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Finite Kuramoto System with Shear and Symmetry

Arindam Chakraborty, Anirban Ray, A. Roy Chowdhury*

High Energy Physics Division, Department of Physics, Jadavpur University, Kolkata, India

Abstract: In this paper we study the locally coupled finite Kuramoto Oscillator system with shear under periodic boundary condition. We also show how analytical solutions can be obtained from symmetry conditions. Existence of attractors and bifurcation patterns are revealed in an elegant way through these solutions. The synchronized regions are identified in the parameter space and critical situations are discussed. an important outcome of present analysis is the derivation of analytical form of the Poincare map.

Keywords: discreet Kuramoto oscillators, bifurcation, Poincare section

Received July 23, 2013; Revised August 02, 2013; Accepted August 05, 2013

Citation: Arindam Chakraborty, Anirban Ray, A. Roy Chowdhury, Finite Kuramoto System with Shear and Symmetry. International Journal of Physics, 1 (4). 94-100. January 2013. doi:10.12691/ijp-1-4-3

1. Introduction

Diverse biological and physical phenomena from the division of a cell to the cycle of an engine piston, can all be modeled by an ensemble of oscillators due to their basic periodic properties [1]. In nature we observe many instances of systems coupled with each other in various manners. Such a collection is best represented in the elegant formulation of Kuramoto oscillators system [2]. There has been a growing interest on globally coupled limit cycle oscillators [3-8]. Of course majority of the analysis related to Kuramoto model deals with the situation of infinite number of such coupled oscillators [9,10]. The natural frequencies of the elements of the population are drawn from predefined prescribed distribution. It has been shown analytically that the system with a mean-field coupling exhibits a phase transition in the thermodynamic limit, if the coupling exceeds a certain

On the other hand, as shown by Strogatz and Mirollo [11], an infinite 1D system does not synchronize. However, in real-life application only finite systems appear. It is, therefore, interesting to see whether finite 1D Kuramoto systems may synchronize. Recently some work has been done in this direction by [12,13,14,15,16]. Another interesting configuration is Kuramoto systems with shear. This was first described by Y. Kuramoto himself [17]. Recently, Diego Paz´o et. al. studied the same system under distributed shear [18,19]. With this point in mind we have studied a finite system of phase scillators(actually N = 3 and latter we have extended it to the case of N = 4) with the effect of shear, and where their configuration is endowed with a property of symmetry. It is quite interesting to observe in many of the situations described in the following the existence of aforesaid symmetry helps to segregate the analytical situations in easier way. Since we started with case of three oscillators, in the later stage the number was increased to N = 4 to have an idea about the increase of the size of the population. We then analyzed the bifurcation pattern as the parameters were varied. In this connection we may mention the work of E.C.Paulo e.t al.[20] who also studied a finite system of such oscillators but without shearing force. Here we have tried to visualize this type of system which is under the influence of such forces.

Figure 1. Variation of real and imaginary part of solution of Eq. (15)

2. Formulation

We consider a set of Kuramoto Oscillators under the influence of shearing force, which can be written as

[picture] | (1)

where is the frequency of the -th oscillator. stands for the coupling between the -th and the -th oscillator. is the corresponding number of such oscillators. is the shear of the -th oscillator which actually express the nonsynchronicity or the dependence of the frequency on the amplitude. Here, stands for

[picture] | (2)

We start by choosing that we have only three such oscillators coupled with each other and consider the fixed points obtained by the condition . The fixed point condition can be written as;

[picture] | (3)

where, stands for

[picture] | (4)

with and

whence Eq. (3) can be written as

[picture] | (5)

so from Eq. (4)

[picture] | (6)

with On the other hand Eq. (6) can also be written as

[picture] | (7)

which can be seen to lead to

[picture] | (8)

Now let us specialize to the case of and impose periodic boundary condition. For this case we have chosen , ; , . and . With this restrictions we can easily observe that Eq. (5) implies

[picture] | (9)

or

[picture] | (10)

If we set ,then we get the second condition;

[picture] | (11)

At this becomes;

[picture] | (12)

[picture] | (13)

We can analyze the function regarding their extremum. For example in case of ,if we set , then

[picture] | (14)

leads to the equation

[picture] | (15)

Solution of Eq. (14) are given by

[picture] | (16)

Then, four roots can be numerically studied as shown in Figure 1. Both the real and imaginary part of the solutions are plotted separately. It shows that for values of ,two roots become purely real. Thus, has two minimum for in the real plane. Similarly, has a single minima at These three minima give birth to a pair of solutions for . There values will help to ascertain the nature of the eigenvalues for the study of the stability. Also for the four roots (16) reduce to two. In the first case, eigenvalues are

[picture] | (17)

[picture] | (18)

where as in the second case they turns out to be

[picture] | (19)

[picture] | (20)

For this system with few oscillators it is not necessary to impose symmetry in frequency distribution to obtain the result numerically. But it is done to make the analytical solution easier. We show the above calculation with asymmetric frequency distribution in the following example. If natural frequencies are randomly chosen, it is always possible to assign positive values to and and set (for . With this configuration fixed point solutions are described by the following set of equations,

[picture] | (21)

[picture] | (22)

We find solutions of above equations in the same fashion. Solutions are given below

[picture] | (23)

[picture] | (24)

Figure 2. (a) Plot of functions where stable solutions are shown in solid black lines and the unstable solutions in dashed red lines and (thin blue lines) for and . (b) Plot of functions of where stable solutions are shown in solid black lines and the unstable solutions in dashed red lines and (thin blue lines) for and . (c)Variation of functions on vs. plane. (d) Stable region(black) and unstable region (gray) of functions on vs. plane. (e) Plot of functions where stable solutions are shown in solid black lines and the unstable solutions in dashed red lines and (thin blue lines) for and . (f) Plot of functions of where stable solutions are shown in solid black lines and the unstable solutions in dashed red lines and (thin blue lines) for and . (g)Variation of functions on vs. plane. (h) Stable region(black) and unstable region (gray) of functions on vs. plane.

