Characterization of the Eigenvalues of the Linearized Tracking Dynamics

Proof of Proposition 6.1

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

Подпись: X1 = a R(1X1 )

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:

Подпись:—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

Подпись:

Подпись: N Подпись: 1 — a image499

n + 1 terms of a geometric series):

it is concluded that

Подпись:Подпись:an — 1

det(a I — A) = a = 1

a—1

Hence, the eigenvalues of A satisfy

an — 1 = 0 and a = 1

Подпись: П(J) = nn
image503

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. □

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