Finite Kuramoto System with Shear and Symmetry

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2. Formulation

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

(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

(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;

(3)

where, stands for

(4)

with and

whence Eq. (3) can be written as

(5)

so from Eq. (4)

(6)

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

(7)

which can be seen to lead to

(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

(9)

(10)

If we set ,then we get the second condition;

(11)

At this becomes;

(12)

(13)

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

(14)

leads to the equation

(15)

Solution of Eq. (14) are given by

(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

(17)

(18)

where as in the second case they turns out to be

(19)

(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,

(21)

(22)

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

(23)

(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

vs.

plane. (d) Stable region(black) and unstable region (gray) of functions

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

vs.

plane. (h) Stable region(black) and unstable region (gray) of functions

vs.

plane.

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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

(25)

(26)

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

(27)

(28)

and obtain the following equations

(29)

(30)

In equation, we have used the following nomenclature,

(31)

The Jacobian pertaining to the equation is

(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

(33)

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

(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

(35)

along with

(36)

From which one gets

(37)

Figure 3. (a) Stable region(black) and unstable region (gray) of functions

vs.

plane. (b) Stable region(black) and unstable region (gray) of functions

vs.

plane

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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

(38)

and

(39)

So implies

(40)

and

(41)

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

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