Abstract
Using the usual phase in plane, we propose a general method to design coupling between systems that will exhibit phase synchronization. Numerical results are shown for Lorenz systems. Phase synchronization and antiphase synchronization are equally probable depending on initial conditions. A new network with Lorenz phase synchronized system is obtained.
1. Introduction
Nonlinear dynamics brought to light the rich dynamics of nonlinear systems including chaotic behavior. By coupling two nonlinear systems a new nonlinear system with an increased dimension and richer dynamics is obtained. The couplings can be suggested by reality (like diffusive coupling) or can result from the design in order to achieve a certain task. Depending on the coupling at least two distinctive cases can occur after the coupling is on: synchronization [1, 2] and Amplitude Death (AD) [3, 4]. Synchronization is a fundamental phenomenon observed in nature and science and used in engineering and medicine [1, 2]. There are several types of synchronization (complete synchronization, lag synchronization, projective synchronization, and generalized synchronization including amplification/attenuation). We mention here just some results that show how to design the couplings for synchronization [5–13]. A weaker type of synchronization is phase synchronization (PS) as locking of the phase of the coupled systems, while their amplitudes remain uncorrelated. The great interest for this kind of synchronization has been triggered by the seminal paper [14]. For chaotic systems it is difficult to introduce a phase. In [14], the phase was introduced by using the analytic signal concept. Many results on PS are observed for chaotic attractors with rather coherent phase dynamics [1, 2]. Phase coherent dynamics is a strong limitation for applications. A step forward is the study of the effect of noncoherence on the onset PS [15] by using a phase based on the general idea of curvature of an arbitrary curve. Also PS was obtained by using active control and by nonlinear state observer algorithm [16, 17] when the difference of two homologous variables is constant. However, a concern has arisen recently related to the use of the analytic signal concept [18].
In this paper we adopt a new strategy. By using the usual definition of phase in plane we deduce a system that governs the dynamics of the phase differences. From this dynamics we search for coupling terms that assure that the phase differences become asymptotically zero. The method is general but it does not assure that we can find coupling terms for any systems. In addition, it is not assured that the dynamics of the coupled system is bounded. For Lorenz system, the method gives good results. Coupling for two Lorenz systems suggests an all-to-all coupling for a network of PS and antiphase synchronized (APS) systems. Such a network can have different outputs depending on initial conditions.
The organization of this paper is as follows. In Section 2, the general theory is presented, where the system that governs the dynamics of the phase differences is deduced. Section 3 gives the results of the theory applied to Lorenz systems with numerical results. Section 4 contains several comments. Finally, conclusions are presented in Section 5.
2. Theory
Without losing generality, we consider 3 dimensional systems with like Lorenz, Chen, Rössler, Sprott, and so forth. , where is a constant matrix and is a vector which contains nonlinear terms. Two systems that can be coupled by terms and have equations:We need to determine and in such a manner that the two systems will reach PS in at least two pairs of variables (let us say ). As phase we adopt the usual phase in plane : . The cross product has the following components:The component depends on phase difference: means (Amplitude Death) or (Amplitude Death) or ; that is, PS or , that is, APS.
The dynamics of can be obtained by using the original dynamics (1a) and (1b) and the components given by (2):orwhere with containing all nonlinear terms, , and coupling terms of and .
Now we can give a partial answer to the question [15] “It is still unknown which type of chaotic oscillators can be synchronized in phase”? The general and partial answer is that system (4a), system (4b), and system (4c) or system (5) contains information on the PS (with phase . This is the main result of this research and is general because (4a), (4b), and (4c) are general. The method of finding and depends on the system and can be more or less complicated (see Section 4).
If we find coupling terms and in such a manner that and , then we have PS or AD as above. If one line in (in (5)) contains just zeros, then we can add and subtract and look for coupling terms to have .
If by choosing and we can have = = 0 and the system is and and if , , assures that and , then all variables are PS (or APS).
If we find and , where all = 0, , and (in (5)) is a Hurwitz matrix (a matrix with negative real part eigenvalues), then , , and all variables will be PS and amplitudes are uncorrelated.
