Bifurcation analysis of visual angle model with anticipated time and stabilizing driving behavior

In the light of the visual angle model (VAM), an improved car-following model considering driver's visual angle, anticipated time and stabilizing driving behavior is proposed so as to investigate how the driver's behavior factors affect the stability of the traffic flow. Based on the model, linear stability analysis is performed together with bifurcation analysis, whose corresponding stability condition is highly fit to the results of the linear analysis. Furthermore, the time-dependent Ginzburg-Landau (TDGL) equation and the modified Korteweg-de Vries (mKdV) equation are derived by nonlinear analysis, and we obtain the relationship of the two equations through the comparison. Finally, parameter calibration and numerical simulation are conducted to verify the validity of the theoretical analysis, whose results are highly consistent with the theoretical analysis.     

Epidemiological models with quadratic equation for endemic equilibria—A bifurcation atlas

The existence and occurrence, especially by a backward bifurcation, of endemic equilibria is of utmost importance in determining the spread and persistence of a disease. In many epidemiological models, the equation for the endemic equilibria is quadratic, with the coefficients determined by the parameters of the model. Despite its apparent simplicity, such an equation can describe an amazing number of dynamical behaviors. In this paper, we shall provide a comprehensive survey of possible bifurcation patterns, deriving explicit conditions on the equation's parameters for the occurrence of each of them, and discuss illustrative examples.     

Bogdanov-Takens Bifurcation in a Host-parasitoid Model

In this paper, we study a host-parasitoid model with Holling II Functional response, where we focus on a special case: the carrying capacity K2 for parasitoids is equal to a critical value r 1η. It is shown that the model can undergo Bogdanov-Takens bifurcation. The approximate expressions for saddle-node, Homoclinic and Hopf bifurcation curves are calculated. Numerical simulations, including bifurcation diagrams and corresponding phase portraits, are also given to illustrate the theoretical results.     

Periodic Solutions of the Duffing Differential Equation Revisited via the Averaging Theory

We use three different results of the averaging theory of first order for studying the existence of new periodic solutions in the two Duffing differential equations $\ddot y+ a \sin y= b \sin t$ and $\ddot y+a y-c y^3=b\sin t$, where $a$, $b$ and $c$ are real parameters.     

参外联合激励下的簇发振荡及其机理

当周期激励频率远小于系统固有频率时，会存在快慢耦合效应，与单项激励不同，参外联合激励不仅会导致快子系统平衡曲线和分岔行为的复杂化，也会产生一些特殊的非线性现象，为此，本文以两耦合Hodgkin-Huxley细胞模型为例，引入周期参外联合激励，探讨在频域不同尺度耦合时该系统的簇发振荡的特点及其分岔机制．通过建立相应的快慢子系统，得到慢变参数变化下的快子系统的各种分岔模式以及相应的分岔行为，结合转换相图，揭示耦合系统随激励幅值变化时的动力学行为及其机理．研究表明，在激励幅值较小时，系统表现为概周期振荡，两频率分别近似于快子系统平衡曲线由Hopf分岔引起的两稳定极限环的振荡频率．概周期解随激励幅值的增加进入簇发振荡，导致这些簇发振荡的主要原因是在慢变参数变化的部分区间内，存在唯一稳定的平衡曲线，使得系统的轨迹逐渐趋向该平衡曲线，产生沉寂态，并随着慢变参数的变化，由分岔进入激发态．同时，快子系统中参与簇发振荡的稳定吸引子随激励幅值的变化也会不同，导致不同形式的簇发振荡．另外，与单项激励下的情形不同，联合激励时快子系统的部分稳定吸引子掩埋在其它稳定吸引子内，从而失去对簇发振荡的影响．     

Delay driven Hopf bifurcation and chaos in a diffusive toxin producing phytoplankton‐zooplankton model

This paper deals with a diffusive toxin producing phytoplankton‐zooplankton model with maturation delay. By analyzing eigenvalues of the characteristic equation associated with delay parameter, the stability of the positive equilibrium and the existence of Hopf bifurcation are studied. Explicit results are derived for the properties of bifurcating periodic solutions by means of the normal form theory and the center manifold reduction for partial functional differential equations. Numerical simulations not only agree with the theoretical analysis but also exhibit the complex behaviors such as the period‐3, 5, 6, 7, 8, 11, and 12 solutions, cascade of period‐doubling bifurcation in period‐2, 4, quasi‐periodic solutions, and chaos. The key observation is that time delay may control harmful algae blooms (HABs). Moreover, numerical simulations show that the chaotic states induced by the period‐doubling bifurcation are purely temporal, which is stationary in space and oscillatory in time. The investigations may provide some new insights on harmful phytoplankton blooms.     

Dynamical analysis of age-structured pertussis model with covert infection

An age-structured pertussis model with covert infection is proposed to understand the effect of covert infection on the recurrence of pertussis. It is found that vaccination only for young children does not have a decisive effect on whooping cough control. It is shown that although the vaccine coverage rate is relatively high, the model has a backward bifurcation for a larger covert infection rate. In addition, sufficient conditions for the disease-free steady state to be globally asymptotically stable are obtained.