- Email: [email protected]
Synchronization effects in networks of stochastic bistable oscillators
Mathematics and Computers in Simulation 58 (2002) 469–476
Synchronization effects in networks of stochastic bistable oscillators V.S. Anishchenko, O.V. Sosnovtseva∗ , A.S. Kopejkin, D.D. Matujshkin, A.V. Klimshin Department of Physics, Saratov State University, Astrakhanskaya Str. 83, Saratov 410026, Russia
Abstract We study the collective dynamics of one- and two-dimensional lattices of coupled stochastic, non-homogeneous oscillators in terms of synchronization. This phenomenon manifests itself as an entrainment of the mean switching frequencies in the form of frequency-locking and frozen synchronized states and as in- or anti-phase hopping dynamics. We analyze the conditions for the onset and existence of these different behavioral regimes. © 2002 Published by Elsevier Science B.V. on behalf of IMACS. Keywords: Synchronization; Bistable systems; Noise
1. Introduction Extended systems serve as a basis for the study and modeling of spatiotemporal behavior in fluid dynamics, chemical reactions, biological systems and so forth. They allow for a wide range of interesting phenomena that cannot occur in dynamical systems with only a few degrees of freedom. One of the most popular models in this area is an ensemble of coupled non-linear oscillators [1]. Arrays of Josephson junctions [2], coupled solid state lasers [3], complex networks of cells and neurons [4] can serve as examples from this large field. When such systems interact, their cooperative behavior can reveal a variety of patterns of which some of the most prominent are clustering, synchronization, and the formation of coherent structures. From an applications point of view, this build-up approach may also be used to create novel systems whose behavior is more flexible and richer than that of the functional units but whose analysis and control remain tractable. Both temporal [5] and spatiotemporal [6] systems can be influenced in a constructive way by the action of external fluctuations. As a classical example of self-organization in systems of non-linear self-sustained oscillators, synchronization in the presence of noise was studied in detail by Stratonovich [7]. The effect of noise on the synchronization of self-sustained oscillators is destructive: increasing noise intensity leads to the loss of phase coherence, i.e. phase slips become more frequent, and the widths of ∗
Corresponding author. E-mail address: [email protected] (O.V. Sosnovtseva). 0378-4754/02/$ – see front matter © 2002 Published by Elsevier Science B.V. on behalf of IMACS. PII: S 0 3 7 8 - 4 7 5 4 ( 0 1 ) 0 0 3 8 4 - 6
470
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
the Arnold tongues shrink. The classical concept of synchronization has subsequently been generalized to a wide class of two-state systems whose time scales are random values [8–11]. The effect of locking of the mean switching frequencies was discovered in noisy bistable systems driven by a strong-amplitude periodic forcing [9]. It was shown that there are regions on the “noise intensity-forcing amplitude” parameter plane where the mean switching frequency remains constant and coincides with the frequency of periodic forcing. The effect of stochastic synchronization of two-coupled bistable systems driven by independent noise sources was discovered by Neiman [8] and it was shown that a bifurcation of the two-dimensional stationary probability density takes place when the coupling is increased. Kramer’s rates for the subsystems are drawn closer to one another with increasing coupling and coincide at the bifurcation. Similar effects were investigated in two-state deterministic systems (Lorenz system [11] and Chua’s circuit [12]) where switchings are caused by the natural chaotic dynamics and may be characterized by the mean frequency switchings as well. The growing interest in synchronization of noisy systems has led researchers to consider the phenomenon of synchronization in large arrays and networks of coupled oscillators. Lindner et al. [13] showed that an array of stochastic bistable elements, subjected to the same periodic signal when appropriately coupled can display global synchronization effects leading to an enhancement of the stochastic resonance. Recently, it has been found that noise improves the signal propagation in extended bistable systems [14,15]. The response of one- and two-dimensional excitable media to external spatiotemporal noise in terms of synchronization was studied by Neiman et al. [16]. The above studies have dealt with the influence of noise on the stochastic resonance characteristics, and the results do not provide information about coherent behavior of switching events. In this paper, we shall dwell on the ordering role of spatiotemporal noise in an important class of extended bistable systems. In particular, we shall look for answers to the following questions. How do the stochastic bistable systems adjust their switching events in accordance with one another? How will the non-uniformities of parameters and different types of coupling influence the synchronization effects?
