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Controlling chaos in systems of coupled oscillators
Physica A 307 (2002) 315 – 330
www.elsevier.com/locate/physa
Controlling chaos in systems of coupled oscillators M. N%un˜ eza , G. Matob; ∗ a Instituto
Balseiro (CNEA and UNC), 8400 San Carlos de Bariloche, R.N., Argentina Nacional de Energ!a Atomica and CONICET, Centro Atomico Bariloche and Instituto Balseiro (CNEA and UNC), 8400 San Carlos de Bariloche, R.N., Argentina
b Comision
Received 6 September 2001; received in revised form 22 October 2001
Abstract We study the dynamical behavior of a one-dimensional network of phase oscillators, displaying a variety of cluster or traveling wave solutions, either stable or unstable according to the type of interaction function. For some range of parameters these solutions coexist with a chaotic attractor. We analyze a mechanism for controlling chaos and stabilizing the periodic solutions. We also analyze the minimal number of controllers needed to stabilize the system and the relation between the location of the controllers and the symmetry of the reference state to be stabilized. c 2002 Elsevier Science B.V. All rights reserved. PACS: ; 05.45.Gg; 87.10.+e; 87.18.Hf Keywords: Chaos; Control; Oscillators
1. Introduction Networks of phase oscillators can display a large variety of dynamical states. Even if there is no spatial structure, such as in fully connected networks, it has been found that these systems can be in an asynchronous state, or in several synchronous states such as cluster states [1,2]. If there is spatial structure the possibilities for complex behavior increase. Traveling waves in one-dimension, spirals in two-dimensions, have
∗
Corresponding author. CNEA and UNC, 8400 San Carlos de Bariloche, R.N., Argentina. Tel.: +54-2944-445100; fax: +54-2944-445299. E-mail address: [email protected] (G. Mato). c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 6 2 4 - 0
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been reported [3]. in some cases analytical expressions for these structures have been found [4]. These spatially and temporally periodic structures can coexist with more complex patterns, that can be even chaotic. These chaotic structures will usually appear when the periodic solutions are unstable. This will depend in a complex way on the details of the interaction function. The transition from periodic to chaotic behavior in spatially extended systems has been found in several systems such as optical systems [6], coupled diode resonators [7] and it has been also proposed as a model for the phenomenon of 6brillation in cardiac tissue. In this phenomenon, the spatial relation between the activities of different regions of the heart is lost, leading to a behavior that is incompatible with normal cardiac functions [8,9]. One possible way to suppress the chaotic dynamics is to re-stabilize one of the periodic solutions by applying external perturbations to some elements of the system. This kind of chaos control [10,17]. is based on the fact that chaotic attractors are embedded with a large number of unstable periodic orbits. If these orbits can be detected and the system taken near one of them, the dynamics can be made non chaotic by making only small perturbations. This idea has been applied in several Eelds [11–15]. See also Ref. [16] for a review on the subject. The main objective of this work is to examine this problem in the context of phase oscillators. In our case, chaos arises due to the coupling between the oscillators. We will focus in the analysis of the conditions under which chaotic solutions can be re-stabilized. We will see which is the minimal number of elements that we have to perturb in order to attain this objective. These elements will be called controllers. We will also analyze the relation between the location of the controllers and the spatial structure of the reference state and the time needed to achieve the stabilization. It is possible to perform this analysis because we know the analytical expression of the periodic solutions. We analyze mainly the case of a one-dimensional system with periodic boundary conditions. Such a system can be easily constructed using electronic circuits such as the ones analyzed in Ref. [7]. Those systems could be used to test the results presented in our paper. Other systems do not have the required symmetry but in some circumstances our results can be applied. For instance, in the cardiac tissue, activity can take place in a ring-like structure around an obstacle (such as a vein or a region of dead tissue) as mentioned in Ref. [8]. The paper is organized as follows: in the next section we introduce the model. In Section 3, we analyze the structure of the cluster states and present some results of numerical simulations showing how the structure of the states depends on the parameters of the interaction function. We also present here the results of the linear stability of the cluster states. In Section 4, we analyze the chaotic regime and in Section 5, we discuss the control mechanism. The results and some extensions are discussed in the last section.
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2. The model We consider a one-dimensional system composed of N phase-coupled oscillators whose dynamics is described by di (t) = !0 − K [sin(i (t) − i+1 (t) + ) dt + sin (i (t) − i−1 (t) + ) − 2 sin ()]
(i = 1; : : : ; N ) ;
(1)
where i (t) denotes the phase of the ith oscillator and !0 the common natural frequency. This will be the frequency in the absence of interaction among oscillators. The coupling is characterized by the coupling strength K and by the phase . As will be shown below, the latter parameter controls the phase diIerence between the oscillators. We chose periodical boundary conditions, i.e., N +i = i . After a change of variables: i = ˜i − !0 t and t˜ = Kt, we can write the dynamics as: d ˜i (t˜ ) = −[sin (˜i − ˜i+1 + ) + sin (˜i − ˜i−1 + ) − 2 sin ()] d t˜
(i = 1; : : : ; N ) : (2)
In what follows, we omit tildes for simplicity. Let us remark that the coupling strength K has disappeared from the dynamics because we are dealing with a homogeneous system. For a heterogeneous system, another relevant parameter would also be present in the equations. This would be the ratio between the coupling strength and the strength of the heterogeneity. Changing this ratio it is possible to observe, for instance, synchronization transitions [1]. In this work, however, we focus our attention on the parameter in order to study its inKuence on collective dynamics of the system.
