nor is the technique difficult to implement. In this review, we
explain the basis of the method and describe how different variants have been used to address
some selected problems in (mostly) stochastic nonlinear dynamics. First, however, we
discuss in more detail why analogue modelling techniques are valuable, and their advantages
and disadvantages as compared to more commonly used digital methods (Mannella 1997).
A major advantage of analogue simulation is that it readily enables large volumes of
a system’s parameter space to be surveyed quickly for interesting phenomena, often by
turning knobs to adjust the relevant parameters while examining the results on a visual
display; the equivalent procedure with a digital system is often slower and more ponderous.
Usually analogue simulations are used to test theoretical predictions. However, there are
many examples of new physical phenomena and ideas first discovered through analogue
simulations. When noise is added to a strongly nonlinear system, for example, it may
improve not only the transmission of signals through the system, but also the signal-to-
noise ratio measured at the output (Fauve and Heslot 1983)—an observation that in a large
part was responsible for the outburst of work on stochastic resonance (see section 4 later).
Other examples include the idea for the Josephson voltage standard (Kautz 1988, 1996), and
the observations of skewing in the stochastic phase portraits of bistable systems driven by
coloured noise (Moss and McClintock 1985, Moss et al 1986), modulation-induced negative
differential resistance (Dow et al 1987) and noise-induced spectral narrowing (Dykman et
al 1990a, b). In each of these cases, the phenomenon had not been predicted theoretically,
was not anticipated prior to the experiments and appeared unexpectedly in the analogue
model.
In a sense, analogue simulations lie in between real experiments and digital simulations,
and combine certain characteristic features of each. In both forms of simulation one
investigates model systems. However, the techniques for data acquisition and processing
used in analogue simulation are often exactly the same as those used in the corresponding
experiments on real physical systems. This feature can sometimes be extremely useful,
for example in that it enables the acquisition/analysis software to be developed, tested
and implemented in studies of an analogue model prior to implementing it in the more
complicated environment of a real experiment. Noise-enhanced heterodyning in a fluctuating
nonlinear optical element, for example, was first simulated in an idealized electronic model
(Dykman et al 1994a) prior to performing the nonlinear optical experiment itself (Dykman
et al 1995a, b).
There is a long-standing and close relationship between studies of fluctuating nonlinear
systems and experiments on electrical circuits. The connections were studied in
radioengineering over several decades, a typical example being the work on fluctuations
in the Thompson generator (Rytov 1956a, b). In fact, it was experiments of this kind
on electronic circuits that were largely responsible for stimulating the development of the
theory of random processes as a whole, yielding a variety of important results some of which
are now being used in quantum electronics. The relationships between electrical circuits
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and the properties of Langevin and Fokker–Planck equations were discussed by Landauer
(1962) and Stratonovich (1963, 1967). In analogue experiments, the static properties of
the electronic system are known (or are assumed to be known), and the problem is to
investigate how it responds to random or periodic forcings. An example of the revealing
role of such simulations can be seen from the analysis (Smythe et al 1983a, McClintock
and Moss 1985, Moss et al 1985b) of the Ito–Stratonovich dilemma (Van Kampen 1990). It
concerns the description of how a system responds to white noise. Any naturally occurring
noise necessarily has a finite correlation time. This is true for the noise used in analogue
simulations. Noise in Langevin equations can be considered effectively white in the limit
where its correlation time is very small compared to the relaxation times of the system, which
corresponds to the Stratonovich prescription. The physical relevance of this prescription
was clearly demonstrated in the analogue simulations. The authors concluded that the
alternative Ito prescription, much used by theorists at that time, was inapplicable to real
physical systems except perhaps in special cases (e.g. equations with delay). In contrast,
when trying to implement similar studies on a digital computer, with the values of the
noise being uncorrelated at different discretized instants of time, it was quickly realized that
any attempt to resolve the dilemma would be fruitless—because it would be necessary for
the programmer to choose either the Ito or Stratonovich stochastic calculus in writing the
simulation code, thus pre-selecting the answer.
In contrast to digital simulations, truncation errors do not accumulate in analogue
simulations. Analogue simulations are therefore especially valuable for use, for example,
with fast oscillating systems where the integration time (the time over which data are
accumulated and perhaps ensemble-averaged) substantially exceeds the vibration period,
as in problems involving quasimonochromatic noise (Dykman et al 1991b, 1993e, and
section 6 later). Although digital techniques can always in principle be made more accurate
than analogue methods, which typically achieve 2–3% accuracy, the relative simplicity
of analogue simulations and their high speed represent significant advantages, particularly
where qualitative results have to be established—as was the case for the ring-laser gyroscope
equation (Vogel et al 1987a), for example, where it was found that analogue results of
satisfactory statistical quality could be acquired in a matter of 15–30 min, whereas numerical
solution of the Fokker–Planck equation took hours of Cray time.
