Stochastic resonance and noise-induced phase coherence
Jan A. Freund, Alexander Neiman and Lutz Schimansky-Geier
Abstract. We prove a close relation between stochastic resonance and the phenomenon of noise-induced synchronization. The method proceeds by assigning instantaneous phases to input and output signals and by reformulating the rates of the standard two-state system in terms of the input-output phase difference. We are able to analytically derive criteria for noise-induced frequency and effective phase locking. The implications of both phenomena for stochastic resonance are discussed.
1. Introduction
1.1. Ice ages and the two-state description of stochastic resonance
While today most stochastic resonance research [1] concentrates on applications to information processing systems, e.g. neurons, detector networks, etc., its discovery [2, 3, 4, 5] originally was made in the context of climatology. Time series of the total volume of ice (inferred from oxygen isotope records of deep sea cores) indicated the existence of a dominant periodic component. Related power spectra revealed a pronounced peak corresponding to a period of about 100 000 years. This time nearly coincides with the periodic variations of the eccentricity of the Earth's orbit. Even though these variations alone are too minor to explain such a strong effect on Earth's climate they were nevertheless considered to entrain the otherwise erratic influences. The essential ingredient for the phenomenon termed stochastic resonance (SR) was the bistability of Earth's climate in the context of the planet's coverage with ice. The average albedo, i.e. the fraction of solar radiation which is reflected back into space, and, hence, the global energy balance [6] are correlated with the percentage of ice covering the Earth's surface: For low temperatures extended ice shields effect a high albedo which in turn stabilizes the cold state; vice versa, high temperatures favour water possessing a higher absorption coefficient than ice, thus, stabilizing the warm state. Transitions between these two metastable states are caused by global temperature fluctuations, either melting parts of the ice shield or creating it from water by freezing. Obviously, the symmetry between these spontaneous transitions is broken by the external signal,
310 Jan A. Freund, Alexander Neiman and Lutz Schimansky-Geier
i.e. the periodic variations in the eccentricity of the Earth's orbit. The enormous amplification of such a small signal is due to the fact that transition rates, in the framework of Kramers theory [7), depend on barrier variations exponentially [ef. Equation (1)).
The core of SR is closely linked to the notion of an optimal noise intensity: Below this value sufficiently large fluctuations, necessary to overcome the subthreshold condition, are much too rare, whereas far beyond optimal noise, fluctuations are dominating the dynamics, thus, erasing the tiny signal characteristics. Of course, it is not clear why climate fluctuations should be related to optimal noise intensity; however, physical systems and rudimentary SR models clearly were supporting the intuitive idea of optimal noise.
The paradigm of noise-induced transitions between bistable states agitated by subthreshold external signals was elaborated theoretically in the seminal paper by McNamara and Wiesenfeld in 1989 [8). In the framework of an adiabatic description they mapped a continuous bistable system onto a two-state model with transition rates
with a(D) denoting the Kramers rate for thermally activated transitions over the barrier 6.U, D the noise intensity, A the signal amplitude, and n the signal frequency.
The probabilistic description of this two-state system started from n+ and n_, denoting the probabilities to observe the system at time t in the states +1, i.e. in the right well, or -1, i.e. in the left well, resp. (conditioned by an initial state Xo at time to). The stochastic dynamics, expressed by the master equation
W(t)n_ - W+(t)n+
W_(t) - [W-(t) + W+(t))n+,
had the solution
n+(t) g-l(t) [n+(to)g(to) + 1: w_(t')g(t') dt']
g(t) = exp [It [W+(t') + w_(t')) dt'] .
(2)
(3)
(4)
(5)
Performing the integrals in (4) and (5) required a linearization of the rates (1). From the explicit solution the output autocorrelation function and, consecutively, the power spectrum could be computed. An averaging with respect to the initial phase of the signal was changing the non-stationary character of the results to a
Stochastic resonance and noise-induced phase coherence 311
stationary one. Eventually, the (one-sided) output power spectrum Sew) was of the following form
with N (D, w) being a Lorentzian spectrum centered at w = 0 characterizing the broadband noise part, R(A, D, D) measuring the transfer from noise to the signal power, the latter being quantified by "I, the spectral power amplification (SPA)
(7)
a widely used measure for SR [9]. Another common measure is the signal-to-noise ratio (SNR), defined as
Both the SPA and the SN R exhibit relative maxima for certain values of noise intensity. The two values do not coincide but are of the same order and can be used to define the range of optimal noise intensity. Contrary to standard resonance the two SR measures, SPA and SN R, do not possess relative maxima as functions of the frequency D, a fact which evoked both, criticism and efforts to prove a bona-fide resonance by employing alternative SR measures, e.g. residence time distributions [10].
