2.5.2 Quasiperiodic solutions of LLG dynamics under circularly polarized field

Let us focus our analysis on the quasiperiodic solutions (limit cycles in the rotating frame). In order to establish the existence, the number and the locations of the limit cycles we can exploit the fact that $\alpha$ is generally a small parameter $\lesssim 0.1$. Thus, we can start our analysis by considering the case $\alpha=0$ which can be easily treated because the dynamical system (2.62) admits the following integral of motion (similar to energy conservation (1.103)):

$$g(\textbf{{m}})= \frac{1}{2}D_z m_z^2 +\frac{1}{2}D_\bot _\bot^2 - _{a\bot} m_x- ( _{az}-\omega) m_z.$$ (2.64)

It is interesting to notice that the function $g(\textbf{{m}})$ satisfies the following equation along the trajectory of the dynamical system

$$\frac{dg}{dt}=\alpha \left[ \omega(\textbf{{m}}\times \mathbf{e}_... ...\textbf{{m}}}{dt}\right\vert^2 \right]= -\alpha {\cal P}(\textbf{{m}}) ,$$ (2.65)

where ${\cal P}(\textbf{{m}})$ is the ``absorbed power'' function which is defined by the opposite of the above expression in square bracket. This function will be instrumental in the following to give an energy interpretation of limit cycles.

The phase portrait for $\alpha=0$ is given by the contour lines of the function $g(\textbf{{m}})$. To give a planar representation of the phase portraits, we use the stereographic variables $w_1,w_2$ introduced in section 2.3. In Fig. 2.16, the phase portrait is represented on the $(w_1,w_2)$-plane for the case of a thin film.

Figure: Phase portrait of conservative system on the stereographic plane $w_1=m_x/(1+m_z)$, $w_2=m_y/(1+m_z)$. Value of the parameters: $\alpha=0$, $D_z=1$, $D_\bot =0$, h$_{az}=0.6$, h$_{a\bot}=0.15$, $\omega =1.1$.

This phase portrait is characterized by three centers C$_1$, C$_2$ and C$_3$ (outside Fig. 2.16) and a saddle $S$ with two homoclinic orbits $\Gamma_1$ and $\Gamma_2$. When the small damping is introduced, almost all closed trajectories around centers are slightly modified and collectively form spiral-shaped trajectories toward attractors. There are only special trajectories which remain practically unchanged under the introduction of the small damping. Two of these trajectories are indicated by $Q_1$ and $Q_2$ in Fig. 2.16. These trajectories can be found by using a perturbation technique which is generally called Poincarè-Melnikov function method [43]. This perturbative approach is reported, for a generic 2D dynamical system, in Appendix B. In order to apply this technique it is convenient to transform Eq. (2.62) in the following perturbative form ($\alpha$ is a small parameter).

$$\frac{d \textbf{{m}}}{d t}= \textbf{f}_0(\textbf{{m}}) +\alpha \textbf{f}_1(\textbf{{m}},\alpha)$$ (2.66)

where

$$\textbf{f}_0(\textbf{{m}})= -\textbf{{m}}\times \left( \textbf{h}... ...a \mathbf{e}_{z}\right)= \textbf{{m}}\times \nabla_\textbf{{m}} g(\textbf{{m}})$$ (2.67)

$$\textbf{f}_1(\textbf{{m}},\alpha)= \frac{\alpha}{1+\alpha^2}\text... ...{1}{1+\alpha^2} \textbf{{m}}\times (\textbf{{m}}\times \textbf{h}_{\text{eff}})$$ (2.68)

For $\alpha=0$ the dynamical system is integrable and trajectories are given by $g(\textbf{{m}})=g_0$ with $g_0$ varying in the appropriate range. In addition, the vector field $\textbf{f}_0(\textbf{{m}})$ is hamiltonian and, as it can be derived from Eq. (2.67), it is divergeless on the unit sphere $\Sigma$ : $\nabla_\Sigma \cdot \textbf{f}_0(\textbf{{m}})=0$.

Figure 2.17: Sketch of a portion of the phase portrait of LLG equation around a center equilibrium.

