Bifurcation diagram

About: Bifurcation diagram is a research topic. Over the lifetime, 8379 publications have been published within this topic receiving 139664 citations.

Criterion to identify Hopf bifurcations in maps of arbitrary dimension

Abstract: 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.

Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations I. Global organization of bistable periodic solutions.

Abstract: The Hodgkin-Huxley equations (HH) are parameterized by a number of parameters and shows a variety of qualitatively different behaviors depending on the parameter values. We explored the dynamics of the HH for a wide range of parameter values in the multiple-parameter space, that is, we examined the global structure of bifurcations of the HH. Results are summarized in various two-parameter bifurcation diagrams with I ext (externally applied DC current) as the abscissa and one of the other parameters as the ordinate. In each diagram, the parameter plane was divided into several regions according to the qualitative behavior of the equations. In particular, we focused on periodic solutions emerging via Hopf bifurcations and identified parameter regions in which either two stable periodic solutions with different amplitudes and periods and a stable equilibrium point or two stable periodic solutions coexist. Global analysis of the bifurcation structure suggested that generation of these regions is associated with degenerate Hopf bifurcations.

Fractional-order PD control at Hopf bifurcations in delayed fractional-order small-world networks

Abstract: Bifurcation and control of fractional-order systems are still an outstanding problem. In this paper, a novel delayed fractional-order model of small-world networks is introduced and several topics related to the dynamics and control of such a network are investigated, such as the stability, Hopf bifurcations, and bifurcation control. The nonlinear interactive strength is chosen as the bifurcation parameter to analyze the impact of the interactive strength parameter on the dynamics of the fractional-order small-world network model. Firstly, the stability domain of the equilibrium is completely characterized with respect to network parameters, delays and orders, and some explicit conditions for the existence of Hopf bifurcations are established for the delayed fractional-order model. Then, a fractional-order Proportional-Derivative (PD) feedback controller is first put forward to successfully control the Hopf bifurcation which inherently happens due to the change of the interactive parameter. It is demonstrated that the onset of Hopf bifurcations can be delayed or advanced via the proposed fractional-order PD controller by setting proper control parameters. Meanwhile, the conditions of the stability and Hopf bifurcations are obtained for the controlled fractional-order small-world network model. Finally, illustrative examples are provided to justify the validity of the control strategy in controlling the Hopf bifurcation generated from the delayed fractional-order small-world network model.

Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates.

Abstract: The transmission of infectious diseases has been studied by mathematical methods since 1760s, among which SIR model shows its advantage in its epidemiological description of spread mechanisms. Here we established a modified SIR model with nonlinear incidence and recovery rates, to understand the influence by any government intervention and hospitalization condition variation in the spread of diseases. By analyzing the existence and stability of the equilibria, we found that the basic reproduction number [Formula: see text] is not a threshold parameter, and our model undergoes backward bifurcation when there is limited number of hospital beds. When the saturated coefficient a is set to zero, it is discovered that the model undergoes the Saddle-Node bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation of codimension 2. The bifurcation diagram can further be drawn near the cusp type of the Bogdanov-Takens bifurcation of codimension 3 by numerical simulation. We also found a critical value of the hospital beds bc at [Formula: see text] and sufficiently small a, which suggests that the disease can be eliminated at the hospitals where the number of beds is larger than bc. The same dynamic behaviors exist even when a ≠ 0. Therefore, it can be concluded that a sufficient number of the beds is critical to control the epidemic.