3. Cluster states The existence of analytical solutions can be easily shown. We give a systematic rule to produce the so called cluster states. These states are characterized by a common frequency and by a Exed phase diIerence between neighbors: i = L ()t + i
(i = 1; : : : ; N ) ;
(3)
where L () is the common frequency of the oscillators. Because of the periodicity, the phase diIerence must verify N = 2 Nc for some integer number Nc . If Nc ¿ 0, the phase of any oscillator will verify i = i+N=Nc = i+2N=Nc = · · · . This means that the system will be divided into N=Nc sets of Nc oscillators called clusters. Oscillators inside the same cluster have the same phase. The phase diIerence between diIerent clusters is constant in time. If Nc = 0 then all the oscillators have the same phase.
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Inserting Eq. (3) into Eq. (2), we obtain for the global frequency of the system () = 2 sin ()[1 − cos ( )] :
(4)
These solutions represent phase traveling waves around the ring of oscillators whose symmetry depends on the phase diIerence between oscillators, . 3.1. Numerical simulations We performed computer simulations to examine the behavior of the system when the parameter is changed. The simulations were performed by integrating Eq. (2) using the second-order Runge–Kutta algorithm with a time step t = 0:03. We have checked that decreasing the time step does not aIect the results. We integrated systems of diIerent numbers of oscillators. As will be shown below, the general behavior does not depend on N . The initial conditions of the oscillators were chosen randomly or in some cluster states according to the objectives of the simulation. We measure the phase diIerence of the oscillators and its frequency in order to characterize the dynamical behavior of the system. One important quantity is the phase diIerence between neighboring oscillators D=
N 1 |i − i+1 | : N
(5)
i=1
Another relevant quantity is the average frequency ˙ = (1=N ) ˙i ;
(6)
i=1
which would be equal to the common frequency of the oscillators Eq. (4) and its standard deviation, , is = (1=N )
N
˙ − ˙i | : |
(7)
i=1
This parameter is zero when all the oscillators of the system are synchronized. We Erst show the results of the simulations for a system with N = 18 oscillators. Initializing the system with randomly chosen phases, we obtain basically two types of solutions, depending on the value of . For || ¡ =2 we obtain a state with = =9 and for || ¿ =2 a state with = . Other types of cluster states can be obtained, but the system does not converge to them often. This suggests that the states Nc = 1 for || ¡ =2 and Nc = 9 for || ¿ =2 have the largest basin of attraction. This point will be clariEed when studying the linear stability of the cluster states. The global behavior of the system can be seen in Fig. 1 where we show the global parameter D for diIerent values of . Except in the range | | ≈ =2, the system stabilizes after a few time steps. The parameter D is in the region around 0 for || ¡ =2 and around for || ¿ =2. This suggests that the solutions with ≈ 0 have the largest basin of attraction for the Erst case and with ≈ for the second case. In the region | | ≈ =2, one Ends an apparently irregular behavior.
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Fig. 1. D vs. . The system has N = 30. The initial phase values are random. With this parameter we can know which cluster state has reached the system.
3.2. Linear stability of the cluster states We now study the linear stability of the cluster states. By linearizing Eq. (2) about the solution given by Eq. (3) we End that the dynamics is given by N
di Aij j ; = dt
(8)
j=1
where the matrix A is A=
−2 cos( ) cos() cos( + ) 0 ::: cos(− + ) cos(− + ) −2 cos( ) cos() cos( + ) 0 ::: : .. .. .. .. .. . . . . . cos( + ) 0 ::: cos(− + ) −2 cos( ) cos() (9)
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The N eigenvalues of this matrix are given by
2 j 2 j − 2i sin () sin () sin j = −2 cos ( ) cos () 1 − cos N N
(10)
for j = 1; : : : ; N . This means that the eigenvalues will have a positive real part (and the state will be unstable) if cos( ) cos() ¡ 0. For small , solutions with small
and Nc will be stable. For ≈ only solutions with large (and large Nc ) will be stable. If = =2 then the states will be marginally stable. In order to analyze the behavior of cluster states when we apply a small but Enite perturbation we Erst generate a cluster state S , and then we introduce a perturbation as i = Si + pi ;
(11)
where pi are random variables with uniform distribution in the range [0; 2 =200]. If the cluster state is restored in spite of a perturbation, then it is stable; otherwise this ˙ is a good indicator because we cluster state is unstable. The time evolution of know its value for all the cluster states. The perturbation is performed after 100 time steps. For a value of such that || ¡ =2, the cluster states with Nc ¿ 4 are unstable and Enally they converge to cluster states with Nc = 1 or to the state with Nc = 0. For the case || ¿ =2, the states with Nc ¡ 5 turn out to be unstable and converge Enally to states with high values of
. We End that for || ¡ =2, the linearly stable states have ¡ =2 and among them the ones with the largest basin of attraction are such that ≈ 0. For || ¿ =2, the linear stable states have ¿ =2 and the ones with the largest basin of attraction are the ones that have ≈ . For ≈ =2 there is coexistence between cluster states and chaotic states. We can see this behavior for a 18 oscillator system in Fig. 2. According to the initial condition the system can settle in a periodic state, characterized by a constant value of d=dt, or in a state where the angular velocity Kuctuates rapidly in time. These results are a generalization of the theoretical ones obtained for the case of two coupled oscillators (see Ref. [3]). Here we End two solutions, for || ¡ =2 the one with oscillations in phase is stable and the other one (anti-phase) unstable; the reverse is true for || ¿ =2.