2. Model With our focus being on synchronization effects of stochastic oscillations, we chose the Schmitt trigger as one of the simplest threshold system [17,18], possessing a static hysteretic non-linearity. As a main advantage compared to the continuous bistable systems, the output of a Schmitt trigger is entirely controlled by the switching mechanism, whereas in a continuous bistable system interwell and intrawell dynamics may be hard to unravel. A Schmitt trigger can be constructed by means of an operational amplifier (Fig. 1a). In the absence of any forcing, it is characterized by two stable states. The input signal is a random function of time Vin (t) while there are only two possible states ±Vout at the output. The system is characterized by a threshold value for the voltage V th = R2 V0 /(R1 + R2 ). If |V in (t)| ≥ V th , it is bistable. Switching events occur as soon as the input voltage reaches the threshold values. In numerical simulations, the ideal Schmitt trigger obeys the equation x(t + t) = sng(Kx(t) − ξ(t)).
(1)
Here, the function sng denotes sign, and K corresponds to the threshold level of the trigger. ξ (t) is exponentially correlated Gaussian noise for which we shall assume a correlation time τ = 0.01 and an
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
471
Fig. 1. (a) The output signal is generated in Vout when Vin exceeds the threshold level, (b) the electrical scheme of the coupled Schmitt triggers.
intensity D is governed by the Ornstein–Uhlenbeck process 2 1 ˙ξ = ξ + D ω(t), ω(t)ω(t + s) = δ(s). τ τ
(2)
A one-dimensional array of such ideal two-state systems demonstrating pure hopping dynamics is given by xj (t + t) = sgn(Kxj (t) − γ Gj (xj −1 , xj , xj +1 ) − ξj (t)),
j = 1, 2, . . . , m
(3)
with γ being the coupling parameter. The noise sources ξ j (t) are assumed to be uncorrelated with D = 0.4. Throughout this work, the local coupling Gj is taken to be either two-way Gj = xj +1 + xj −1 or one-directional Gj = xj −1 . The boundary conditions in Eq. (3) are free-end, and the initial conditions are distributed non-uniformly in two states. Fig. 1b illustrates an implementation of two-coupled Schmitt triggers. Two-way coupling provides interaction of each functional unit with the previous and next elements. Since the input signal from the previous functional unit is applied to inverting input (−) of operational amplifier, and Vout is an anti-phase, neighboring units operate in anti-phase for γ > 0. For γ < 0, on the other hand, all functional units tend to be at the same state. At the output of the bistable system, we have a dichotomous stochastic process that can be characterized by the mean durations of the upper state Tu and lower state Tl . The mean “period” of switching is therefore, T s = T u + T l . This is related to the mean switching frequency f = 1/T s . Stochastic bistable systems
472
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
have a characteristic time scale which corresponds to the mean frequency of the transition from one potential well to the other. This averaged switching rate, referred to as Kramer’s rate, is fully controlled by noise and is characterized by the exponential Arrhenius law [19] −V f ∝ exp (4) D with V being the height of the potential barrier. Hence, a mismatch between the mean switching frequencies along an array is provided by the random distribution of the threshold parameter K over the interval [1.0, 1.1]. We measured the mean switching frequency between two states in each system as described in our previous publication [9] M
1 1 fj = lim , M→∞ M t − tk k=1 k+1
(5)
where tk is the moment of crossing of the potential barrier and tk+1 − tk is the residence time between two switchings.