It should be mentioned that in addition to standard SR, which deals with periodic signals, also aperiodic SR (ASR) has been described successfully [11] employing measures which are tailored to quantify the correlations between input and output signals, e.g. coherence function [12] or information measures [13].
Besides bistable systems the class of threshold systems serves as another useful paradigm for SR, especially in the context of neuronal and excitable dynamics [14]. In fact, the minimal requirement for SR is the existence of a modulated barrier which can only be reached or crossed in the presence of fluctuations.
Recently, a close connection between the phenomenon of SR and synchronization was elaborated. As was demonstrated in experiments with a noisy Schmitt-trigger [15] SR can be understood as a locking of the mean frequency of 8witchings between the bistable states. Later on it was shown that this frequency locking was also accompanied by a locking of the instantaneous input-output phase difference of the overdamped stochastic bistable oscillator [16, 17]. These experimental observation were verified numerically and allowed to interpret SR in the context of forced synchronization. In the present publication we will justify this view by analytic results (see also [18]).
312 Jan A. Freund, Alexander Neiman and Lutz Schimansky-Geier
1.2. Forced synchronization of noisy nonlinear oscillators The notion of effective synchronization and phase locking naturally occurs when considering a noisy nonlinear oscillator forced by an external harmonic signal A cos(Ot). The asymptotic dynamics of the nonlinear oscillator alone gives rise to a limit cycle with natural frequency no. Changing the description of the oscillator state from the phase space coordinates x and x to a time dependent amplitude a(t) and an instantaneous phase <I>(t) [19] also implies equations of motion for the new dynamic variables. For a sufficiently large signal amplitude A the oscillator will effectively synchronize with the external signal, i.e. the phases of the signal <l>s(t) = nt and the oscillator <I>(t) will constantly obey a relation like
n<l>(t) = m<l>s(t) + <1>0 (9)
with a constant phase shift <1>0 and natural numbers nand m which depend on the frequency mismatch ~ = no - n and on the signal amplitude A. A graphical evaluation of this phenomenon in the ~ vs. A plane yields the well known Arnold tongues. Defining the phase difference cp(t) = n<l>(t) - m<l>s(t) - <1>0 allows to reformulate the locked dynamics by equations of motion for cp(t) and a(t).
The influence of noise on the mechanism of phase locking was first considered by Stratonovich [20] (restricting to the case n = m = 1). Under suitable conditions! the stochastic dynamics of cp is described by the following Langevin type Adler equation
rp = ~ - €sincp + ~ (10)
with the nonlinearity parameter (synchronization bandwidth) € and ~ representing white noise. Employing the potential V(cp) = -~. cp - €coscp the noisy dynamics can be identified with the stochastic motion of a mass point along a corrugated inclined plane (cf. Figure 1). The slope is caused by the frequency mismatch ~
-~<P-ECOS<p
----------~~~~--------<p
FIGURE 1 The noisy dynamics of the phase difference r.p can be identified with the stochastic motion of a mass point along a corrugated inclined plane where the slope is given by the frequency mismatch Ll. and the undulation amplitude by the synchronization bandwidth E
IThe correlation time of noise must be small compared to the relaxation time of the oscillator and the noise intensity has to be sufficiently small [20].
Stochastic resonance and noise-induced phase coherence 313
whereas the nonlinearity parameter E corresponds to the undulation amplitude. A necessary condition for locking is ~ < E: In terms of the tilted washboard potential (cf. Figure 1) this simply means that the tilt must be less than the undulation amplitude to guarantee a well structure. This requirement motivates the interpretation of E as the synchronization bandwidth. Noise agitates the phase difference, thus, moving it around inside one of the wells; now and then a sufficiently large fluctuation will cause a jump to a neighbouring well: a phase slip occurs which terminates the preceding locking episode. Synchronization is not perfect but only effective.
1.3. Instantaneous phase
In this section we discuss various methods of assigning an instantaneous phase 4.>(t) to a real signal x(t).