4. Chaos We have characterized the behavior of the ring of oscillators in ranges of values of such that || ¿ =2 and || ¡ =2. In this section, we study the system in the region || ≈ =2 and show the presence of chaos in the dynamics. We prove the existence of chaos by evaluating the maximum Lyapunov exponent as a function of using Wolf’s method [5]. The results can be seen in Fig. 3 where the maximum exponent has positive values when || ≈ =2 indicating the presence of chaos in that range.
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Fig. 2. Oscillation frequency d=dt for several initial conditions. There is coexistence between cluster states where the frequency is constant and a chaotic state.
We noted before (Section 3.1) that when || ≈ =2, the system does not converge to any stable solution after being randomly initialized. However, we can still obtain the cluster states with the proper initial conditions. The next step is to study their linear ˙ before and after stability. For this purpose, we can analyze the time evolution of the perturbation. We use the same perturbation as before (Eq. (11)) which is turned on at time step 1000. The states with 0 or with remain stable, while the rest of the solutions are unstable and converge to a state where the oscillators are ˙ changes very fast. We have a new not synchronized and the average frequency attractor in the system with no temporal or spatial order. We checked that when the system is initialized randomly it converges to this state. This suggests that it has the largest basin of attraction. Unless is exactly equal to =2 stable cluster states do exist, but as we approach this value their basins of attraction become smaller and eventually collapse. In this way we see that the chaotic state coexists with stable and unstable cluster states.
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Fig. 3. Maximal Lyapunov exponent vs. . We used an algorithm introduced by Wolf. In the range of close to ≈ =2 the exponent has positive values and the dynamics is chaotic.
5. Chaos control The aim of controlling chaos is to suppress it by stabilizing a certain unstable reference state. For this purpose, we need to perturb some or all of the oscillators. Our goal is to perturb the minimum number of them, exploiting the spatial symmetry of the reference state. Our control method is an adaptation of a well-known feedback scheme [17]. The diIerence is based on the periodic nature of the variables, the phases. We can interpret the control of chaos as the synchrony between the oscillators of the system and the oscillators of the reference state. Next, we show the concept with two oscillators, and afterwards we give the algorithm we used for controlling chaos in the ring and Enally we present the results. 5.1. Synchronizing two oscillators The phases i of the oscillators are deEned in the range [0; 2 ]. We can imagine the periodic path of an oscillator as a circle and the phase as the angle in radians that deEnes its position on it.
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Let suppose that we have two oscillators, whose phases 1 and 2 evolve as: di (12) = fi (i ); i = 1; 2 : dt Our goal is to get the second oscillator synchronized with the Erst one. For that purpose, we can modify its dynamics in the following way: d2 = f2 (2 ) + P() ; (13) dt where P is a perturbation proportional to where = 1 − 2 :
(14)
This means that the second oscillator will change its instantaneous frequency as a function of the angular deviation relative to the Erst oscillator. For optimizing the perturbation we need P to be proportional to the smallest angular distance on the circumference between the two oscillators. We deEned P such that || ¡ → P = ; || ¿ →
¿ 0 → P = ( − 2 ) ; ¿ 0 → P = ( + 2 ) ;
(15)
where is a parameter that Ex the strength of the perturbation. If the second oscillator has the same phase as the Erst one, the perturbation (or feedback) goes to zero and the feedback is activated every time that the phase diIerence between them is diIerent from zero. However, if the second term on the right hand side of Eq. (13) is very small compared to the Erst, then the oscillators are synchronized. 5.2. Controlling chaos in the ring Controlling chaos in a system of oscillators, stabilizing one of the unstable solutions embedded in the chaotic attractor, can be interpreted as synchronizing the N oscillators of the system, each one of which oscillates at its own instantaneous frequency, with the other oscillators sharing a common frequency (the reference state). For this purpose it may not be necessary to perturb all the oscillators of the system, this depends speciEcally on the spatial pattern of the reference state. For the ring of oscillators we used the following algorithm: di = −[sin(i − i+1 + ) + sin(i − i−1 + ) − 2 sin ] dt + ik P(Sk (t); i (t));
(i = 1 : : : N ); k ∈ C ;
(16)
where C represents the set of all the oscillators to be perturbed and S (t) is the reference state. The parameter P is deEned as 15 and = Sk − k . 5.3. Results If ≈ =2, a chaotic attractor is present in the system (Section 4). We Erst apply the algorithm introduced in the last section, in such a way that all the oscillators are
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perturbed. In this situation, we can easily stabilize any of the unstable solutions. The transient time between the switch on of the feedback and the stabilization time of the solution is short (≈ 102 steps for a system with N =18). Although this result is positive, it is easy to perturb all the oscillators when we deal with relative small systems. For larger systems we may not have access to all the oscillators of the system. In this case, we want to control the dynamics with the minimum number of controllers. For this purpose we have to choose the oscillators to be perturbed in a way that is related to the spatial pattern of the reference state. We consider a system N = 18 with = 1:53 and we want to stabilize a reference state with Nc = 6. We can see its phase spatial pattern in Fig. 4(a). In Figs. 4(b) – (e), we show the possible distribution for six controllers and in Fig. 4(f) for three controllers. In cases (b) and (d), where the controllers are concentrated in sets that are separated from each other by distances of the order of the characteristic length of the spatial pattern of the reference state, we could stabilize the solution after a transient of the order of 103 time steps. After the transient, the feedback reaches a maximal value of 1% with respect to the Erst term of Eq. (16). In case (c) where we put the controllers in symmetric positions but not in concentrated sets, we stabilize the system after a transient of 107 steps and we End that the maximal feedback value reaches a 5% of the Erst term. Locating the controllers in an asymmetric way as is shown in case (e), the solution could not be stabilized. In case (f) we used just three controllers separated by a distance equal to the characteristic length of the reference state. The reference state is stabilized with a longer transient. In all the cases we had to tune the value of . In Fig. 5, we show in a space-time diagram the results. The control is switched on after 1000 time steps. Once the unstable solution has been stabilized, all the oscillators have a common frequency and the value of is zero. The feedback reaches its maximal value at the moment it is turned on. In Fig. 5(a) we can see clearly the transition from chaos to the periodic reference state for Nc = 9. In Fig. 5(b) we control the same system but using as the reference state the solution with Nc = 3. From our experience controlling chaos in systems of diIerent sizes, we infer that for stabilizing an unstable state, using the minimum number of oscillators in the shortest possible time, we must locate the controllers in equal concentrated sets that are separated between them by distances of the order of the characteristic length of the spatial pattern of the reference state. Furthermore, as the number of oscillators in each set is smaller, the transient is longer. As expected, for larger systems, we would have longer transients. We illustrate the result showing a system of N =24 where we control chaos stabilizing a reference state with Nc = 4. There are 4 controllers separated by a characteristic length of the reference state as shown in Fig. 6(a). In Figs. 6(b) and (c) we show the average feedback and the standard deviation of the average frequency. The control is switched on after 1000 time steps. In Fig. 7, we show the results for a larger system (N = 120). In this case the reference state has a value of Nc = 20 and the controllers are separated by the characteristic length of the reference state (the same as the previous case). We can see that the transient time is longer than the smaller system. Finally, we put the controllers in random positions (see Fig. 8(a)). The system
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325
Fig. 4. vs. number of the oscillator in the ring. (a) Space pattern of the solution with Nc = 3. (b) – (e) Possible positions for six controllers. A big circle points the position of a controller. (f) Distribution for 3 controllers.
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Fig. 5. Space–time graph. The values of the phases of the oscillators are represented in gray levels. Its minimum value is white and the maximum is black. The control is switched on at the time step 1000. (a) Chaos is controlled with the reference state with Nc = 9 and (b) with Nc = 3.
could not be synchronized as we can see from the behavior of the standard deviation (see Fig. 8(c)) which remains always diIerent from 0. Comparing the cases of N = 24 with N = 120, we can see that it is possible to control the system in a similar time scale keeping constant the fraction of controllers among all the elements of the network and the spatial symmetry of the reference state. We checked this in larger systems (up to N = 300). We also note that if we have more oscillators in the ring, we have more unstable solutions embedded in the chaotic attractor. This means that we have a larger number of possible reference states that could replace the chaotic behavior. 6. Discussion In this paper, we studied basically the possibility of controlling chaos in a system of oscillators with Erst neighbor coupling. We used an adaptation of the well known feedback scheme for controlling chaos, Erst introduced in [17]. Although we study a ring of oscillators, the control method can be easily generalized to higher dimensional systems. We characterized the system by its solutions or cluster states, which can be deEned by the phase diIerence between neighboring oscillators. As the number of
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Fig. 6. System with N = 24 oscillators. (a) Reference state with Nc = 4 and positions of the controllers with big circles. (b) Average feedback vs. time steps. The system is initialized randomly and the control is switched on at the time step 1000. (c) Standard deviation of the frequency vs. time steps.
oscillators in the system grows we have more cluster states. We found that for given ranges of values of the dynamics of the system become chaotic. We can use any of the cluster states as a reference periodic state to be stabilized using our control scheme in such a way that the chaos is controlled. We can also interpret chaos control in the oscillator system as the synchrony between the oscillators of the system and those of the reference state. For this purpose, we need to perturb all or some of the oscillators of the system. Our goal is to perturb a minimum number. We found that we have to choose the oscillators to be perturbed in a way that takes into account the spatial symmetry of the reference state. We End that the minimum number of controllers is equal to the number of oscillators in one cluster, Nc . In order to minimize the time necessary to control the system, one can introduce more controllers with the same geometry, for instance increasing the number of controllers in each set of oscillators.
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Fig. 7. System with N = 120 oscillators. (a) Reference state with Nc = 20 and positions of the controllers with big circles. (b) Average feedback vs. time steps. The system is initialized randomly and the control is switched on at the time step 1000. (c) Standard deviation of the frequency vs. time steps.
The question of the relation between the symmetry of the system and the target state and the number and location of controllers has already been discussed in Ref. [18] using methods of linear feedback control theory. It is found that for coupled map lattices in one-dimensional systems, two control parameters are enough to control the system independently of the symmetry of the target state. In our method the feedback is not linear. We need a larger number of controllers but we can control the target state beginning from an arbitrary initial condition, which is not guaranteed by the linear feedback control theory. The method we have presented in this paper can be easily applied to higher dimensional systems, if one knows the structure of the target states. In some cases this is known, e.g. the spiral waves in bi-dimensional systems [4]. Preliminary results indicate that our results still hold qualitatively for those systems. We can also mention the possibility of using singular value decomposition of the spatio temporal
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Fig. 8. (a) Reference state with Nc = 20. Random positions of the controllers with big circles. (b) Average feedback vs. time steps. The system is initialized randomly and the control is switched on at the time step 1000. (c) Standard deviation of the frequency vs. time steps.
solution [19]. This method would allow us to generalize the results to situations without symmetries. Acknowledgements We thank H. Wio and V. Grunfeld for a careful reading of the manuscript. This work was partially supported by grant PICT97 03-00000-00131 from ANPCyT for G.M. References [1] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, New York, 1984. [2] D. Hansel, G. Mato, C. Meunier, Phys. Rev. E 48 (1993) 3470.