3. Spatiotemporal order According to recent results [8–10], synchronization phenomena can take place for interacting stochastic systems whose mean switching frequency determines a clearly distinguishable time scale. Synchronization is defined as the coincidence of the identified time scales. This approach offers a glimpse of the richness of behaviors that are possible in large arrays of coupled noisy oscillators. For two-way interaction, an array of oscillators with different parameters and, as a consequence, varying time scales is interesting both conceptually and because it is encountered in practice in various situations. When a mismatch between the Kramer’s frequencies is introduced, increasing of the coupling causes a reduction of the individual switching frequencies that tend to a vanishing value (Fig. 2a). Moreover, an adjustment of the individual frequencies fj takes place and is considered the effect of stochastic synchronization. However, with one-way coupling (i.e. the dynamics of each element does not depend on the evolution processes in the next units but it is affected by the signal only from the previous oscillators), individual frequencies do not tend to a vanishing value, but approach the switching frequency of the first oscillator (Fig. 2b). Note, that the effect of frequency-locking is observed within some range of coupling parameter. At the critical value, the coherence dynamics of the hopping events breaks down. A retarding effect of noise-induced switching events appears to be general for bistable systems with two-way coupling. This effect seems to be analogous to the effect observed in systems of coupled self-sustained oscillators and referred to as a “frozen state” [20]. In the deterministic case, due to the mutual interaction there may be dramatic shifts of the synchronization frequency. For this reason, the frozen state denotes a stable synchronized state with vanishing (or at least small) synchronization frequency. In our case, for strong interaction, the switching frequencies vanish and retarding of hopping events take place. The above effect of retarding of stochastic oscillations leading finally to their “death” plays an ordering role on the array dynamics. Let us consider how the phase coherence behavior manifests itself at such a transition. In the deterministic case, the appearance of an in-phase regime where the phase difference between any two
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
473
Fig. 2. Individual mean switching frequency for the system (2) with j = 1, 20, 40, 60, 80, 100 as the function of (a) two-way, and (b) one-way coupling. The effects of frequency-locking are clearly distinguishable in both cases.
oscillators vanishes (modulo 2π) corresponds to the strongest form of synchronization, referred to as full synchronization [21,22]. For an array of bistable systems driven by uncorrelated random forcing, full synchronization is not achieved. Fig. 3 illustrates the spatiotemporal evolution of an array of m = 100 oscillators. An oscillator is indicated as white if it is in the left well and as black if it is in the right well.
Fig. 3. Spatiotemporal dynamics of an array of 100 Schmitt triggers (2) for two-way coupling with (a) γ = 0.02, and (b) γ = 0.4.
474
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
Fig. 4. Entrainment of the mean switching frequencies averaged over elements of a two-dimensional lattice at (a) γ = 0.015, and (b) γ = 0.2.
Fig. 5. Spatiotemporal evolution of a two-dimensional lattice of 50 × 50 Schmitt triggers at (a) γ = 0.0, (b) γ = 0.015, (c) γ = 0.2, and (d) γ = −0.2.
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
475
Originally, all oscillators were started with a random spread in their potential wells. For weak coupling, the oscillators can be in each potential well with equal probability as reflected by the equal amounts of black and white in Fig. 3a. With increasing coupling, alternating of colored regions is observed (Fig. 3b). This corresponds to a self-organization of the stochastic oscillators. With increasing system size, qualitatively the same behavior in the sense of the averaged characteristics is observed. Fig. 4 shows how the mean switching frequencies are decreased to a constant value demonstrating retarding effects and frequency-locking when the coupling strength becomes strong enough. Being randomly distributed in system states (Fig. 5a), the stochastic oscillators remain non-synchronous for weak interaction (Fig. 5b). With increasing coupling strength they finally form well-pronounced anti-phase (Fig. 5c) or in-phase (Fig. 5d) coherence pattern.