In Figure 2 explicit form of for the first set of solution. For example, in Figure we see the explicit structure of as functions of for some value . The region of stability is shown in the surface plot given in . The similar situation for is given in Figure and Figure . On the other hand the second set solution leads to the situations for and depicted respectively in Figure . To make the result more transparent we have given the two dimensional plots of plane for the and second kind of solution in Figure 3 and Figure 4. In this figure the black region gives the stable zones, whereas grey one indicate the instability and the white zones represents the negative values. One may add that the first maximum at represents a saddle(stable)node bifurcation. Atthis point two solutions are born, one stable with and one unstable solution with in the limit . The solutions generated at the minima and are unstable as the real parts of the eigenvalues are positive at this points. One should note that is a saddle node bifurcation and is a node node bifurcation. Unstable bifurcation points are indicated with the rightmost branch of . At this point a pitch-fork bifurcation takes place. To have more insight in the bifurcation pattern we consider the two equations

[picture] | (25)

[picture] | (26)

where , under the assumption and . Also we have set . To proceed further we substitute

[picture] | (27)

[picture] | (28)

and obtain the following equations

[picture] | (29)

[picture] | (30)

In equation, we have used the following nomenclature,

[picture] | (31)

The Jacobian pertaining to the equation is

[picture] | (32)

It is now convenient to define the trace and determinant of as, the help of grey shedding, while stable one by black. It is now possible to observe that solutions are starting from

[picture] | (33)

From which after some algebraic manipulation we can deduce that the following relation holds when the synchronization condition is satisfied

[picture] | (34)

which is an ellipse in the parameter plane . Since the determinant of the Jacobian is less than zero at this fixed point, the point is a saddle which changes from stable to unstable direction when the boundary of the ellipse is crossed. On the otherhand when we get

[picture] | (35)

along with

[picture] | (36)

From which one gets

[picture] | (37)

Figure 3. (a) Stable region(black) and unstable region (gray) of functions on vs. plane. (b) Stable region(black) and unstable region (gray) of functions on vs. plane

This means that for all synchronization states loose stability due to Andronov-Hopf bifurcation. On the otherhand saddle node bifurcation lines are determined implicitly by

[picture] | (38)

and

[picture] | (39)

So implies

[picture] | (40)

and

[picture] | (41)

Figure 4. Variation of Eq. (33) in parameter plane

So may imply either or . Similar conclusion can be drawn for . So for any value of q.

3. Poincare Map

Further we calculate the Poincare section analytically of coupled Kuramoto equations of dimension . We transfer the system in to co-ordinate using relation .

[picture] | (42)

[picture] | (43)

Where

[picture] | (44)

[picture] | (45)

[picture] | (46)

[picture] | (47)

[picture] | (48)

where and . Hence the Poincare map becomes

[picture] | (49)

4. N=4 with Specular Symmetry

Based on the same assumption as the previous case, we analyzed a ring of oscillators where the configuration of frequency follow the prescribed symmetry. We assumed the situation when oscillator oscillator present specular symmetry on the natural frequency with and both and positives. Now, we have a set of equations

[picture] | (50)

[picture] | (51)

We replace , and , . Thus Eqns. (??) transform into a new set of coupled equations in and .

[picture] | (52)

[picture] | (53)

For

[picture] | (54)

[picture] | (55)

The eigenvalues of the Jacobian is given by

[picture] | (56)

For

[picture] | (57)

[picture] | (58)

The eigenvalues of the Jacobian is given by

[picture] | (59)

5. Two Natural Frequencies

Here

[picture] | (60)

[picture] | (61)

[picture] | (62)

[picture] | (63)

[picture] | (64)

and

[picture] | (65)

6. Conclusion

In the present work, we studied a exhaustive description of locally coupled Kuramoto model with shear under presence of symmetries and explored to a large extent the analyticity limits of the system. In the following we summarize the salient features. Starting with the analysis of small system, we gave a full description of synchronized region, along with determination of critical coupling for some case. We showed how fixed point may be obtained from critical synchronization coupling for some case, and we showed how the fixed points may be obtained from general configuration of natural frequency independent of system size. We also deduced the bifurcation scenario along with Poincare section analytically. But it becomes difficult to pursue the analytical result for higher dimensional system when shear is present in the system.

The author ARC is thankful to UGC(University Grant Commission) BSR(Basic Science Research) grant of Govt. of India and AR is thankful to CSIR(Council of Scientific and Industrial Research) for SRF(Senior Research Fellowship).

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