1.3.1. ANALYTIC SIGNAL AND HILBERT PHASE A rather general way to define an instantaneous phase employs the concept of the analytic signal [21]. The idea is to extend the real signal x(t) to an analytic signal z(t) by adding a related imaginary part y(t) which is linked to x(t) by some transformation, i.e. y = H 0 x
z(t) = x(t) + iy(t) = a(t) ei<p(t). (11)
As indicated by Equation (11) the analytic signal z(t) equivalently can be expressed through a time dependent amplitude (envelope signal) a(t) and an instantaneous phase 4.>(t) implying the following relations
a(t) = ";x2(t) + y2(t) and [y(t) ] tan 4.>(t) = x(t) . (12)
So far we have not specified the transformation H which allows to compute y(t) from a given x(t). Using the ansatz [22]
00
y(t) = J x(u) K(t - u) du (13) -00
the convolution theorem of Fourier transforms .1'[.] states that
F[y(t)] = F[x(t)] . F[K(t)]. (14)
The kernel K (t) is now specified by the demand that the concept should identically render the phase of a purely harmonic signal, i.e. for x(t) = acos(wt). From Equation (12) we immediately see that y(t) = asin(wt). As becomes clear from Figure 2 the effect of the transform H has to be a pure phase shift of ± 7r /2 depending on the sign of the frequency sgn(w).
314 Jan A. Freund, Alexander Neiman and Lutz Schimansky-Geier
1
o
-1
-2 -1 o tlx
1 2
FIGURE 2 The tmnsformation H[cos(wt)] = sin(wt) corresponds to a pure phase shift of ± 7r /2 depending on the sign of the frequency: w = - 2~ (left) and w = 2~ (right)
This implies that F[K(t)) = -sgn(w)i and, by applying the inverse Fourier transform, that K(t) = ';t. Hence, we see that the Hilbert transform2
00
y(t) = J x(u) du 7r(t - u)
-00
(15)
is most appropriate for phase descriptions of harmonic signals. Nonetheless, it was also used for describing the phase synchronization of chaotic oscillators [23).
In the following we will restrict ourselves to dichotomic signals, i.e. a discontinuous flipping between two values + 1 and -1. As an example we consider a dichotomic Markovian process (DMP) - also named random telegraph signal -which can be regarded as the result of a Poissonian jump process. A typical realization is shown in the top row of Figure 3 together with its Hilbert transform (second row), its related time dependent amplitude (third row) and instantaneous phase (bottom row). We can see that the Hilbert transform and the time dependent amplitude diverge at the jump times whereas the evolution of the Hilbert phase (plotted in units of 7r) is smoothly growing with time.
1.3.2. ALTERNATIVE PHASE DEFINITIONS For dichotomic signals there exist various definitions of an instantaneous phase: For t with tk :::; t :::; tk+1:
t - tk • Linear interpolation between jumps: 4>linear(t) = k7r + 7r
tk+l - tk • Discrete jumps: 4>discrete(t) = k7r
In Figure 4 we show all three phases for the DMP of Figure 3 (top row). All three definitions capture the qualitative growth feature of the phase with elapsing
2The integral is meant in the sense of the Cauchy principal value.
Stochastic resonance and noise-induced phase coherence 315
1 x
-1
Y 0
A 1
<I> 10 5 0 -30 -20 -10 0 10 20 30
t FIGURE 3 A typical mndom telegmph signal (top row) together with its Hilbert tmnsform (second row) and the related time dependent amplitude (third row) and instantaneous phase (bottom row)
10
8
6
4
2
o -30 -10 10 30
t
FIGURE 4 All three phase definitions (see text) capture the basic growth feature
316 Jan A. Freund, Alexander Neiman and Lutz Schimansky-Geier
time. For its simplicity and since x(t) = exp[i<Pdiscrete(t)] is an exact description of the DMP we choose this phase description for the remainder of this publication.
2. Stochastic resonance in the framework of synchronization phenomena
2.1. Stochastic resonance and phase descriptions
The connection between SR and the phase representation of dichotomic signals is illustrated in Figure 5. Note that the phase difference plot for optimal noise (middle box of bottom line) exhibits frequency locking - through vanishing average slope -as well as effective phase synchronization - through comparatively long phase locking episodes.
Interestingly enough, the time dependent rates (1) of the two-state model treated by McNamara and Wiesenfeld [8] can be recast to show an explicit dependence on the phase difference t.p(t) between the dichotomic output phase <Pout(t) = k(t)rr and the input phase <Pin(t) = nt. Applying a trigonometric theorem we find
From this point on we will consider dichotomic signals for the input as well as for the output. This restriction defines a 2x2-state system [17] sketched schematically in Figure 6. Thus, we are left with two noise dependent rates
corresponding to the in-synchrony and out-of-synchrony transitions due to output switching events resp. The rate "f (for the DMP) accounts for transitions caused by switchings of the input.