Finally, experience suggests that digital and analogue methods should be regarded as
complementary techniques for the study of stochastic nonlinear problems. Each has its
own advantages and disadvantages; which of these is emphasized or de-emphasized will
depend on the nature of the system being investigated. As in any experimental study, it is
possible to make mistakes and generate artifacts using either form of simulation. For really
complicated problems it is therefore desirable to use both techniques if possible, with one
acting as a check on the other, to eliminate mistakes or hidden flaws in the algorithm used
for the digital simulation, or misconnections and experimental uncertainties in the analogue
circuit.
In what follows, although we will concentrate mostly on analogue experimental
investigations by the Lancaster group, we would emphasize that the technique has been,
and continues to be, used successfully in many other laboratories. For example, electronic
models have been employed extensively over many years for studies of deterministic
phenomena such as heart rate variability (Van der Pol and Van der Mark 1928), chaos
(Holmes 1979, Linsay 1981, Testa et al 1982, Jeffries and P’erez 1982, Yeh and Kao 1982,
D’Humieres et al 1982, Robinson 1990, King and Gaito 1992, Gomes and King 1992, Heagy
et al 1994), phenomena near period-doubling bifurcations (Jeffries and Wiesenfeld 1985,
Bryant and Wiesenfeld 1986, Vohra and Bucholtz 1993, Vohra et al 1994), and the three-
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photon Josephson junction parametric amplifier (Bryant et al 1987). Modelling techniques
for noise-driven systems were developed by a number of scientists and mathematicians
including, particularly, Morton and Corrsin (1969). The first applications of the technique
in its contemporary form were probably those reported by Sancho et al (1982), Arecchi et
al (1982), Fauve and Heslot (1983) and Smythe et al (1983a, b). Subsequently, stochastic
analogue modelling has been used by numerous research groups including, for example,
Mitschke et al (1985), Fronzoni et al (1985), Gammaitoni et al (1989), Anishchenko et al
(1993) and Carroll and Pecora (1993).
The particular topics chosen as examples for discussion are intended to be representative.
We point out, however, that we have successfully applied the analogue approach to the study
of many other interesting stochastic nonlinear problems, in addition to those mentioned,
including: noise-induced phase transitions (Smythe et al 1983a), stochastic postponements
of critical onsets (Robinson et al 1985); noise-induced changes in the relaxation times of
bistable systems driven by both white (Sancho et al 1985) and coloured (Casademunt et al
1987) parametric (i.e. multiplicative) noise; bistability driven by coloured noise (Hanggi et
al 1985, Fronzoni et al 1986); noise-induced postponements of bifurcations in a ring laser
model (Mannella et al 1987a, c) and in the Brusselator (Fronzoni et al 1987a); the effect of
noise on a Hopf bifurcation (Fronzoni et al 1987b); fluctuation spectra of the double-well
Duffing oscillator (Mannella et al 1987b, Dykman et al 1988); Fokker–Planck descriptions
of coloured noise-driven processes (Grigolini et al 1988); relaxation of fluctuations in the
steady state of the Stratonovich model (Mannella et al 1988); the effect of noise on the
Fr
´
eedericksz transition (Stocks et al 1989b); quantum phenomena, via the Ricatti equation
(Stocks et al 1989a, 1993a); relaxation near a predicted noise-induced transition, falsifying
(Jackson et al 1989) an earlier theory and leading to a deeper understanding encompassing
coloured noise (Mannella et al 1990); supernarrow spectral peaks near a kinetic phase
transition (Dykman et al 1990e); velocity spectra for Brownian motion driven by coloured
noise in a periodic potential (Igarashi et al 1992); and noise-induced chaos in the Lorenz
model (Fedchenia et al 1992). Earlier reviews of the application of analogue techniques to
problems in stochastic nonlinear dynamics include those by Fronzoni (1989), McClintock
and Moss (1989) and Mannella and McClintock (1990).
We start, in section 2, by describing the basis of the technique, with the intention of
providing sufficient detail to enable other scientists to apply it in practice. In sections 3–7 we
discuss the use of analogue simulation to illuminate the understanding of several physical
phenomena selected, in part, to demonstrate the ease with which the technique may be
applied to a wide diversity of challenging problems. Finally, in section 8, we summarize
the present status of the work and offer some speculative comments about future directions.
2. Technical details
The aim is to build an electronic model of the system to be investigated, and then to
study its properties, usually while being driven by randon fluctuations (noise) and/or other
external forces. In this section, we sketch the basic principles of circuit design, discuss
noise generators and outline the steps to be taken in analysing the signal(s) coming from
the circuit model.
2.1. Circuit design
The work-horse of modern analogue circuit design is the operational amplifier. For present
purposes, it enables most of the arithmetic operations needed to model the equations of