4. Summary We have demonstrated the spatiotemporal stochastic synchronization in one- and two-dimensional media modeled by coupled bistable functional units driven by external noise. The synchronization effects manifest themselves in two ways. The first manifestation consists in the formation of synchronized frequency-locked states with the same average frequency or frozen states with vanishing synchronization frequency. In the latter case, a retardation of the switching dynamics takes place. The second manifestation produces in- or anti-phase hopping dynamics with well-defined phase coherence structures. Our results indicate that noise can play an important role in the self-organization of coupled stochastic elements. The correspondence to well-known classical effects of synchronization emphasizes the universal aspects of these phenomena.
Acknowledgements This paper is based upon work supported by the Naval Research Laboratory under Contract no N68171-00-M-5430. A.K. acknowledges Award no. REC-006 of the US Civilian Research & Development Foundation for the Independent States of the Former Soviet Union (CRDF). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
V. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, Berlin, 1984. S. Watanabe, S.H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica 74D (1994) 197. R. Roy, K.S. Thornburg Jr., Experimental synchronization of chaotic lasers, Phys. Rev. Lett. 72 (1994) 2009. A.T. Winfree, The Geometry of Biological Time, Springer, Berlin, 1980. W. Horsthemke, R. Lefever, Noise-Induced Transitions, Springer, Berlin, 1984. J. Garcia-Ojalvo, J.M. Sancho, Noise in Spatially Extended Systems, Springer, New York, 2000. R.L. Stratonovich, Topics in the Theory of Random Noise, Gordon and Breach, New York, 1967. A.B. Neiman, Synchronization like phenomena in coupled stochastic bistable systems, Phys. Rev. E 49 (1994) 3484. B.V. Shulgin, A.B. Neiman, V.S. Anishchenko, Mean switching frequency-locking in stochastic bistable system driven by periodic force, Phys. Rev. Lett. 75 (1995) 4157. [10] V.S. Anishchenko, A.B. Neiman, F. Moss, L. Schimansky-Geier, Stochastic resonance: noise-enhanced order, Physics-Uspekhi 42 (1999) 7.
476
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
[11] V.S. Anishehenko, A.N. Silchenko, I.A. Khovanov, Synchronization of switching processes in coupled Lorenz systems, Phys. Rev. E 57 (1997) 316. [12] V.S. Anishchenko, A.B. Neiman, M.A. Safonova, Stochastic resonance in chaotic systems, J. Stat. Phys. 70 (1993) 183. [13] J.F. Lindner, B.K. Meadows, W.L. Ditto, M.E. Inchiosa, A.R. Bulsara, Array enhanced stochastic resonance and spatiotemporal synchronization, Phys. Rev. Lett. 75 (1995) 3. [14] Y. Zhang, G. Hu, L. Gammaitoni, Signal transmission in one-way coupled bistable systems: noise effects, Phys. Rev. E 58 (1998) 2952. [15] J.F. Lindner, S. Chandramouli, A.R. Bulsara, M. Löcher, W.L. Ditto, Noise-enhanced propagation, Phys. Rev. Lett. 81 (1998) 5048. [16] A. Neiman, L. Schimansky-Geier, A. Cornell-Bell, F. Moss, Noise-enhanced phase synchronization in excitable media, Phys. Rev. Lett. 83 (1999) 4896. [17] S. Fauve, F. Heslot, Stochastic resonance in a bistable system, Phys. Lett. A 97 (1983) 5. [18] B. McNamara, K. Wiesenfeld, Theory of stochastic resonance, Phys. Rev. A 39 (1989) 4854. [19] H.A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (1940) 284. [20] P. Tass, Phase and frequency shifts in a population of phase oscillators, Phys. Rev. E 56 (1997) 2043. [21] H. Fujisaka, T. Yamada, Stability theory of synchronized motions in coupled oscillatory systems, Prog. Theor. Phys. 69 (1983) 32. [22] A.S. Pikovsky, On the interaction of strange attractors, Z. Physik B 55 (1984) 149.