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www.elsevier.com/locate/physa
Controlling chaos in systems of coupled oscillators M. N%un˜ eza , G. Matob; ∗ a Instituto
Balseiro (CNEA and UNC), 8400 San Carlos de Bariloche, R.N., Argentina Nacional de Energ!a Atomica and CONICET, Centro Atomico Bariloche and Instituto Balseiro (CNEA and UNC), 8400 San Carlos de Bariloche, R.N., Argentina
b Comision
Received 6 September 2001; received in revised form 22 October 2001
Abstract We study the dynamical behavior of a one-dimensional network of phase oscillators, displaying a variety of cluster or traveling wave solutions, either stable or unstable according to the type of interaction function. For some range of parameters these solutions coexist with a chaotic attractor. We analyze a mechanism for controlling chaos and stabilizing the periodic solutions. We also analyze the minimal number of controllers needed to stabilize the system and the relation between the location of the controllers and the symmetry of the reference state to be stabilized. c 2002 Elsevier Science B.V. All rights reserved. PACS: ; 05.45.Gg; 87.10.+e; 87.18.Hf Keywords: Chaos; Control; Oscillators
1. Introduction Networks of phase oscillators can display a large variety of dynamical states. Even if there is no spatial structure, such as in fully connected networks, it has been found that these systems can be in an asynchronous state, or in several synchronous states such as cluster states [1,2]. If there is spatial structure the possibilities for complex behavior increase. Traveling waves in one-dimension, spirals in two-dimensions, have
∗
Corresponding author. CNEA and UNC, 8400 San Carlos de Bariloche, R.N., Argentina. Tel.: +54-2944-445100; fax: +54-2944-445299. E-mail address: [email protected] (G. Mato). c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 6 2 4 - 0
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been reported [3]. in some cases analytical expressions for these structures have been found [4]. These spatially and temporally periodic structures can coexist with more complex patterns, that can be even chaotic. These chaotic structures will usually appear when the periodic solutions are unstable. This will depend in a complex way on the details of the interaction function. The transition from periodic to chaotic behavior in spatially extended systems has been found in several systems such as optical systems [6], coupled diode resonators [7] and it has been also proposed as a model for the phenomenon of 6brillation in cardiac tissue. In this phenomenon, the spatial relation between the activities of different regions of the heart is lost, leading to a behavior that is incompatible with normal cardiac functions [8,9]. One possible way to suppress the chaotic dynamics is to re-stabilize one of the periodic solutions by applying external perturbations to some elements of the system. This kind of chaos control [10,17]. is based on the fact that chaotic attractors are embedded with a large number of unstable periodic orbits. If these orbits can be detected and the system taken near one of them, the dynamics can be made non chaotic by making only small perturbations. This idea has been applied in several Eelds [11–15]. See also Ref. [16] for a review on the subject. The main objective of this work is to examine this problem in the context of phase oscillators. In our case, chaos arises due to the coupling between the oscillators. We will focus in the analysis of the conditions under which chaotic solutions can be re-stabilized. We will see which is the minimal number of elements that we have to perturb in order to attain this objective. These elements will be called controllers. We will also analyze the relation between the location of the controllers and the spatial structure of the reference state and the time needed to achieve the stabilization. It is possible to perform this analysis because we know the analytical expression of the periodic solutions. We analyze mainly the case of a one-dimensional system with periodic boundary conditions. Such a system can be easily constructed using electronic circuits such as the ones analyzed in Ref. [7]. Those systems could be used to test the results presented in our paper. Other systems do not have the required symmetry but in some circumstances our results can be applied. For instance, in the cardiac tissue, activity can take place in a ring-like structure around an obstacle (such as a vein or a region of dead tissue) as mentioned in Ref. [8]. The paper is organized as follows: in the next section we introduce the model. In Section 3, we analyze the structure of the cluster states and present some results of numerical simulations showing how the structure of the states depends on the parameters of the interaction function. We also present here the results of the linear stability of the cluster states. In Section 4, we analyze the chaotic regime and in Section 5, we discuss the control mechanism. The results and some extensions are discussed in the last section.
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2. The model We consider a one-dimensional system composed of N phase-coupled oscillators whose dynamics is described by di (t) = !0 − K [sin(i (t) − i+1 (t) + ) dt + sin (i (t) − i−1 (t) + ) − 2 sin ()]
(i = 1; : : : ; N ) ;
(1)
where i (t) denotes the phase of the ith oscillator and !0 the common natural frequency. This will be the frequency in the absence of interaction among oscillators. The coupling is characterized by the coupling strength K and by the phase . As will be shown below, the latter parameter controls the phase diIerence between the oscillators. We chose periodical boundary conditions, i.e., N +i = i . After a change of variables: i = ˜i − !0 t and t˜ = Kt, we can write the dynamics as: d ˜i (t˜ ) = −[sin (˜i − ˜i+1 + ) + sin (˜i − ˜i−1 + ) − 2 sin ()] d t˜
(i = 1; : : : ; N ) : (2)
In what follows, we omit tildes for simplicity. Let us remark that the coupling strength K has disappeared from the dynamics because we are dealing with a homogeneous system. For a heterogeneous system, another relevant parameter would also be present in the equations. This would be the ratio between the coupling strength and the strength of the heterogeneity. Changing this ratio it is possible to observe, for instance, synchronization transitions [1]. In this work, however, we focus our attention on the parameter in order to study its inKuence on collective dynamics of the system.
