[en] This paper presents new results related to the coexistence of function-based hybrid synchronization types between non-identical incommensurate fractional-order systems characterized by different dimensions and orders. Specifically, a new theorem is illustrated, which ensures the coexistence of the full-state hybrid function projective synchronization (FSHFPS) and the inverse full-state hybrid function projective synchronization (IFSHFPS) between wide classes of three-dimensional master systems and four-dimensional slave systems. In order to show the capability of the approach, a numerical example is reported, which illustrates the coexistence of FSHFPS and IFSHFPS between the incommensurate chaotic fractional-order unified system and the incommensurate hyperchaotic fractional-order Lorenz system.

[en] In this paper, a two-dimensional discrete fractional reduced Lorenz map is achieved by utilizing discrete fractional calculus. By adopting the bifurcation diagrams, chaos diagram, and phase portraits, the chaotic dynamics of the two-dimensional discrete fractional reduced Lorenz map are analyzed. Complexity of this fractional map versus parameters is discussed by employing the $C_0$ algorithm. It is found that this fractional map has rich dynamical behaviors. In addition, it also shows that the $C_0$ algorithm provides a parameter choice method for practice applications of discrete fractional maps. Finally, some numerical simulations are given to demonstrate the effectiveness of the proposed results.

[en] In this paper, stochastic effect on the spread of the infectious disease with saturated incidence rate and the special transfer from infectious is discussed. The threshold dynamics is explored for the case of relatively small noise. Our results show that large noise will cause the elimination of the disease, which will help suppress the spread of the disease.

[en] We investigate the Riemann problem for one-dimensional Temple class, obtain the Riemann solutions containing delta-shock wave, and discover that each variable of this system contains the Dirac delta function in different region. By the interaction of the delta-shock wave with the elementary waves we present the generalized Riemann problem for this system and construct the global solutions. By studying the limits of the solutions as perturbed parameter ε approaches zero we observe that the Riemann solutions are stable for such perturbations of the initial data.

[en] In this paper, an adaptive neural network (NN) synchronization controller is designed for two identical strict-feedback chaotic systems (SFCSs) subject to dead-zone input. The dead-zone models together with the system uncertainties are approximated by NNs. The dynamic surface control (DSC) approach is applied in the synchronization controller design, and the traditional problem of “explosion of complexity” that usually occurs in the backstepping design can be avoided. The proposed synchronization method guarantees the synchronization errors tend to an arbitrarily small region. Finally, this paper presents two simulation examples to confirm the effectiveness and the robustness of the proposed control method.

[en] A spatiotemporal discrete predator–prey system is investigated for understanding the pattern self-organization on the route to chaos. The discrete system is modelled by a coupled map lattice and shows advection of populations in space. Based on the conditions of stable stationary states and Hopf bifurcation, Turing pattern formation conditions are determined. As the parameter value is changed, self-organization of diverse patterns and complex phase transition among the patterns on the route to chaos are observed in simulations. Ordered patterns of stripes, bands, circles, and various disordered states are revealed. When we zoom in to observe the pattern transition in smaller and smaller parameter ranges, subtle structures for transition process are found: (1) alternation between self-organized structured patterns and disordered states emerges as the main nonlinear characteristic; (2) when the parameter value varies in the level from 10⁻³ to 10⁻⁴, a cyclic pattern transition process occurs repeatedly; (3) when the parameter value shifts in the level of 10⁻⁵ or below, stochastic pattern fluctuation dominates as essential regularity for pattern variations. The results obtained in this research promote comprehending pattern self-organization and pattern transition on the route to chaos in spatiotemporal predator–prey systems.

[en] In this paper, the deterministic and stochastic eco-epidemiological models with modified Leslie–Gower functional response are studied. For a deterministic system, the stability of disease-free equilibrium and positive equilibrium is studied. For a stochastic system, we verify that the system admits a unique positive global solution starting from any positive initial value, and we establish the conditions of extinction for infected prey population and strong persistence in mean for all species. We also show the system has a stationary distribution under some conditions. Finally, some numerical simulations are carried out to illustrate the main results.

[en] We investigate general HIV infection models with three types of infected cells: latently infected cells, long-lived productively infected cells, and short-lived productively infected cells. We incorporate three discrete or distributed time delays into the models. Moreover, we consider the effect of humoral immunity on the dynamical behavior of the HIV. The HIV-target incidence rate, production/proliferation, and removal rates of the cells and HIV are represented by general nonlinear functions. We show that the solutions of the proposed model are nonnegative and ultimately bounded. We derive two threshold parameters which fully determine the existence and stability of the three steady states of the model. Using Lyapunov functionals, we establish the global stability of the steady states of the model. The theoretical results are confirmed by numerical simulations.

[en] The paper is devoted to investigating a class of neutral stochastic integro-differential equations with impulses driven by fractional Brownian motion. By establishing two new impulsive integral inequalities which improve the inequalities established by Li (Neurocomputing 177:620-627, 2016) and Long et al. (Stat. Probab. Lett. 82(9):1699-1709, 2012), attracting and quasi-invariant sets of the system are obtained. Moreover, exponential stability of the mild solution is established with sufficient conditions.

[en] In this paper, a new SIRS epidemic model which considers the influence of information intervention and environmental noise is studied. The study shows that information intervention and white noise have great effects on the disease. First, we show that there is global existence and positivity of the solution. Then, we prove that the stochastic basic production ${\mathcal{R}}_s$ is a threshold which determines the extinction or persistence of the disease. When the intensity of noise is large, we obtain ${\mathcal{R}}_s<1$ and the disease will die out. When the intensity of noise is small, then ${\mathcal{R}}_s>1$ and a sufficient condition for the existence of stationary distribution is obtained, which means the disease is prevalent. Finally, the main results are illustrated by numerical simulations.