To investigate stability of the linear stochastic delay differential equations (12) consider the conditions for a matrix A = [parallel][a.sub.ij][parallel] to be the Hurwitz matrix.

A 3x 3-matrix A is the Hurwitz matrix if and only if

have established an approach to generate multiscroll attractors via destabilization of piecewise linear systems based on

Hurwitz matrix. First the authors present some results about the abscissa of stability of characteristic polynomials from linear differential equations systems; that is, they consider Hurwitz polynomials.

If A is a Hurwitz matrix, then there are some real constants K [greater than or equal to] 1 and [lambda] > 0 such that

In addition, since P is a Hurwitz matrix, it follows from Lemma 6 that there exist constants [K.sub.1], [K.sub.2] > 1 and [[lambda].sub.1] > 0 and [[lamda].sub.2] > 0 such that

As A is Hurwitz matrix, there exists only positive definite matrix P with respect to the following Lyapunov function, which satisfies

where A is Hurwitz matrix and [xi] = [E.sup.T]PB and B = [I.sub.m] satisfy (26).

[A.sub.0] is a

Hurwitz matrix, the papers [13, 14] proposed sufficient conditions for the [alpha]-stability of system (1) in terms of the solution of a scalar inequality involving the eigenvalues, the matrix measures and the spectral radius of the system matrices.

Thus, given a matrix gain L, such that A--LC is a

Hurwitz matrix, it is always possible to asymptotically stabilize prediction error (12) for a constant time delay [tau] < [[tau].sup.*].

Assume that the system [??](t) = Ax(t) is asymptotically stable (i.e., A [member of] [R.sup.nxn] is a Hurwitz matrix); then one has the following:

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Because D > 0 and A is a Hurwitz matrix, so Y > 0 and

(1) [A.sub.011] is

Hurwitz matrix, and the last column of [A.sub.0] is zero vector, which is [A.sub.012] = 0 and [A.sub.022] = 0.