Synchronization effects in networks of stochastic bistable oscillators V.S. Anishchenko, O.V. Sosnovtseva∗ , A.S. Kopejkin, D.D. Matujshkin, A.V. Klimshin Department of Physics, Saratov State University, Astrakhanskaya Str. 83, Saratov 410026, Russia
Abstract We study the collective dynamics of one- and two-dimensional lattices of coupled stochastic, non-homogeneous oscillators in terms of synchronization. This phenomenon manifests itself as an entrainment of the mean switching frequencies in the form of frequency-locking and frozen synchronized states and as in- or anti-phase hopping dynamics. We analyze the conditions for the onset and existence of these different behavioral regimes. © 2002 Published by Elsevier Science B.V. on behalf of IMACS. Keywords: Synchronization; Bistable systems; Noise
1. Introduction Extended systems serve as a basis for the study and modeling of spatiotemporal behavior in fluid dynamics, chemical reactions, biological systems and so forth. They allow for a wide range of interesting phenomena that cannot occur in dynamical systems with only a few degrees of freedom. One of the most popular models in this area is an ensemble of coupled non-linear oscillators [1]. Arrays of Josephson junctions [2], coupled solid state lasers [3], complex networks of cells and neurons [4] can serve as examples from this large field. When such systems interact, their cooperative behavior can reveal a variety of patterns of which some of the most prominent are clustering, synchronization, and the formation of coherent structures. From an applications point of view, this build-up approach may also be used to create novel systems whose behavior is more flexible and richer than that of the functional units but whose analysis and control remain tractable. Both temporal [5] and spatiotemporal [6] systems can be influenced in a constructive way by the action of external fluctuations. As a classical example of self-organization in systems of non-linear self-sustained oscillators, synchronization in the presence of noise was studied in detail by Stratonovich [7]. The effect of noise on the synchronization of self-sustained oscillators is destructive: increasing noise intensity leads to the loss of phase coherence, i.e. phase slips become more frequent, and the widths of ∗
Corresponding author. E-mail address: [email protected] (O.V. Sosnovtseva). 0378-4754/02/$ – see front matter © 2002 Published by Elsevier Science B.V. on behalf of IMACS. PII: S 0 3 7 8 - 4 7 5 4 ( 0 1 ) 0 0 3 8 4 - 6
470
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
the Arnold tongues shrink. The classical concept of synchronization has subsequently been generalized to a wide class of two-state systems whose time scales are random values [8–11]. The effect of locking of the mean switching frequencies was discovered in noisy bistable systems driven by a strong-amplitude periodic forcing [9]. It was shown that there are regions on the “noise intensity-forcing amplitude” parameter plane where the mean switching frequency remains constant and coincides with the frequency of periodic forcing. The effect of stochastic synchronization of two-coupled bistable systems driven by independent noise sources was discovered by Neiman [8] and it was shown that a bifurcation of the two-dimensional stationary probability density takes place when the coupling is increased. Kramer’s rates for the subsystems are drawn closer to one another with increasing coupling and coincide at the bifurcation. Similar effects were investigated in two-state deterministic systems (Lorenz system [11] and Chua’s circuit [12]) where switchings are caused by the natural chaotic dynamics and may be characterized by the mean frequency switchings as well. The growing interest in synchronization of noisy systems has led researchers to consider the phenomenon of synchronization in large arrays and networks of coupled oscillators. Lindner et al. [13] showed that an array of stochastic bistable elements, subjected to the same periodic signal when appropriately coupled can display global synchronization effects leading to an enhancement of the stochastic resonance. Recently, it has been found that noise improves the signal propagation in extended bistable systems [14,15]. The response of one- and two-dimensional excitable media to external spatiotemporal noise in terms of synchronization was studied by Neiman et al. [16]. The above studies have dealt with the influence of noise on the stochastic resonance characteristics, and the results do not provide information about coherent behavior of switching events. In this paper, we shall dwell on the ordering role of spatiotemporal noise in an important class of extended bistable systems. In particular, we shall look for answers to the following questions. How do the stochastic bistable systems adjust their switching events in accordance with one another? How will the non-uniformities of parameters and different types of coupling influence the synchronization effects?
