Pair Of Imaginary Roots Research Articles

Bifurcation analysis and chaos for a double-strains HIV coinfection model with intracellular delays, saturated incidence and Logistic growth

In this paper, a class of virus-to-cell HIV model with intracellular delays, saturated incidence and Logistic growth is proposed to characterize the interaction between two types HIV strains, i.e., wild-type and drug-resistant strains. First, a series of threshold criteria on the locally and globally asymptotic stability of (infection-free, dominant, coexistence) equilibria are discussed based on the basic reproduction number R0. Furthermore, a detailed Hopf bifurcation analysis is performed on the coexistence equilibrium using two delays as bifurcation parameters. We find that the Hopf bifurcations induced by double-strains are evidently different and more complicated than that of single strain, the former switches from stability (periodic branches) to un-stability (chaos) more frequently and earlier than the latter since double-strains would yield more pairs of imaginary roots in the characteristic equations. Meanwhile, the total viral load of double-strains would be higher than that of single-strain as well. The emergence of drug resistance imposes either negative or positive influences on the survival of wild-type strain, which would further facilitate the transmission of HIV.

Bifurcation of a feed forward neural network with delay and application in image contrast enhancement.

This paper is concerned with how the singularity and delay in a feed forward neural network affect generic dynamics and bifurcations. By computation of Hopf-pitchfork point in a two-parameter nonlinear problem, the mode interactions in two parameters bifurcations with a single zero and a pair of imaginary roots are considered. The codimension two normal form with Hopf-pitchfork bifurcations are given. Then, the bifurcation diagrams and phase portraits are obtained by computing the normal form. Furthermore, we find some interesting dynamical behaviors of the original system, such as the coexistence of two unstable nontrivial equilibria and a pair of stable periodic orbits, which are verified both theoretically and numerically. Through numerical simulation, we also find that this model has a special signal enhancement property in Hopf bifurcation state. Using this feed-forward neural network, we show that the gray scale picture contrast is strongly enhanced even if this one is initially very small.

Equivariant Hopf-Pitchfork Bifurcation of Symmetric Coupled Neural Network with Delay

This paper is concerned with how the symmetry and singularity of a system of differential equations affect generic dynamics and bifurcations. By computation of Hopf-pitchfork point in a two-parameter nonlinear problem satisfying a [Formula: see text]-symmetry condition, the mode interactions in two-parameter bifurcations with a single zero and two pairs of imaginary roots are considered. The codimension two normal form with equivariant Hopf-pitchfork bifurcations are given. Through analyzing the unfolding structure, local classification in the neighborhood of equivariant Hopf-pitchfork bifurcations point for the [Formula: see text]-symmetric is undertaken. A rich variety of dynamical and bifurcation behaviors is pointed out. Beyond a stable fixed point or a pair of stable fixed points, some interesting phenomena are also found, such as the coexistence of two periodic solutions which are verified both theoretically and numerically.