![]() In order to demonstrate how this can be accomplished, let us first recall that all the essential information about a generic conservative portrait can be coded by listing the number of saddles, central regions, and fixed points present in this portrait. Therefore, one expects that the classification of conservative phase portraits discussed in Chapter 4 can be instrumental in the description of spin-transfer effects as well. This fact implies that the phase portraits of the dynamics can be viewed as perturbations of the corresponding conservative phase portraits obtained for the same value of the field and α = β = 0. It has been previously mentioned that an essential role in the analysis of spin-transfer-driven magnetization dynamics is played by the fact that α and β are small quantities, of the order of 10 −2 or less. The final result of this study will be the theoretical stability diagram shown in Figs 9.3 and 9.4. The knowledge of the location and nature of these lines is essential for the interpretation of spin-transfer experiments. These bifurcation conditions result in bifurcation lines in the ( h a x, β / α )-plane. However, there exist bifurcation conditions where the control-parameter change leads to qualitative changes in the phase portrait, namely changes in the number and stability of fixed points and limit cycles. When a small change in the control parameters induces correspondingly small, continuous modifications in the phase portrait, the portrait is said to be structurally stable. For each point in the ( h a x, β / α ) control plane, we shall determine the corresponding phase portrait for the magnetization dynamics and follow the evolution of this portrait as the field or the current is varied. The parameter β / α will be used to represent the current density. Claudio Serpico, in Nonlinear Magnetization Dynamics in Nanosystems, 2009 9.4 Phase Portraits and Bifurcationsīased on the discussion presented in the previous sections, we shall now determine the stable stationary states as well as the steady-state self-oscillations present in a spin-transfer device subject to the external field h a x along the free-layer easy axis and the spin-polarized current density J e. Again, trajectories cannot cross except at the origin.Giorgio Bertotti. Also, they cannot pass through the origin yet must be continuous lines- the only way that can happen is if the curve close to the origin so that a line toward the origin turns into a line away from the origin. ![]() In that case, trajectories close to the line with negative eigenvalue must be toward the origin and trajectories close to the line with positive eigenvalue must be away form the origin. The "interesting" case is when one eigenvalue is positive and the other negative. ![]() If both eigenvalues were positive, you would have the same thing but with arrows pointing away from the origin. Essentially you just have lines through the origin with arrows pointing toward the origin. If you wanted to be very accurate, you could draw more lines close to the line with eigenvalue -2 to indicate that but probably your teacher will not require it. Flow close to the line with eigenvalue -2 must be twice as fast as that near the line with eigenvalue -1. Here, that means that trajectories must all move toward the origin. Also solutions are "continuous" in the sense that trajectories through near by points must be similar. Except where the right side of the system is 0 (at the origin, the equilibrium point) there must be only a single trajectory through any point other than the origin. The crucial point here is the "existance and uniqueness" theorem.
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