2. Model With our focus being on synchronization effects of stochastic oscillations, we chose the Schmitt trigger as one of the simplest threshold system [17,18], possessing a static hysteretic non-linearity. As a main advantage compared to the continuous bistable systems, the output of a Schmitt trigger is entirely controlled by the switching mechanism, whereas in a continuous bistable system interwell and intrawell dynamics may be hard to unravel. A Schmitt trigger can be constructed by means of an operational amplifier (Fig. 1a). In the absence of any forcing, it is characterized by two stable states. The input signal is a random function of time Vin (t) while there are only two possible states ±Vout at the output. The system is characterized by a threshold value for the voltage V th = R2 V0 /(R1 + R2 ). If |V in (t)| ≥ V th , it is bistable. Switching events occur as soon as the input voltage reaches the threshold values. In numerical simulations, the ideal Schmitt trigger obeys the equation x(t + t) = sng(Kx(t) − ξ(t)).
(1)
Here, the function sng denotes sign, and K corresponds to the threshold level of the trigger. ξ (t) is exponentially correlated Gaussian noise for which we shall assume a correlation time τ = 0.01 and an
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
471
Fig. 1. (a) The output signal is generated in Vout when Vin exceeds the threshold level, (b) the electrical scheme of the coupled Schmitt triggers.
intensity D is governed by the Ornstein–Uhlenbeck process 2 1 ˙ξ = ξ + D ω(t), ω(t)ω(t + s) = δ(s). τ τ
(2)
A one-dimensional array of such ideal two-state systems demonstrating pure hopping dynamics is given by xj (t + t) = sgn(Kxj (t) − γ Gj (xj −1 , xj , xj +1 ) − ξj (t)),
j = 1, 2, . . . , m
(3)
with γ being the coupling parameter. The noise sources ξ j (t) are assumed to be uncorrelated with D = 0.4. Throughout this work, the local coupling Gj is taken to be either two-way Gj = xj +1 + xj −1 or one-directional Gj = xj −1 . The boundary conditions in Eq. (3) are free-end, and the initial conditions are distributed non-uniformly in two states. Fig. 1b illustrates an implementation of two-coupled Schmitt triggers. Two-way coupling provides interaction of each functional unit with the previous and next elements. Since the input signal from the previous functional unit is applied to inverting input (−) of operational amplifier, and Vout is an anti-phase, neighboring units operate in anti-phase for γ > 0. For γ < 0, on the other hand, all functional units tend to be at the same state. At the output of the bistable system, we have a dichotomous stochastic process that can be characterized by the mean durations of the upper state Tu and lower state Tl . The mean “period” of switching is therefore, T s = T u + T l . This is related to the mean switching frequency f = 1/T s . Stochastic bistable systems
472
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
have a characteristic time scale which corresponds to the mean frequency of the transition from one potential well to the other. This averaged switching rate, referred to as Kramer’s rate, is fully controlled by noise and is characterized by the exponential Arrhenius law [19] −V f ∝ exp (4) D with V being the height of the potential barrier. Hence, a mismatch between the mean switching frequencies along an array is provided by the random distribution of the threshold parameter K over the interval [1.0, 1.1]. We measured the mean switching frequency between two states in each system as described in our previous publication [9] M
1 1 fj = lim , M→∞ M t − tk k=1 k+1
(5)
where tk is the moment of crossing of the potential barrier and tk+1 − tk is the residence time between two switchings.