3. Cluster states The existence of analytical solutions can be easily shown. We give a systematic rule to produce the so called cluster states. These states are characterized by a common frequency and by a Exed phase diIerence between neighbors: i = L ()t + i
(i = 1; : : : ; N ) ;
(3)
where L () is the common frequency of the oscillators. Because of the periodicity, the phase diIerence must verify N = 2 Nc for some integer number Nc . If Nc ¿ 0, the phase of any oscillator will verify i = i+N=Nc = i+2N=Nc = · · · . This means that the system will be divided into N=Nc sets of Nc oscillators called clusters. Oscillators inside the same cluster have the same phase. The phase diIerence between diIerent clusters is constant in time. If Nc = 0 then all the oscillators have the same phase.
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Inserting Eq. (3) into Eq. (2), we obtain for the global frequency of the system () = 2 sin ()[1 − cos ( )] :
(4)
These solutions represent phase traveling waves around the ring of oscillators whose symmetry depends on the phase diIerence between oscillators, . 3.1. Numerical simulations We performed computer simulations to examine the behavior of the system when the parameter is changed. The simulations were performed by integrating Eq. (2) using the second-order Runge–Kutta algorithm with a time step t = 0:03. We have checked that decreasing the time step does not aIect the results. We integrated systems of diIerent numbers of oscillators. As will be shown below, the general behavior does not depend on N . The initial conditions of the oscillators were chosen randomly or in some cluster states according to the objectives of the simulation. We measure the phase diIerence of the oscillators and its frequency in order to characterize the dynamical behavior of the system. One important quantity is the phase diIerence between neighboring oscillators D=
N 1 |i − i+1 | : N
(5)
i=1
Another relevant quantity is the average frequency ˙ = (1=N ) ˙i ;
(6)
i=1
which would be equal to the common frequency of the oscillators Eq. (4) and its standard deviation, , is = (1=N )
N
˙ − ˙i | : |
(7)
i=1
This parameter is zero when all the oscillators of the system are synchronized. We Erst show the results of the simulations for a system with N = 18 oscillators. Initializing the system with randomly chosen phases, we obtain basically two types of solutions, depending on the value of . For || ¡ =2 we obtain a state with = =9 and for || ¿ =2 a state with = . Other types of cluster states can be obtained, but the system does not converge to them often. This suggests that the states Nc = 1 for || ¡ =2 and Nc = 9 for || ¿ =2 have the largest basin of attraction. This point will be clariEed when studying the linear stability of the cluster states. The global behavior of the system can be seen in Fig. 1 where we show the global parameter D for diIerent values of . Except in the range | | ≈ =2, the system stabilizes after a few time steps. The parameter D is in the region around 0 for || ¡ =2 and around for || ¿ =2. This suggests that the solutions with ≈ 0 have the largest basin of attraction for the Erst case and with ≈ for the second case. In the region | | ≈ =2, one Ends an apparently irregular behavior.
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Fig. 1. D vs. . The system has N = 30. The initial phase values are random. With this parameter we can know which cluster state has reached the system.
3.2. Linear stability of the cluster states We now study the linear stability of the cluster states. By linearizing Eq. (2) about the solution given by Eq. (3) we End that the dynamics is given by N
di Aij j ; = dt
(8)
j=1
where the matrix A is A=
−2 cos( ) cos() cos( + ) 0 ::: cos(− + ) cos(− + ) −2 cos( ) cos() cos( + ) 0 ::: : .. .. .. .. .. . . . . . cos( + ) 0 ::: cos(− + ) −2 cos( ) cos() (9)
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The N eigenvalues of this matrix are given by
2 j 2 j − 2i sin () sin () sin j = −2 cos ( ) cos () 1 − cos N N
(10)
for j = 1; : : : ; N . This means that the eigenvalues will have a positive real part (and the state will be unstable) if cos( ) cos() ¡ 0. For small , solutions with small
and Nc will be stable. For ≈ only solutions with large (and large Nc ) will be stable. If = =2 then the states will be marginally stable. In order to analyze the behavior of cluster states when we apply a small but Enite perturbation we Erst generate a cluster state S , and then we introduce a perturbation as i = Si + pi ;
(11)
where pi are random variables with uniform distribution in the range [0; 2 =200]. If the cluster state is restored in spite of a perturbation, then it is stable; otherwise this ˙ is a good indicator because we cluster state is unstable. The time evolution of know its value for all the cluster states. The perturbation is performed after 100 time steps. For a value of such that || ¡ =2, the cluster states with Nc ¿ 4 are unstable and Enally they converge to cluster states with Nc = 1 or to the state with Nc = 0. For the case || ¿ =2, the states with Nc ¡ 5 turn out to be unstable and converge Enally to states with high values of
. We End that for || ¡ =2, the linearly stable states have ¡ =2 and among them the ones with the largest basin of attraction are such that ≈ 0. For || ¿ =2, the linear stable states have ¿ =2 and the ones with the largest basin of attraction are the ones that have ≈ . For ≈ =2 there is coexistence between cluster states and chaotic states. We can see this behavior for a 18 oscillator system in Fig. 2. According to the initial condition the system can settle in a periodic state, characterized by a constant value of d=dt, or in a state where the angular velocity Kuctuates rapidly in time. These results are a generalization of the theoretical ones obtained for the case of two coupled oscillators (see Ref. [3]). Here we End two solutions, for || ¡ =2 the one with oscillations in phase is stable and the other one (anti-phase) unstable; the reverse is true for || ¿ =2.
