The dynamic behaviors for fractional-order Cohen–Grossberg neural networks with time-varying delays (FCGNND) are studied in this paper. By introducing the Mittag-Leffler (ML) function, based on properties of fractional calculus, the differential mean-value theorem and Arzela–Ascoli theorem, we give some sufficient theorems to determine the boundedness, global Mittag-Leffler stability (GMLS) and global asymptotical [Formula: see text]-periodicity (GAP) for FCGNND. Finally, a numerical example is given to verify the effectiveness of the theorems.

The exponential synchronization (ES) of Cohen–Grossberg stochastic neural networks with inertial terms (CGSNNIs) is studied in this paper. It is investigated in two ways. The first way is using variable substitution to transform the system to another one and then based on the properties of i t ^ o integral, differential operator, and the second Lyapunov method to get a sufficient condition of ES. The second way is based on the second-order differential equation, the properties of calculus are used to get a sufficient condition of ES. At last, results of the theoretical derivation are verified by virtue of two numerical simulation examples.

The dynamic behavior of anti-periodic solutions for fractional-order inertia Cohen-Grossberg neural networks is investigated in the article. First, the fractional derivative with different orders is transformed to that with the same order by properly variable substitution; Second, a sufficient condition can ensure the solution is global Mittag-Leffler stability by using properties of fractional calculus and characteristics of Mittag-Leffler function; Moreover, a sufficient condition for the existence of an anti-periodic solution is given by constructing a system sequence solution that converges to a continuous function using Arzela-Asolitheorem. In the final, we verify the correctness of the conclusion by numerical simulation.

In this paper, we adopt two methods to study the problem. Initially, directly from the second-order differential equation, we obtain a sufficient condition (SC) for the mean square exponential stability (MSES) of the system at the equilibrium point by constructing a suitable function and applying some properties of calculus. Thereafter, the system is transformed into a vector form, using the basic solution matrix of linear differential equation, constructing a piecewise function and using the generalized Halanay one-dimensional delay differential inequality, another SC is given for the P-moment exponential stability (PMES) of the system at the equilibrium point. Finally, two examples are used to investigate the correctness and demonstrate that each SC has own advantage, the suitable theorem can be selected according to the parameters.

<abstract><p>The dynamic signal transmission process can be regarded as an anti-periodic process, and fractional-order inertial neural networks are widely used in signal processing and other fields, so anti-periodicity is also regarded as an important dynamic feature of inertial neural networks. This paper mainly studies the existence and Mittag-Leffler stability of anti-periodic solutions for a class of fractional-order inertial BAM neural networks with time-delays. By introducing variable substitution, the model with two different fractional-order derivatives is transformed into a model with only one fractional-order derivative of the same order. Using the properties of fractional-order calculus, the relationship between the fractional-order integral of the state function with and without time-delays is given. Firstly, the sufficient conditions for the boundedness and the Mittag-Leffler stability of the solutions for the system are derived. Secondly, by constructing the sequence solution of the function for the system and applying Ascoli-Arzela theorem, the sufficient conditions for the existence and Mittag-Leffler stability of the anti-periodic solution are given. Finally, the correctness of the conclusion is verified by a numerical example.</p></abstract>