2.3. Stochastic phase dynamics
As the stochastic evolution of the phase difference in the 2x2-state system corresponds to a birth-death process we briefly denote Pk(t) "the probability to observe a phase difference t.p = krr at time t (conditioned by t.po = 0 at time to = 0)" and,
Stochastic resonance and noise-induced phase coherence
Inpu t Signal (Dichotomic Markovian Prace 's) & Phase cPin
FIGURE 5 The connection between stochastic resonance and the phase representation of dichotomic input and output signals: suboptimal (left), optimal (center) and superoptimal noise intensity (right). Note that the phase difference plot for optimal noise (middle box of bottom line) exhibits frequency locking - through vanishing average slope - as well as effective phase synchronization - through comparatively long phase locking episodes
correspondingly, gk = gk1r [cf. Equation (18)]. Hence, the stochastic dynamics is governed by the following master equation
(19)
318 Jan A. Freund, Alexander Neiman and Lutz Schimansky-Geier
-1,-1 a, +1 ,-1
• • • • a2
y y y y
a2
• • • • - 1,+1 a, +1,+1
FIGURE 6 The restriction to dichotomic input and output variables defines the 2x2-state system; the noise dependent rates Ul and U2 relate to the in-synchrony and out-oj-synchrony transitions resp. due to output switching events. The rate 'Y (Jor the D M P) corresponds to transitions due to switchings oj the input signal
with the input part
W DMP ±
""n=~-oo J: (t _ n1f: '19) " ~ U H (non-stationary)
n (stationary)
(20)
(21)
(22)
Here, WfPP (t,'I9) (21) is the non-stationary rate related to a dichotomic periodic process (DPP) and '19 denotes the initial phase with reference to the signal. Averaging over this initial phase, i.e. considering an equidistributed ensemble, eventually yields a stationary process (22).
2.4. Noise-induced frequency locking
From the master Equation (19) we can compute the mean frequency (r.p) and find
(r.p) -(Win) + (wout) 1f 1f
-(Win) + "2(a1 + a2) - "2(a2 - al)(cos<P)
(23)
(24)
with (Win) = "I1f for the DMP and ((Win)) = n for the DPP (the double brackets for the DPP indicate the initial phase average).
Equation (24) is in close analogy to the averaged Adler Equation (10) . By comparison we see that the frequency mismatch ~ corresponds to 1f al !a2 whereas the difference 1f a2;al is the analogue of the nonlinearity parameter e. Note that
Stochastic resonance and noise-induced phase coherence 319
now both parameters are noise dependent which explains the fact that frequency locking is noise-induced. Moreover, we stress the fact that the dependence of Eon the ratio ~ is nonlinear, in contrast to many linear response descriptions of SR [1].
The term (cos cp), in general, is time dependent and its evolution is determined by the initial preparation, e.g. the choice CPo = cp(to) = 0 means (coscp}(to) = 1, and by the dynamics. However, the asymptotic value is unique and can be derived as the stationary value of the related master equation
(25)
As was shown in [17] this value equals the stationary correlation coefficient of the input and output of the 2x2-state model.
In Figure 7 we show the mean output frequency (wout) (in units of 71") as a function of noise intensity D for the DMP with 'Y = 0.001 and for three amplitudes A = 0 (vanishingly small), 0.1,0.2. Note that the case of A = 0 yields the well known Kramers rate.
0.003
A
0.002
.!;
.A
88 V
0.001
0.2
0.1
0.0 0.00 0.04 0.08
0.04 D
0.06
, , , , , , ,
, , , ,
, , , , ,
0.08
FIGURE 7 The mean frequency of the output [cf. Equations (23) and (24)] in units of 11' for the DMP with 'Y = 0.001 and for the DPP with 0./11' = 0.001 for three signal amplitudes: A = 0, i.e. vanishingly small, (solid), A = 0.1 (dotted) and A = 0.2 (dashed). For sufficiently large amplitudes (with 0<0 = 1 and f1U = 0.25) a plateau around'Y = 0./11' is forming which - in connection with a suitable definition - gives rise to the tongue like structures shown in the inset
For sufficiently large3 signal amplitude A we see the formation of a plateau around the frequency of the input signal. The range of noise intensities yielding
3In order to avoid switchings in the absence of noise we have to obey A < f1U = 0.25.
