DEUDAS A CORTO PLAZO POR PRÉSTAMOS RECIBIDOS Y OTROS CONCEPTOS

Pω = ηαK ◦ P + η(R_zK) ◦ P −ηRxy(P ◦ cos(Θ)) ◦ P ◦ K + ηαRxy(P ◦ sin(Θ)) (7.34) where ◦ denotes the Hadamard product between two vectors.

This way we can have a system with 2N independent equations and 2N unknowns. This way, with the numerical solution of equation 7.33 and 7.34 I can calculate the phase between the oscillators, without the time consuming simulation of the differential equa- tions.

Figure 7.5: In this image we can see the oscillations in every component, when the coupling in Mxand My are different. As we can see the coupling in the third component is three-four

orders of magnitude smaller, than the oscillations in the first two components, this way it’s elimination will not affect the results significantly, and with the solution of equations 7.33 and 7.34 I can get the same results as I have obtained by the simulations.

7.3

Two coupled oscillator

7.3.1 Coupling only in the Mz component

As a special case I have examined two STOs coupled only by M_z. The equilibrium is determined by the following equation:

  areM_{z 1}∗ M_{z 2}∗  = − γ αη   −δ 1 l 1 r −δ     A1 A2   (7.35)

Given two different input current A1 and A2, the two uncoupled STOs have different

angular frequencies (see equation (7.13)). By exploiting the couplings lzand rzit is possible

to drive the STOs at the same angular frequency despite their different currents. This is easily achieved by imposing M_{z 1}∗ = M_{z 2}∗ , i.e. from the previous equation the following condition is derived:

A2 A1 = 1 r + δ 1 l + δ (7.36) I have tested this case of two coupled STOs also with the simulator and the measure- ments of the simulations are matching the theoretical data.

Even in case of large STO arrays, it is possible to design the coupling matrix P (or equivalently its inverse P−1) and the input currents w in such a way that the STOs drop into different clusters. Each cluster results to be defined by the frequency of the oscillations within a group. This may enhance the synchronization between oscillators in the same groups and prevent the synchronization between oscillators in different groups. Similarly, with the design of the matrix P I can create different clusters of STOs, with different frequencies considering a cellular array with input (different currents) only on the border cells, when the range of the inputs is known.

A simple example containing six STOs and their division into two groups can be seen in Fig. 7.6. Fig. 7.7 shows the limit cycles associated to each cluster.

Figure 7.6: The Mz(t) variables of six different STOs converge into two different clusters.

7.3.2 General coupling of two Oscillators

I would like to investigate the general coupling of two STOs, especially the effect of the input current and the coupling strength on the phase-shift after synchronization. As we have seen previously it is enough to solve equations 7.33 and 7.34 two calculate every parameters of an oscillator after the transient behavior. However these equations are not good to design an array with given properties, because they do not reveal the dependency of the phase-shifts on the input current. But as we can seen the input current can be found only in equation 7.33. If I substitute the proper parameters in this equation for the case of two, generally coupled oscillator we will have the following equations:

Figure 7.7: The two different limit cycles of the M vector can be seen according to the designed clusters. After transient time, the Mz(t) variables of six different STOs converge

into two different groups, i.e. the STOs synchronize in two clusters having different angular frequencies.

Figure 7.8: The dependency of the phase-shift based on the current. As it can be seen the dependency of the phase-shift is linear with respect to the change of the current difference between the oscillators.

Figure 7.9: The dependency of the phase-shift based on the coupling strength. I have ignored the case when the coupling strength is zero. As it can be seen the dependency of the phase-shift is hyperbolic.

αηδK1P1− cA1P1− ηαrzK1P1 =

sin(θ)rxyηP2+ cos(θ)rxyαηK2P1

(7.37)

Because θij = −θji, for the other oscillator I can write: αηδK2P2− cA2P2− ηαrzK2P2 =

−sin(θ)r_xyηP1+ cos(θ)rxyαηK1P2

(7.38)

Since, with an oscillation with a common frequency the planes of oscillation is close to each other I can approximate K₁ = K₂ as K and P₁ = P₂ as P Noting the difference of the input current on the two oscillators as A2− A1 = A∆ and subtract equation 7.3.2

from 7.3.2. We will have:

θ = asin( A∆

2rxyMs

) (7.39)

Since A∆

2rxyMs is between ±0.1 according to the physical parameters and the sinus func- tion is quasi-linear on this interval, I can also write:

θ = A∆

2r_xyMs

(7.40) This way I can examine the phase-shift as a function of the coupling strength and the input current, this equation is also suitable to design phase shift for previously given coupling strengths and/or input currents [8]. As it can be seen on Figure 7.3.2 and 7.3.2 the phase-shift depends linearly on the input current and hyperbolically on the coupling weight.