Design of impulsive controllers and impulsive control strategy for the Mittag-Leffler stability behavior of fractional gene regulatory networks

Lyapunov approach to manifolds stability for impulsive Cohen-Grossberg-type conformable neural network models

In this paper, motivated by the advantages of the generalized conformable derivatives, an impulsive conformable Cohen-Grossberg-type neural network model is introduced. The impulses, which can be also considered as a control strategy, are at fixed instants of time. We define the notion of practical stability with respect to manifolds. A Lyapunov-based analysis is conducted, and new criteria are proposed. The case of bidirectional associative memory (BAM) network model is also investigated. Examples are given to demonstrate the effectiveness of the established results.

Random periodic sequence of globally mean-square exponentially stable discrete-time stochastic genetic regulatory networks with discrete spatial diffusions

<abstract><p>This paper regards the dual effects of discrete-space and discrete-time in stochastic genetic regulatory networks via exponential Euler difference and central finite difference. Firstly, the global exponential stability of such discrete networks is investigated by using discrete constant variation formulation. In particular, the optimal exponential convergence rate is explored by solving a nonlinear optimization problem under nonlinear constraints, and an implementable computer algorithm for computing the optimal exponential convergence rate is given. Secondly, random periodic sequence for such discrete networks is investigated based on the theory of semi-flow and metric dynamical systems. The researching findings show that the spatial diffusions with nonnegative intensive coefficients have no influence on global mean square boundedness and stability, random periodicity of the networks. This paper is pioneering in considering discrete spatial diffusions, which provides a research basis for future research on genetic regulatory networks.</p></abstract>

Stability analysis and synchronized control of fuzzy Mittag-Leffler discrete-time genetic regulatory networks with time delays

Exponential Euler differences for semi-linear differential equations of first order have got rapid development in the past few years and a variety of exponential Euler difference methods have become very significant researching topics. In allusion to fuzzy genetic regulatory networks of fractional order, this paper firstly establishes a novel difference method called Mittag-Leffler Euler difference, which includes the exponential Euler difference. In the second place, the existence of a unique global bounded solution and equilibrium point, global exponential stability and synchronization of the derived difference models are investigated. Compared with the classical fractional Euler differences, fuzzy Mittag-Leffler discrete-time genetic regulatory networks can better depict and retain the dynamic characteristics of the corresponding continuous-time models. What’s more important is that it starts a new avenue for studying discrete-time fractional-order systems and a set of theories and methods is constructed in studying Mittag-Leffler discrete models.

Synchronization in a Multiplex Network of Nonidentical Fractional-Order Neurons

Fractional-order neuronal models that include memory effects can describe the rich dynamics of the firing of the neurons. This paper studies synchronization problems in a multiple network of Caputo–Fabrizio type fractional order neurons in which the orders of the derivatives in the layers are different. It is observed that the intralayer synchronization state occurs in weaker intralayer couplings when using nonidentical fractional-order derivatives rather than integer-order or identical fractional orders. Furthermore, the needed interlayer coupling strength for interlayer near synchronization decreases for lower fractional orders. The dynamics of the neurons in nonidentical layers are also considered. It is shown that in lower fractional orders, the neurons’ dynamics change to periodic when the near synchronization state occurs. Moreover, decreasing the derivative order leads to incrementing the frequency of the bursts in the synchronization manifold, which is in contrast to the behavior of the single neuron.

Lyapunov Approach for Almost Periodicity in Impulsive Gene Regulatory Networks of Fractional Order with Time-Varying Delays

This paper investigates a class of fractional-order delayed impulsive gene regulatory networks (GRNs). The proposed model is an extension of some existing integer-order GRNs using fractional derivatives of Caputo type. The existence and uniqueness of an almost periodic state of the model are investigated and new criteria are established by the Lyapunov functions approach. The effects of time-varying delays and impulsive perturbations at fixed times on the almost periodicity are considered. In addition, sufficient conditions for the global Mittag–Leffler stability of the almost periodic solutions are proposed. To justify our findings a numerical example is also presented.