320 Jan A. Freund, Alexander Neiman and Lutz Schimansky-Geier
values belonging to this plateau - a criterion which we pinpoint by the demand of a tiny slope, - in dependence on the signal amplitude A shapes the tongue like structures shown in the inset of Figure 7. These results identically apply to the DPP when setting n/7r = "t.
2.5. Noise-induced phase synchronization
As we have argued in Section 2.1 effective phase synchronization reveals itself by the extended duration of locking episodes. Defining a phase diffusion coefficient by virtue of
(26)
we can determine the average duration of a locking episode (1Jock) from the ansatz
In principle the computation of the diffusion coefficient is straightforward, however, for the DPP a mathematical subtlety occurs. This is caused by the fact that, while the mean of functions of cp for the DMP are smooth in time, related averages in the case of the DPP are still discontinuous across jumping times tk' For this reason derivatives with respect to time are at variance with the standard product rule of differentiation
d dt [(f(cp» . (g(cp») [
/)./ /).g /)./ /).g] lim _.g+/.-+--.Il t--+O /). t /).t /). t
Stochastic resonance and noise-induced phase coherence 321
The resulting expressions for the diffusion coefficient read
Its general structure is V = Vin + V out - Vcoh where only the last coherence term scales with (a*)k (k = 1,2,3) and which alone can reduce the effective phase difference diffusion for optimal noise intensity.
In Figure 8 we plot the dependence of 7iock on the noise intensity D for the DMP (left) and the DPP (right).
0,02 0.04 D
0.06 0.08 0.02 0.04 D
0.06
FIGURE 8 The average duration of locking episodes [cf. Equation (28)) for the DMP (left) and the DPP (right) both for signal amplitudes: A = 0, i.e. vanishingly small, (solid), A = 0.1 (dotted) and A = 0.2 (dashed). Pronounced maxima - note the logarithmic scale - occur around noise intensities which coincide with the frequency locking condition (cf. Figure 7). For the DPP we find a maximum even for A -> 0+ around the noise intensity at which the Kramers rate coincides with the signal frequency because the phase diffusion coefficient increases rather slowly
0.08
In both plots we find huge maxima (note the logarithmic scale) around noise intensities where also frequency locking occurs indicating the region of effective
322 Jan A. Freund, Alexander Neiman and Lutz Schimansky-Geier
phase synchronization. Let us note that the onset of phase synchronization is triggered by a sharp increase of (0-*). However, even beyond the region of effective phase synchronization the correlator maintains a large value, whence, it is not an equivalent measure.
3. Conclusions
We have argued that SR and phase synchronization can be closely linked by an appropriate assignment of instantaneous phases to input and output signals. The rates of a two-state system could be recast to explicitly depend on the phase difference between input and output signal. Restricting to twice dichotomic signals we were able to derive criteria for effective frequency and phase synchronization analytically. Both phenomena are noise-induced because frequency mismatch and synchronization bandwidth can be tuned by variation of noise intensity. Our description offers a scheme for both SR and ASR. In contrast to many linearized descriptions of SR we could proceed with a nonlinear driving force. The phenomenon of phase locking is a much stronger effect than usual SR because it goes beyond spectral properties by considering the mean duration of locking episodes.
Acknowledgement
A.N. acknowledges support from the Fetzer Institute.
References [1] L. Gammaitoni, P. Hanggi, P. Jung and F. Marchesoni, Stochastic resonance. Rev.
Mod. Phys., 70(1998), 223-288. [2] C. Nicolis, Solar variability and stochastic effects on climate. Solar Physics, 74(1981),
473-478. [3] R. Benzi, A. Sutera and A. Vulpiani, The mechanism of Stochastic Resonance.
J. Phys. A: Math. Gen., 14(1981), L453-L457. [4] C. Nicolis, Stochastic aspects of climatic transitions - response to a periodic forcing.