At the eye inspection of the temporal pattern of two systems PS or APS (let us say and and and ) we will observe (see Figures 1 and 2; numerical results) that the homologous variables cross the time axis at the same time as expected from (2). This conclusion can have practical importance at the investigation of real data.
(a)
(b)
(a)
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To check numerically PS we can use coefficients (, , and ) with and similar expressions for and . For , we have For PS, and means APS. Again, this result can be of practical use. There can be identified two limitations of this general strategy. Firstly, the coupling terms do not assure that the dynamics of the coupled systems are bounded. Secondly, the coupling terms are not unique, but this can be an advantage when the first limitation occurs.
3. Phase Synchronized Lorenz Systems
We apply the above strategy to Lorenz systems , , and In this case, Equation (4c) with (9) gives withWe need to find coupling terms to have .
Three cases can be identified.
Case 1. One hasIn this case, subtracting the 3rd equations of (1a) and (1b), we have so and, from (12), and, from (11), . The final state will be PS or APS depending on initial conditions like in Figure 1. Numerical results are given in Figure 1 with Two examples are given: one PS and one APS from different initial conditions. means PS or APS; means PS and means APS.
Case 2. One hasNumerical results are given in Figure 2.
Case 3. One hasNumerical results (not shown) are qualitatively similar to Case 2.
4. Discussions
In our numerical experiments, we did not find any AD and we do not have any indications how or if it can be found. In some cases one oscillation is much smaller than the other but not AD. This is worthy to be deeply investigated.
For Chen systems, the results are very similar to Lorenz system.
Unfortunately, we did not find reasonable coupling terms for Rössler system using a direct method like we used for Lorenz system. Other methods should be tried. Here we need to clarify this problem. We report here that we did not find coupling terms for Rössler by simple eye inspection as we obtained for Lorenz system. In fact, in general, we need to find coupling terms in (4a), (4b), and (4c) or (5) which assures that components of go to zero. As an ultimate method the OPCL (open-plus-closed-loop) method [19] can be used which can offer driving for any system to reach any desired dynamics. The coupling terms can be complicated but for engineering applications it is important that they can be possible. This will be elaborated (with examples) elsewhere.
For system from Sprott collection [20] (, , and ), , , and . Equation (4c) in this case is ; . So we subtract and add and find the coupling terms from as Numerical results (not shown) for initial condition give bounded dynamics just for .
Initial conditions in connection with PS and APS were considered also in [21]. Antiphase synchronization (for diffusive coupling) was investigated in [22].
Case 1 for Lorenz systems (13) suggests a network of PS and APS systems given by
The coupling here is a nonlinear global coupling used recently [23]. Here is the number of systems coupled all-to-all. In this case all systems of the network organize in 2 PS clusters and APS between them. Such a network can have a very interesting output. If the output is , , and , then the output will depend on the initial conditions. If all systems are PS, then the output will be maximum, and if the two PS clusters (that are APS one to another) are with nearly equal numbers of systems, then the output will be very small in comparison to the previous case. Figure 3 gives numerical results for Lorenz systems starting from different initial conditions. For similar initial conditions the output is very different. As a common characteristic, all variables cross the time axis at the same moment.
(a)
(b)
(c)
This property of crossing time axis at the same moment can be searched by eye inspection in real signal (like EKG, EEG, EMG, or others) and if it happens then a model with this type of coupling can be used.
Based on this general strategy other results will be presented elsewhere.
5. Conclusions
In this paper new strategy for phase synchronization is used a. Phase is not calculated effectively but a coupling is searched which assures that the two current phases are equal. Phase is the usual definition in plane: (. The main result with possible practical importance is that the PS or APS signals pass zero value at the same moment at an eye inspection. The present phase synchronized Lorenz systems can be used as a tool for signal processing of real data as complete synchronized Lorenz systems have been used recently [24]. Deviation from measures the difference between two signals; results can be much better because is adimensional.
Competing Interests
The authors declare that they have no competing interests.