The tracking dynamics is described by the following system of differential equations
Xn = 0
Xn-1 = аR(1 – –Xn-2)
xn xn—1
Xn-2 = aR(1 — X7-—Xn—3) (G.1)
xn xn 1
X1 )
Xn—Xn — 1 h
The first equation is already linear and provides an eigenvalue at the origin. Linearizing the other n — 1 equations around the equilibrium point defined by
r
Xi = — i r = 1
n
yields the (n — 1) x (n — 1) Jacobian matrix:
—2n n 0 0 … 0 n nn 0 …0 —n 0 — n n … 0
—n 0 ………….. 0 n
In order to compute the eigenvalues of this Jacobian matrix, start by observing that it can be written as
J = n(—I + A) (G.3)
where I is the identity of order n — 1 and A[(n — 1) x (n — 1)] is the matrix
The characteristic polynomial of J is obtained from
1
det(sl — J ) = det(sl + I — A) n
= det((s + 1) I — A)
= det(a I — A),
where the change of variable
a := s + 1 (G.5)
has been made.
Since A is a matrix in companion form, the coefficients of its characteristic polynomial (in a) are the symmetric of the entries of the first column of A. Therefore:
det (a I — A) = an—1 + an—2 + ••• + 1
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n + 1 terms of a geometric series):
it is concluded that
an — 1
det(a I — A) = a = 1
a—1
Hence, the eigenvalues of A satisfy
an — 1 = 0 and a = 1
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Recalling the change of variable (G.5), it is concluded that the eigenvalues of 1J are located over a circumference of radius 1, centered at —1. Furthermore, since
where П(M) denotes the eigenvalues of a generic matrix M, it follows that the eigenvalues of J are located over a circumference of radius n centered at —n.
In order to get the eigenvalues of the tracking dynamics, to the n — 1 eigenvalues of J one must add another eigenvalue, located at the origin, associated with xn.
In conclusion, there are n eigenvalues of the tracking dynamics, that divide the circumference in equal parts. □
