Patterns governed by chemotaxis and time delay.

In this paper, sufficient conditions of Turing instability are established for general delayed reaction-diffusion-chemotaxis models with no-flux boundary conditions. In particular, we address the difficulty brought about by the time delay in investigating the Turing instability. These models involve the time delay parameter τ and the general density (concentration) function in chemotaxis terms with the chemotaxis parameter χ. Theoretical results reveal that the time delay parameter τ could determine the stability of positive equilibrium for ordinary differential equations, while the chemotaxis parameter χ could describe the stability of positive equilibrium for partial differential equations. In this fashion, the general conditions of Turing instability are presented. To confirm their validity, the delayed chemotaxis-type predator-prey model and the phytoplankton-zooplankton model are considered. It is found that these two models admit Turing instability, and the numerical results are in good agreement with the theoretical analysis. The obtained results are helpful in the application of the Turing pattern in models with time delay and chemotaxis.

Hopf bifurcation in delayed nutrient-microorganism model with network structure.

In this paper, we introduce and deal with the delayed nutrient-microorganism model with a random network structure. By employing time delay τ as the main critical value of the Hopf bifurcation, we investigate the direction of the Hopf bifurcation of such a random network nutrient-microorganism model. Noticing that the results of the direction of the Hopf bifurcation in a random network model are rare, we thus try to use the method of multiple time scales (MTS) to derive amplitude equation and determine the direction of the Hopf bifurcation. It is showed that the delayed random network nutrient-microorganism model can exhibit a supercritical or subcritical Hopf bifurcation. Numerical experiments are performed to verify the validity of the theoretical analysis.

Delay-dependent criterion for asymptotic stability of a class of fractional-order memristive neural networks with time-varying delays.

The Lyapunov-Krasovskii functional approach is an important and effective delay-dependent stability analysis method for integer order system. However, it cannot be applied directly to fractional-order (FO) systems. To obtain delay-dependent stability and stabilization conditions of FO delayed systems remains a challenging task. This paper addresses the delay-dependent stability and the stabilization of a class of FO memristive neural networks with time-varying delay. By employing the FO Razumikhin theorem and linear matrix inequalities (LMI), a delay-dependent asymptotic stability condition in the form of LMI is established and used to design a stabilizing state-feedback controller. The results address both the effects of the delay and the FO. In addition, the upper bound of the absolute value of the memristive synaptic weights used in previous studies are released, leading to less conservative conditions. Three numerical simulations illustrate the theoretical results and show their effectiveness.

Turing-Hopf bifurcation analysis in a superdiffusive predator-prey model.

The predator-prey model with superdiffusion is investigated in this paper. Here, the existence of Turing-Hopf bifurcation and the resulting dynamics are studied. To understand such a degenerate bifurcation in the anomalously diffusive system, the weakly nonlinear analysis is employed and the amplitude equations at the Turing-Hopf bifurcation point are obtained. Moreover, by analyzing the amplitude equations under suitable conditions, the abundant spatiotemporal dynamics are presented. In addition, to illustrate the theoretical analysis, some numerical simulations are carried out.

Patterns induced by super cross-diffusion in a predator-prey system with Michaelis-Menten type harvesting.

Turing instability and pattern formation in a super cross-diffusion predator-prey system with Michaelis-Menten type predator harvesting are investigated. Stability of equilibrium points is first explored with or without super cross-diffusion. It is found that cross-diffusion could induce instability of equilibria. To further derive the conditions of Turing instability, the linear stability analysis is carried out. From theoretical analysis, note that cross-diffusion is the key mechanism for the formation of spatial patterns. By taking cross-diffusion rate as bifurcation parameter, we derive amplitude equations near the Turing bifurcation point for the excited modes by means of weakly nonlinear theory. Dynamical analysis of the amplitude equations interprets the structural transitions and stability of various forms of Turing patterns. Furthermore, the theoretical results are illustrated via numerical simulations.

Stability and synchronization of fractional-order memristive neural networks with multiple delays.

The paper presents theoretical results on the global asymptotic stability and synchronization of a class of fractional-order memristor-based neural networks (FMNN) with multiple delays. First, the asymptotic stability of fractional-order (FO) linear systems with single or multiple delays is discussed. Delay-independent stability criteria for the two types of systems are established by using the maximum modulus principle and the spectral radii of matrices. Second, new testable algebraic criteria for ensuring the existence and global asymptotic stability of the system equilibrium point are obtained by employing the Kakutani's fixed point theorem of set-valued maps, the comparison theorem, and the stability criterion for FO linear systems with multiple delays. Third, the synchronization criterion for FMNN is presented based on the linear error feedback control. Finally, numerical examples are given demonstrating the effectiveness of the proposed results.

Design and implementation of grid multi-scroll fractional-order chaotic attractors.

This paper proposes a novel approach for generating multi-scroll chaotic attractors in multi-directions for fractional-order (FO) systems. The stair nonlinear function series and the saturated nonlinear function are combined to extend equilibrium points with index 2 in a new FO linear system. With the help of stability theory of FO systems, stability of its equilibrium points is analyzed, and the chaotic behaviors are validated through phase portraits, Lyapunov exponents, and Poincaré section. Choosing the order 0.96 as an example, a circuit for generating 2-D grid multiscroll chaotic attractors is designed, and 2-D 9 × 9 grid FO attractors are observed at most. Numerical simulations and circuit experimental results show that the method is feasible and the designed circuit is correct.

Stability and synchronization of memristor-based fractional-order delayed neural networks.

Global asymptotic stability and synchronization of a class of fractional-order memristor-based delayed neural networks are investigated. For such problems in integer-order systems, Lyapunov-Krasovskii functional is usually constructed, whereas similar method has not been well developed for fractional-order nonlinear delayed systems. By employing a comparison theorem for a class of fractional-order linear systems with time delay, sufficient condition for global asymptotic stability of fractional memristor-based delayed neural networks is derived. Then, based on linear error feedback control, the synchronization criterion for such neural networks is also presented. Numerical simulations are given to demonstrate the effectiveness of the theoretical results.

Linear matrix inequality criteria for robust synchronization of uncertain fractional-order chaotic systems.

This paper is devoted to synchronization of uncertain fractional-order chaotic systems with fractional-order α: 0 < α < 1 and 1 ≤ α < 2, respectively. On the basis of the stability theory of fractional-order differential system and the observer-based robust control, two sufficient and necessary conditions for synchronizing uncertain fractional-order chaotic systems with parameter perturbations are presented in terms of linear matrix inequality, which is an efficient method and could be easily solved by the toolbox of MATLAB. Finally, fractional-order uncertain chaotic Lü system with fractional-order α = 0.95 and fractional-order uncertain chaotic Lorenz system with fractional-order α = 1.05 are taken as numerical examples to show the validity and feasibility of the proposed method.