Tellus, 34(1982), 1-9. [5] R. Benzi, G. Parisi, A. Sutera and A. Vulpiani, Stochastic resonance in climatic
change. Tellus, 34(1982), 10-18. [6] C. Nicolis, Long term climatic transitions and stochastic resonance. J. Stat. Phys.,
70(1993), 3-14. [7] P. Hanggi, P. Talkner and M. Borkovec, Reaction-rate theory: fifty years after
Kramers. Rev. Mod. Phys., 62(1990), 251-341. [8] B. McNamara and K. Wiesenfeld, Theory of stochastic resonance. Phys. Rev. A,
39(1989), 4854-4869. [9] P. Jung and P. Hanggi, Stochastic Nonlinear Dynamics Modulated by External Perio
dic Forces. Europhys. Lett., 8(1989), 505-510; P. Jung and P. Hanggi, Amplification of Small Signals via Stochastic Resonance. Phys. Rev. A, 44(1991), 8032-8042.
[10] L. Gammaitoni, F. Marchesoni and S. Santucci, Stochastic Resonance as a Bona Fide resonance. Phys. Rev. Lett., 74(1995), 1052-1056.
Stochastic resonance and noise-induced phase coherence 323
[11] A. Neiman and L. Schimansky-Geier, Stochastic Resonance in Bistable Systems Driven by Harmonic Noise. Phys. Rev. Lett., 72(1994), 2988-2991; J.J. Collins, C.C. Chow and T.T. Imhoff, Aperiodic stochastic resonance in excitable systems. Phys. Rev. E, 52(1995), R3321-R3324.
[12] A. Neiman and L. Schimansky-Geier, Stochastic resonance in two coupled bistable systems. Phys. Lett. A, 197(1995), 379-386.
[13] A. Neiman, B. Shulgin, V.S. Anishchenko, W. Ebeling, L. Schimansky-Geier and J. Freund, Dynamic Entropies Applied to Stochastic Resonance. Phys. Rev. Lett., 76(1996), 4299-4302; L. Schimansky-Geier, J. Freund, A. Neiman and B. Shulgin, Noise Induced Order: Stochastic Resonance. Intern. Journal of Bifurc. and Chaos, 8(1998), 869-879.
[14] A. Longtin, Stochastic resonance in neuron models. J. Stat. Phys., 70(1993) 309-327. [15] B. Shulgin, A. Neiman and V. Anishchenko, Mean switching frequency locking in
stochastic bistable systems driven by a periodic force. Phys. Rev. Lett., 75(1995), 4157-4160.
[16] A. Neiman, A. Silchenko, V. Anishchenko and L. Schimansky-Geier, Stochastic resonance: Noise enhanced phase coherence. Phys. Rev. E, 58(1998), 7118-7125; V.S. Anishchenko, A.B. Neiman, F. Moss and L. Schimansky-Geier, Stochastic resonance as noise induced phenomenon increasing order. Physics-Uspekhi, 42(1999), 7-36.
[17] A. Neiman, L. Schimansky-Geier, F. Moss, B. Shulgin and J.J. Collins, Synchronization in noisy systems by stochastic signals. Phys. Rev. E, 60(1999), 284-292.
[18] J.A. Freund, A.B. Neiman and L. Schimansky-Geier, Analytic description of noise-induced phase synchronization. Europhys. Lett., 50(2000), 8-14; J.A. Freund, A. Neiman and L. Schimansky-Geier, Stochastic resonance and noise-induced synchronization, in D.S. Broomhead, E.A. Luchinskaya, P.V.E. McClintock and T. Mullin, eds., Stochaos: Stochastic and Chaotic Dynamics in the Lakes. American Institute of Physics, Melville, NY, 2000, 422-427.
[19] N.N. Bogolyubov and Yu.A. Mitropolski, Asymptotic methods in the theory of nonlinear oscillators, 4th ed.(in Russian), 1974, Nauka, Moscow.
[20] R.L. Stratonovich, Topics in the Theory of Random Noise, Vol. 2(1967), Gordon and Breach, New York.
[21] D. Gabor, Theory of communication. J. lEE London, 93(1946),429-457. [22] L.A. Vainstein and D.E. Vakman, Frequency analysis in the theory of oscillations
and waves (in Russian), 1983, Nauka, Moscow. [23] M.G. Rosenblum, A.S. Pikovsky and J. Kurths, Phase synchronization of chaotic
Jan A. Freund and Lutz Schimansky-Geier and Lutz Schimansky-Geier Institute of Physics Humboldt-University at Berlin Invalidenstr. 110 10115 Berlin Germany E-mail: [email protected]
Alexander Neiman Center for Neurodynamics University of Missouri at St. Louis 8001 Natural Bridge Road St. Louis, MO 63121 USA