3. Spatiotemporal order According to recent results [8–10], synchronization phenomena can take place for interacting stochastic systems whose mean switching frequency determines a clearly distinguishable time scale. Synchronization is defined as the coincidence of the identified time scales. This approach offers a glimpse of the richness of behaviors that are possible in large arrays of coupled noisy oscillators. For two-way interaction, an array of oscillators with different parameters and, as a consequence, varying time scales is interesting both conceptually and because it is encountered in practice in various situations. When a mismatch between the Kramer’s frequencies is introduced, increasing of the coupling causes a reduction of the individual switching frequencies that tend to a vanishing value (Fig. 2a). Moreover, an adjustment of the individual frequencies fj takes place and is considered the effect of stochastic synchronization. However, with one-way coupling (i.e. the dynamics of each element does not depend on the evolution processes in the next units but it is affected by the signal only from the previous oscillators), individual frequencies do not tend to a vanishing value, but approach the switching frequency of the first oscillator (Fig. 2b). Note, that the effect of frequency-locking is observed within some range of coupling parameter. At the critical value, the coherence dynamics of the hopping events breaks down. A retarding effect of noise-induced switching events appears to be general for bistable systems with two-way coupling. This effect seems to be analogous to the effect observed in systems of coupled self-sustained oscillators and referred to as a “frozen state” [20]. In the deterministic case, due to the mutual interaction there may be dramatic shifts of the synchronization frequency. For this reason, the frozen state denotes a stable synchronized state with vanishing (or at least small) synchronization frequency. In our case, for strong interaction, the switching frequencies vanish and retarding of hopping events take place. The above effect of retarding of stochastic oscillations leading finally to their “death” plays an ordering role on the array dynamics. Let us consider how the phase coherence behavior manifests itself at such a transition. In the deterministic case, the appearance of an in-phase regime where the phase difference between any two
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
473
Fig. 2. Individual mean switching frequency for the system (2) with j = 1, 20, 40, 60, 80, 100 as the function of (a) two-way, and (b) one-way coupling. The effects of frequency-locking are clearly distinguishable in both cases.
oscillators vanishes (modulo 2π) corresponds to the strongest form of synchronization, referred to as full synchronization [21,22]. For an array of bistable systems driven by uncorrelated random forcing, full synchronization is not achieved. Fig. 3 illustrates the spatiotemporal evolution of an array of m = 100 oscillators. An oscillator is indicated as white if it is in the left well and as black if it is in the right well.
Fig. 3. Spatiotemporal dynamics of an array of 100 Schmitt triggers (2) for two-way coupling with (a) γ = 0.02, and (b) γ = 0.4.
474
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
Fig. 4. Entrainment of the mean switching frequencies averaged over elements of a two-dimensional lattice at (a) γ = 0.015, and (b) γ = 0.2.
Fig. 5. Spatiotemporal evolution of a two-dimensional lattice of 50 × 50 Schmitt triggers at (a) γ = 0.0, (b) γ = 0.015, (c) γ = 0.2, and (d) γ = −0.2.
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
475
Originally, all oscillators were started with a random spread in their potential wells. For weak coupling, the oscillators can be in each potential well with equal probability as reflected by the equal amounts of black and white in Fig. 3a. With increasing coupling, alternating of colored regions is observed (Fig. 3b). This corresponds to a self-organization of the stochastic oscillators. With increasing system size, qualitatively the same behavior in the sense of the averaged characteristics is observed. Fig. 4 shows how the mean switching frequencies are decreased to a constant value demonstrating retarding effects and frequency-locking when the coupling strength becomes strong enough. Being randomly distributed in system states (Fig. 5a), the stochastic oscillators remain non-synchronous for weak interaction (Fig. 5b). With increasing coupling strength they finally form well-pronounced anti-phase (Fig. 5c) or in-phase (Fig. 5d) coherence pattern.
