The Landau-Lifshitz-Gilbert equation is nonlinear. Therefore, the
magnetic systems will present phenomena associated with nonlinearity
such as stochastic resonance, nonlinear bifurcations or chaos [Fernández 00].
Furthermore, it is not possible to find exact analytical solutions
except in very simple cases. The dissipation term is explicitly
nonlinear, but even in the absence of this term the dynamics are
nonlinear, because the effective field is dependent on the
To illustrate we can examine the case of a single domain particle
with uniaxial anisotropy and easy axis in the Z direction under
constant applied field. The energy of such particle is:
where the first term is uniaxial anisotropy and the external field is
measured in units of .
We can choose
without loss of generality and the applied field becomes where is the field angle
with the axis Z. The energy minimum makes an angle with the Z axis. We
can change from the laboratory reference frame to a reference frame where the minimum position always
corresponds to the Z' axis. The transformation is a rotation of angle around the Y axis.
In the small amplitude approximation, where the components and of the magnetization
vector are supposed to be small in comparison to
component , the equation of motion in the
new frame of reference
and the variables ,
, and are determined by equations:
In the frictionless case, the solution up to second harmonic of Eq. (2.44) is
where is the nonlinear instantaneous frequency, is the amplitude of motion, is the zero harmonic
and is the second
harmonic. The zero harmonic ,
which value can be obtained from
describes the displacement of the center of the magnetization vector
rotation due to the nonlinear effect and is always positive.
In the considered case of constant applied field, , we
arrive to an expression for the instantaneous frequency
where . The comparison of Eq. (2.49) with numerical simulations shows good
for small values of the amplitude of the precessional cycle (see Fig. 2.10).
This can be expected since our approximation is only valid for small
amplitudes. In the case of
we have an
axially symmetric case and Eq. (2.49) reduces
The nonlinear rotation frequency of the
magnetization vector for a small ferromagnetic particle under the
applied magnetic field versus
the square of
the analytical result, Eq. (2.49), and the
numerical simulation of the LLG
The applicability of the solution can be extended including
additional terms. The existence of corrections to the orbit
center allows us to conclude that under numerical simulation of the
LLG equation the precessional terms should not be neglected at long
simulation times in the micromagnetic simulations.