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We introduce a discretization process to discretize a modified fractional-order optically injected semiconductor lasers model and investigate its dynamical behaviors. More precisely, a sufficient condition for the existence and uniqueness of the solution is obtained, and the necessary and sufficient conditions of stability of the discrete system are investigated. The results show that the system’s fractional parameter has an effect on the stability of the discrete system, and the system has rich dynamic characteristics such as Hopf bifurcation, attractor crisis, and chaotic attractors.

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The paper presents a new urban public traffic supernetwork model by using the existing bus network modeling method, consisting of the conventional bus traffic network and the urban rail traffic network. We investigate the synchronization problem of urban public traffic supernetwork model by using the coupled complex network’s outer synchronization theory. Analytical and numerical simulations are given to illustrate the impact of traffic dispatching frequency and traffic lines optimization to the urban public traffic supernetwork balance.

A set of deterministic SIS models with density-dependent treatments are studied to understand the disease dynamics when different treatment strategies are applied. Qualitative analyses are carried out in terms of general treatment functions. It has become customary that a backward bifurcation leads to bistable dynamics. However, this study finds that finds that bistability may not be an option at all; the disease-free equilibrium could be globally stable when there is a backward bifurcation. Furthermore, when a backward bifurcation occurs, the fashion of bistability could be the coexistence of either dual stable equilibria or the disease-free equilibrium and a stable limit cycle. We also extend the formula for mean infection period from density-independent treatments to density-dependent ones. Finally, the modeling results are applied to the transmission of gonorrhea in China, suggesting that these gonorrhea patients may not seek medical treatments in a timely manner.

The paper studies the dynamical behaviors of a discrete predator-prey system with Holling type III functional response. More precisely, we investigate the local stability of equilibriums, flip bifurcation and Neimark-Sacker bifurcation of the model by using the center manifold theorem and the bifurcation theory. And analyze the dynamic characteristics of the system in two-dimensional parameter-spaces, one can observe the "cluster" phenomenon. Numerical simulations not only illustrate our results, but also exhibit the complex dynamical behaviors of the model. The results show that we can more clearly and directly observe the chaotic phenomenon, period-adding and Neimark-Sacker bifurcation from two-dimensional parameter-spaces and the optimal parameters matching interval can also be found easily. c 2016 all rights reserved.

The paper investigated the existence and stability of the Stochastic Hopf Bifurcation for a novel finance chaotic system with noise by the orthogonal polynomial approximation method, which reduces the stochastic nonlinear dynamical system into its equal deterministic nonlinear dynamical system. And according to the Gegenbauer polynomial approximation in Hilbert space, the financial system with random parameter can be reduced into the deterministic equivalent system. The parameter condition to ensure the appearance of Hopf bifurcation in this novel finance chaotic system is obtained by the Hopf bifurcation theorem. We show that a supercritical Hopf bifurcation occurs at systems' unique equilibriums s 0 . In addition, the stability and direction of the Hopf bifurcation is investigated by the calculation of the first Lyapunov coefficient. And the critical value of stochastic Hopf bifurcation is determined by deterministic parameters and the intensity of random parameter in stochastic system. Finally, the simulation results are presented to support the analysis.

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