4. Chaos We have characterized the behavior of the ring of oscillators in ranges of values of such that || ¿ =2 and || ¡ =2. In this section, we study the system in the region || ≈ =2 and show the presence of chaos in the dynamics. We prove the existence of chaos by evaluating the maximum Lyapunov exponent as a function of using Wolf’s method [5]. The results can be seen in Fig. 3 where the maximum exponent has positive values when || ≈ =2 indicating the presence of chaos in that range.
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Fig. 2. Oscillation frequency d=dt for several initial conditions. There is coexistence between cluster states where the frequency is constant and a chaotic state.
We noted before (Section 3.1) that when || ≈ =2, the system does not converge to any stable solution after being randomly initialized. However, we can still obtain the cluster states with the proper initial conditions. The next step is to study their linear ˙ before and after stability. For this purpose, we can analyze the time evolution of the perturbation. We use the same perturbation as before (Eq. (11)) which is turned on at time step 1000. The states with 0 or with remain stable, while the rest of the solutions are unstable and converge to a state where the oscillators are ˙ changes very fast. We have a new not synchronized and the average frequency attractor in the system with no temporal or spatial order. We checked that when the system is initialized randomly it converges to this state. This suggests that it has the largest basin of attraction. Unless is exactly equal to =2 stable cluster states do exist, but as we approach this value their basins of attraction become smaller and eventually collapse. In this way we see that the chaotic state coexists with stable and unstable cluster states.
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Fig. 3. Maximal Lyapunov exponent vs. . We used an algorithm introduced by Wolf. In the range of close to ≈ =2 the exponent has positive values and the dynamics is chaotic.
5. Chaos control The aim of controlling chaos is to suppress it by stabilizing a certain unstable reference state. For this purpose, we need to perturb some or all of the oscillators. Our goal is to perturb the minimum number of them, exploiting the spatial symmetry of the reference state. Our control method is an adaptation of a well-known feedback scheme [17]. The diIerence is based on the periodic nature of the variables, the phases. We can interpret the control of chaos as the synchrony between the oscillators of the system and the oscillators of the reference state. Next, we show the concept with two oscillators, and afterwards we give the algorithm we used for controlling chaos in the ring and Enally we present the results. 5.1. Synchronizing two oscillators The phases i of the oscillators are deEned in the range [0; 2 ]. We can imagine the periodic path of an oscillator as a circle and the phase as the angle in radians that deEnes its position on it.
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Let suppose that we have two oscillators, whose phases 1 and 2 evolve as: di (12) = fi (i ); i = 1; 2 : dt Our goal is to get the second oscillator synchronized with the Erst one. For that purpose, we can modify its dynamics in the following way: d2 = f2 (2 ) + P() ; (13) dt where P is a perturbation proportional to where = 1 − 2 :
(14)
This means that the second oscillator will change its instantaneous frequency as a function of the angular deviation relative to the Erst oscillator. For optimizing the perturbation we need P to be proportional to the smallest angular distance on the circumference between the two oscillators. We deEned P such that || ¡ → P = ; || ¿ →
¿ 0 → P = ( − 2 ) ; ¿ 0 → P = ( + 2 ) ;
(15)
where is a parameter that Ex the strength of the perturbation. If the second oscillator has the same phase as the Erst one, the perturbation (or feedback) goes to zero and the feedback is activated every time that the phase diIerence between them is diIerent from zero. However, if the second term on the right hand side of Eq. (13) is very small compared to the Erst, then the oscillators are synchronized. 5.2. Controlling chaos in the ring Controlling chaos in a system of oscillators, stabilizing one of the unstable solutions embedded in the chaotic attractor, can be interpreted as synchronizing the N oscillators of the system, each one of which oscillates at its own instantaneous frequency, with the other oscillators sharing a common frequency (the reference state). For this purpose it may not be necessary to perturb all the oscillators of the system, this depends speciEcally on the spatial pattern of the reference state. For the ring of oscillators we used the following algorithm: di = −[sin(i − i+1 + ) + sin(i − i−1 + ) − 2 sin ] dt + ik P(Sk (t); i (t));
(i = 1 : : : N ); k ∈ C ;
(16)
where C represents the set of all the oscillators to be perturbed and S (t) is the reference state. The parameter P is deEned as 15 and = Sk − k . 5.3. Results If ≈ =2, a chaotic attractor is present in the system (Section 4). We Erst apply the algorithm introduced in the last section, in such a way that all the oscillators are
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perturbed. In this situation, we can easily stabilize any of the unstable solutions. The transient time between the switch on of the feedback and the stabilization time of the solution is short (≈ 102 steps for a system with N =18). Although this result is positive, it is easy to perturb all the oscillators when we deal with relative small systems. For larger systems we may not have access to all the oscillators of the system. In this case, we want to control the dynamics with the minimum number of controllers. For this purpose we have to choose the oscillators to be perturbed in a way that is related to the spatial pattern of the reference state. We consider a system N = 18 with = 1:53 and we want to stabilize a reference state with Nc = 6. We can see its phase spatial pattern in Fig. 4(a). In Figs. 4(b) – (e), we show the possible distribution for six controllers and in Fig. 4(f) for three controllers. In cases (b) and (d), where the controllers are concentrated in sets that are separated from each other by distances of the order of the characteristic length of the spatial pattern of the reference state, we could stabilize the solution after a transient of the order of 103 time steps. After the transient, the feedback reaches a maximal value of 1% with respect to the Erst term of Eq. (16). In case (c) where we put the controllers in symmetric positions but not in concentrated sets, we stabilize the system after a transient of 107 steps and we End that the maximal feedback value reaches a 5% of the Erst term. Locating the controllers in an asymmetric way as is shown in case (e), the solution could not be stabilized. In case (f) we used just three controllers separated by a distance equal to the characteristic length of the reference state. The reference state is stabilized with a longer transient. In all the cases we had to tune the value of . In Fig. 5, we show in a