We write the generalized eigenvalue equation
in the form of a standard eigenvalue equation
From (7), (8), and (4), if the system has friction involved, then the complex eigenvalue equation
is attained as follows:
The eigenmodes are defined by the following standard eigenvalue equation
Let us now look at the stationary eigenvalue equation
, where we recall that n = 2m,
In Section 2 we present the model, compute equilibria, and derive the nth order eigenvalue equation
which we need in order to perform stability and bifurcation analysis.
Constructing the Jacobian matrix from (5) and evaluating it at the two equilibrium points, we obtain the eigenvalue equation
It is obvious that [[mu].sub.1] = 0 is the first eigenvalue and [[mu].sub.2] = [[pi].sup.2] is the second eigenvalue of the eigenvalue equation
For the upper component (i = 1), we write the potential parameter as [m.sub.1] = [V.sub.0] + [S.sub.0] which gives the following energy eigenvalue equation
from (14) as
Because the eigenvalue equation
for a given frequency, [omega], is quadratic in k, it can be transformed to a linear eigenvalue problem of the form:
Then, the eigenvalue equation
can be simplified as follows:
Next applying the Method of Moments gives a linear eigenvalue equation
that gives all the multi-band solutions simultaneously for a point in the first Brillouin zone.
The eigenvalue equation
of the equilibrium point is locally asymptotically stable.