The technique is based on the extraction of a Poincarè map [43] (associated to an arbitrarly chosen line $S$ transveral to the vector field, as sketched in Fig. 2.17) of the perturbed system by using an expansion in terms of the perturbation parameter $\alpha$, around $\alpha=0$. The zero order term of this expansion is the identity since for $\alpha=0$ all trajectories (except separitrices) goes back to the initial point (see Fig. 2.16). The first order term of the expansion with respect to $\alpha$ is proportional to the Melnikov function which, in the case of divergenceless unperturbed vector field, is given by the following integral along the trajectories of the unperturbed system (see Eq. (B.39) in Appendix B)

$$M(g_0)=\int_0^{T_{g_0}} \!\!\!\! \textbf{f}_0(\textbf{{m}}_{g_0}(t)) \wedge \textbf{f}_1(\textbf{{m}}_{g_0}(t),0) dt$$ (2.69)

where $\textbf{{m}}_{g_0}(t)$ is the trajectory of the unperturbed system with $g(\textbf{{m}}_{g_0}(t))=g_0$, $T_{g_0}$ is its period and

$$\textbf{f}_0(\textbf{{m}}) \wedge \textbf{f}_1(\textbf{{m}},0)=\t... ... \textbf{f}_0(\textbf{{m}}) \times \textbf{f}_1(\textbf{{m}},0) \right) \quad.$$ (2.70)

By using the expressions of $\textbf{f}_0(\textbf{{m}})$ and $\textbf{f}_1(\textbf{{m}},0)$ and appropriate algebraic manipulations one can derive that:

$$M(g_0)=-\int_0^{T_{g_0}}\left[ \omega(\textbf{{m}}_{g_0} \times \... ..._0}}{dt}-\left\vert \frac{d\textbf{{m}}_{g_0}}{dt}\right\vert^2 \right]dt .$$ (2.71)

The last equation can be also transformed in the following line-integral form which permits one to compute $M(g_0)$ without deriving the time dependence of $\textbf{{m}}_{g_0}(t)$:

$$M(g_0)=-\oint_{g=g_0} \textbf{{m}}\times \textbf{h}_{\text{eff}}\cdot d \textbf{{m}} .$$ (2.72)

Figure: Two branches of the Melnikov function vs the value of $g_0$: $g_{0;Q_1}$ and $g_{0;Q_2}$ correspond to conservative trajectories $g=g_{0;Q_1}$ and $g=g_{0;Q_2}$ in Fig. 2.16, which become limit cycles $Q_1$ and $Q_2$ in the perturbed system.

Periodic orbits of the dissipative system are given by the zeros of the Melnikov function. In Fig. 2.18, the Melnikov function computed from Eq. (2.69) is plotted versus the value of $g_0$ and the zeros of $M(g_0)$, which correspond to the trajectories $g=g_{0;Q_1}$ and $g=g_{0;Q_2}$ in Fig. 2.16, are emphasized. In Fig. 2.19, by sketching the phase portrait for the dissipative case ( $\alpha=0.05$), we have then verified that the limit cycles ($Q_1$, $Q_2$) predicted by the theory are preserved under the introduction of the damping. Let us notice that the introduction of damping transformed centers in foci $F_1$(unstable), $F_2$(stable) and $F_3$(unstable) and disconneted the homoclinic trajectories ($L_1$ and $L_2$ are the separatrices). It is interesting to notice that the Melnikov function given by Eq. (2.71) can be rewritten as

$$M(g_0)=\int_0^{T_{g_0}} {\cal P}(\textbf{{m}}_{g_0}(t)) dt \quad.$$ (2.73)

In this respect, it is possible to give a physical interpretation of limit cycles: the limit cycle arise from those unperturbed trajectories on which there is an average balance between ``dissipation'' ( ${\cal P}(\textbf{{m}}) \geq 0$) and ``absorption'' ( ${\cal P}(\textbf{{m}}) \leq 0$) of energy.

By using the technique we have just illustrated, it is possible to predict the existence and the number of the limit cycles in a certain interval of values of $\alpha$ around $\alpha=0$. The stability of the limit cycles can be obtained by studying the sign of the derivative of the Melnikov function at its zeros [43]: a limit cycle is stable for positive derivative (in our case $Q_1$ is stable, see Figs. 2.18-2.19), unstable for negative derivative (in our case $Q_2$ is unstable, see Figs. 2.18-2.19). Finally the shape of the limit cycles can be estimated by taking, as first order approximation, the unperturbed trajectories corresponding to the values of the energy function $g(\textbf{{m}})$ where the Melnikov function vanishes.

Figure: Phase portrait of dissipative system. The parameters are the same as in Fig. 2.16 except for $\alpha=0.05$.