space-time diagram the results. The control is switched on after 1000 time steps. Once the unstable solution has been stabilized, all the oscillators have a common frequency and the value of is zero. The feedback reaches its maximal value at the moment it is turned on. In Fig. 5(a) we can see clearly the transition from chaos to the periodic reference state for Nc = 9. In Fig. 5(b) we control the same system but using as the reference state the solution with Nc = 3. From our experience controlling chaos in systems of diIerent sizes, we infer that for stabilizing an unstable state, using the minimum number of oscillators in the shortest possible time, we must locate the controllers in equal concentrated sets that are separated between them by distances of the order of the characteristic length of the spatial pattern of the reference state. Furthermore, as the number of oscillators in each set is smaller, the transient is longer. As expected, for larger systems, we would have longer transients. We illustrate the result showing a system of N =24 where we control chaos stabilizing a reference state with Nc = 4. There are 4 controllers separated by a characteristic length of the reference state as shown in Fig. 6(a). In Figs. 6(b) and (c) we show the average feedback and the standard deviation of the average frequency. The control is switched on after 1000 time steps. In Fig. 7, we show the results for a larger system (N = 120). In this case the reference state has a value of Nc = 20 and the controllers are separated by the characteristic length of the reference state (the same as the previous case). We can see that the transient time is longer than the smaller system. Finally, we put the controllers in random positions (see Fig. 8(a)). The system
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Fig. 4. vs. number of the oscillator in the ring. (a) Space pattern of the solution with Nc = 3. (b) – (e) Possible positions for six controllers. A big circle points the position of a controller. (f) Distribution for 3 controllers.
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Fig. 5. Space–time graph. The values of the phases of the oscillators are represented in gray levels. Its minimum value is white and the maximum is black. The control is switched on at the time step 1000. (a) Chaos is controlled with the reference state with Nc = 9 and (b) with Nc = 3.
could not be synchronized as we can see from the behavior of the standard deviation (see Fig. 8(c)) which remains always diIerent from 0. Comparing the cases of N = 24 with N = 120, we can see that it is possible to control the system in a similar time scale keeping constant the fraction of controllers among all the elements of the network and the spatial symmetry of the reference state. We checked this in larger systems (up to N = 300). We also note that if we have more oscillators in the ring, we have more unstable solutions embedded in the chaotic attractor. This means that we have a larger number of possible reference states that could replace the chaotic behavior. 6. Discussion In this paper, we studied basically the possibility of controlling chaos in a system of oscillators with Erst neighbor coupling. We used an adaptation of the well known feedback scheme for controlling chaos, Erst introduced in [17]. Although we study a ring of oscillators, the control method can be easily generalized to higher dimensional systems. We characterized the system by its solutions or cluster states, which can be deEned by the phase diIerence between neighboring oscillators. As the number of
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Fig. 6. System with N = 24 oscillators. (a) Reference state with Nc = 4 and positions of the controllers with big circles. (b) Average feedback vs. time steps. The system is initialized randomly and the control is switched on at the time step 1000. (c) Standard deviation of the frequency vs. time steps.
oscillators in the system grows we have more cluster states. We found that for given ranges of values of the dynamics of the system become chaotic. We can use any of the cluster states as a reference periodic state to be stabilized using our control scheme in such a way that the chaos is controlled. We can also interpret chaos control in the oscillator system as the synchrony between the oscillators of the system and those of the reference state. For this purpose, we need to perturb all or some of the oscillators of the system. Our goal is to perturb a minimum number. We found that we have to choose the oscillators to be perturbed in a way that takes into account the spatial symmetry of the reference state. We End that the minimum number of controllers is equal to the number of oscillators in one cluster, Nc . In order to minimize the time necessary to control the system, one can introduce more controllers with the same geometry, for instance increasing the number of controllers in each set of oscillators.
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Fig. 7. System with N = 120 oscillators. (a) Reference state with Nc = 20 and positions of the controllers with big circles. (b) Average feedback vs. time steps. The system is initialized randomly and the control is switched on at the time step 1000. (c) Standard deviation of the frequency vs. time steps.
The question of the relation between the symmetry of the system and the target state and the number and location of controllers has already been discussed in Ref. [18] using methods of linear feedback control theory. It is found that for coupled map lattices in one-dimensional systems, two control parameters are enough to control the system independently of the symmetry of the target state. In our method the feedback is not linear. We need a larger number of controllers but we can control the target state beginning from an arbitrary initial condition, which is not guaranteed by the linear feedback control theory. The method we have presented in this paper can be easily applied to higher dimensional systems, if one knows the structure of the target states. In some cases this is known, e.g. the spiral waves in bi-dimensional systems [4]. Preliminary results indicate that our results still hold qualitatively for those systems. We can also mention the possibility of using singular value decomposition of the spatio temporal
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Fig. 8. (a) Reference state with Nc = 20. Random positions of the controllers with big circles. (b) Average feedback vs. time steps. The system is initialized randomly and the control is switched on at the time step 1000. (c) Standard deviation of the frequency vs. time steps.
solution [19]. This method would allow us to generalize the results to situations without symmetries. Acknowledgements We thank H. Wio and V. Grunfeld for a careful reading of the manuscript. This work was partially supported by grant PICT97 03-00000-00131 from ANPCyT for G.M. References [1] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, New York, 1984. [2] D. Hansel, G. Mato, C. Meunier, Phys. Rev. E 48 (1993) 3470.
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