Hopf Bifurcation Criterion Research Articles

The complexity of an investment competition dynamical model with imperfect information in a security market

We present a nonlinear discrete dynamical model of investment competition with imperfect information for N heterogeneous oligopolists in a security market. In this paper, our focus is on a given three-dimensional model which exhibits highly rich dynamical behaviors. Based on Wen’s Hopf bifurcation criterion [Wen GL. Criterion to identify Hopf bifurcations in maps of arbitrary dimension. Phys Rev E 2005;72:026201–3; Wen GL, Xu DL, Han X. On creation of Hopf bifurcations in discrete-time nonlinear systems. Chaos 2002;12(2):350–5] and Kuznetsov’s normal form theory [Kuznetsov YA. Elements of applied bifurcation theory. New York: Springer-Verlag; 1998. p. 125–37], we study the model’s stability, criterion and direction of Neimark–Sacker bifurcation. Moreover, we numerically simulate a complexity evolution route: fixed point, closed invariant curve, double closed invariant curves, fourfold closed invariant curves, strange attractor, period-3 closed invariant curve, period-3 2-tours, period-4 closed invariant curve, period-4 2-tours.

An SIRS model with a nonlinear incidence rate

Abstract The global dynamics of an SIRS model with a nonlinear incidence rate is investigated. We establish a threshold for a disease to be extinct or endemic, analyze the existence and asymptotic stability of equilibria, and verify the existence of bistable states, i.e., a stable disease free equilibrium and a stable endemic equilibrium or a stable limit cycle. In particular, we find that the model admits stability switches as a parameter changes. We also investigate the backward bifurcation, the Hopf bifurcation and Bogdanov–Takens bifurcation and obtain the Hopf bifurcation criteria and Bogdanov–Takens bifurcation curves, which are important for making strategies for controlling a disease.

Criterion to identify Hopf bifurcations in maps of arbitrary dimension

The classical Hopf bifurcation criterion is stated in terms of the properties of eigenvalues. In this paper, a criterion without using eigenvalues is proposed for maps of arbitrary dimension. The parameter mechanism of Hopf bifurcation may be explicitly formulated on the basis of the criterion. A numerical example demonstrates that the proposed criterion is preferable to the classical Hopf bifurcation criterion in theoretical analysis and practical applications.