The proposed method of setting the PI controller for IPDT and UFOPTD processes turned out to be extremely simple to apply and efficient while using the boundary conditions for selecting the conjugate complex
If 8[w.sub.1] (1 + [theta]) -3(2 + [theta]) < 0, the violation occurs with the modulus of a pair of conjugate complex eigenvalues being one.
If the characteristic polynomial has conjugate complex numbers, the moduli of conjugate complex numbers are less than one.
Although a single eigenvalue becoming minus one and the modulus of a pair of conjugate complex eigenvalues being equal to one are necessary conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation, respectively, they constitute strong evidence combined with numerical simulations which show that such bifurcations do occur .
where: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [[bar.Y].sub.0]] & [[bar.Y].sub.1]]: two initial conditions, k & l: the real and imaginary part of the conjugate complex roots respectively with k = [bar.
This expectation is proved right since D=-2, indicating by this way the existence of two conjugate complex characteristic roots, the magnitude of which equals to [[lambda].sub.1,2]=
When the characteristic equation has two conjugate complex roots, namely [[lambda].sub.1, 2] = k [+ or -] li, the complementary solution of the basic difference [E.sub.q].
We also assume that the zeros of [[Omega].sub.m] are either real or occurring in conjugate complex pairs, and we assume that [[Omega].sub.m] has constant sign on supp(d[Lambda]).
the sum being extended over all pairs of conjugate complex poles, where
The poles are all simple and occur in conjugate complex pairs on the line Im [Zeta] = [Eta] at odd integer multiples of [Pi] away from the real axis.