4. Summary We have demonstrated the spatiotemporal stochastic synchronization in one- and two-dimensional media modeled by coupled bistable functional units driven by external noise. The synchronization effects manifest themselves in two ways. The first manifestation consists in the formation of synchronized frequency-locked states with the same average frequency or frozen states with vanishing synchronization frequency. In the latter case, a retardation of the switching dynamics takes place. The second manifestation produces in- or anti-phase hopping dynamics with well-defined phase coherence structures. Our results indicate that noise can play an important role in the self-organization of coupled stochastic elements. The correspondence to well-known classical effects of synchronization emphasizes the universal aspects of these phenomena.
Acknowledgements This paper is based upon work supported by the Naval Research Laboratory under Contract no N68171-00-M-5430. A.K. acknowledges Award no. REC-006 of the US Civilian Research & Development Foundation for the Independent States of the Former Soviet Union (CRDF). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
V. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, Berlin, 1984. S. Watanabe, S.H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica 74D (1994) 197. R. Roy, K.S. Thornburg Jr., Experimental synchronization of chaotic lasers, Phys. Rev. Lett. 72 (1994) 2009. A.T. Winfree, The Geometry of Biological Time, Springer, Berlin, 1980. W. Horsthemke, R. Lefever, Noise-Induced Transitions, Springer, Berlin, 1984. J. Garcia-Ojalvo, J.M. Sancho, Noise in Spatially Extended Systems, Springer, New York, 2000. R.L. Stratonovich, Topics in the Theory of Random Noise, Gordon and Breach, New York, 1967. A.B. Neiman, Synchronization like phenomena in coupled stochastic bistable systems, Phys. Rev. E 49 (1994) 3484. B.V. Shulgin, A.B. Neiman, V.S. Anishchenko, Mean switching frequency-locking in stochastic bistable system driven by periodic force, Phys. Rev. Lett. 75 (1995) 4157. [10] V.S. Anishchenko, A.B. Neiman, F. Moss, L. Schimansky-Geier, Stochastic resonance: noise-enhanced order, Physics-Uspekhi 42 (1999) 7.
476
V.S. Anishchenko et al. / Mathematics and Computers in Simulation 58 (2002) 469–476
[11] V.S. Anishehenko, A.N. Silchenko, I.A. Khovanov, Synchronization of switching processes in coupled Lorenz systems, Phys. Rev. E 57 (1997) 316. [12] V.S. Anishchenko, A.B. Neiman, M.A. Safonova, Stochastic resonance in chaotic systems, J. Stat. Phys. 70 (1993) 183. [13] J.F. Lindner, B.K. Meadows, W.L. Ditto, M.E. Inchiosa, A.R. Bulsara, Array enhanced stochastic resonance and spatiotemporal synchronization, Phys. Rev. Lett. 75 (1995) 3. [14] Y. Zhang, G. Hu, L. Gammaitoni, Signal transmission in one-way coupled bistable systems: noise effects, Phys. Rev. E 58 (1998) 2952. [15] J.F. Lindner, S. Chandramouli, A.R. Bulsara, M. Löcher, W.L. Ditto, Noise-enhanced propagation, Phys. Rev. Lett. 81 (1998) 5048. [16] A. Neiman, L. Schimansky-Geier, A. Cornell-Bell, F. Moss, Noise-enhanced phase synchronization in excitable media, Phys. Rev. Lett. 83 (1999) 4896. [17] S. Fauve, F. Heslot, Stochastic resonance in a bistable system, Phys. Lett. A 97 (1983) 5. [18] B. McNamara, K. Wiesenfeld, Theory of stochastic resonance, Phys. Rev. A 39 (1989) 4854. [19] H.A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (1940) 284. [20] P. Tass, Phase and frequency shifts in a population of phase oscillators, Phys. Rev. E 56 (1997) 2043. [21] H. Fujisaka, T. Yamada, Stability theory of synchronized motions in coupled oscillatory systems, Prog. Theor. Phys. 69 (1983) 32. [22] A.S. Pikovsky, On the interaction of strange attractors, Z. Physik B 55 (1984) 149.