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In this paper, we present multi-almost periodicity of a general class of discrete-time neural networks derived from a well-known semi-discretization technique, that is, coexistence and exponential stability of 2 N almost periodic sequence solutions of discrete-time neural networks subjected to external almost periodic stimuli. By using monotonicity and boundedness of activation functions, we construct 2 N close regions to attain the existence of almost periodic sequence solutions. Meanwhile, some new and simple criteria are derived for the networks to converge exponentially toward 2 N almost periodic sequence solutions. As special cases, our results can extend to discrete-time analogues of periodic or autonomous neural networks and hence complement or improve corresponding existing ones. Finally, computer numerical simulations are performed to illustrate multi-almost periodicity of semi-discretizations of neural networks.

In this paper, we employ a fixed point theorem due to Krasnosel’skii to attain the existence of periodic solutions for neutral-type neural networks with delays on a periodic time scale. Some new sufficient conditions are established to show that there exists a unique periodic solution by the contraction mapping principle.

In this paper, we investigate convergence dynamics of $2^N$ almost periodic encoded patterns of general neural networks (GNNs) subjected to external almost periodic stimuli, including almost periodic delays. Invariant regions are established for the existence of $2^N$ almost periodic encoded patterns under two classes of activation functions. By employing the property of $\mathscr{M}$-cone and inequality technique, attracting basins are estimated and some criteria are derived for the networks to converge exponentially toward $2^N$ almost periodic encoded patterns. The obtained results are new, they extend and generalize the corresponding results existing in previous literature.

An analytical scheme to determine the statistical behavior of a stochastic system including two terms of fractional derivative with real, arbitrary, fractional orders is proposed. In this approach, Green’s functions obtained are based on a Laplace transform approach and the weighted generalized Mittag–Leffler function. The responses of the system can be subsequently described as a Duhamel integral-type close-form expression. These expressions are applied to obtain the statistical behavior of a dynamical system excited by stationary stochastic processes. The numerical simulation based on the modified Euler method and Monte Carlo approach is developed. Three examples of single-degree-of-freedom system with fractional derivative damping under Gaussian white noise excitation are presented to illustrate application of the proposed method.

In this paper, we investigate a class of impulsive bidirectional associative memory (BAM) networks with both periodic coefficients and finite distributed delays. New criteria are established for the existence of an exponential periodic attractor, which generalize and improve the previously known results. Our criteria are less restrictive and can be applied to impulsive or non-impulsive BAM networks with a broad range of activation functions without differentiability and strict monotonicity. Moreover, our results are given in terms of system parameters and finite delay kernels of impulsive BAM networks by employing inequality technique, M -matrix and spectral theory. Finally, an example is given to show the feasibility and effectiveness of our results.

By employing a fixed point theorem in cones, this paper investigates the existence of almost periodic solutions for an impulsive logistic equation with infinite delay. A set of sufficient conditions on the existence of almost periodic solutions of the equation is obtained.

In this paper, by using the contraction principle and Gronwall–Bellman’s inequality, some sufficient conditions are obtained for checking the existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks (SICNNs) with impulse. Our results are essentially new. It is the first time that the existence of almost periodic solutions for the impulsive neural networks are obtained.

In the present paper, several sufficient conditions are obtained for the existence and exponential attractivity of a unique κ-almost periodic sequence solution of discrete time neural network. Our results generalize the corresponding results about almost periodic sequence solution in common sense. It is shown that discretization step κ affects the dynamical characteristics of discrete-time analogues of continuous time neural networks and exponential convergence is dependent on small discretization step size. Our results on exponential attractivity of κ-almost periodic sequence solution can provide us with relevant estimates on how precise such networks can perform during real-time computations. Finally, computer simulations are performed in the end to show the feasibility of our results.

A procedure for designing a feedback control to asymptoticallystabilize, with probability one, a quasi-partially integrableHamiltonian system is proposed. First, the averaged stochasticdifferential equations for controlled r first integrals are derived fromthe equations of motion of a given system by using the stochasticaveraging method for quasi-partially integrable Hamiltonian systems.Second, a dynamical programming equation for the ergodic control problemof the averaged system with undetermined cost function is establishedbased on the dynamical programming principle. The optimal control law isderived from minimizing the dynamical programming equation with respectto control. Third, the asymptotic stability with probability one of theoptimally controlled system is analyzed by evaluating the maximalLyapunov exponent of the completely averaged Ito equations for the rfirst integrals. Finally, the cost function and optimal control forces aredetermined by the requirements of stabilizing the system. An example isworked out in detail to illustrate the application of the proposedprocedure and the effect of optimal control on the stability of thesystem.

A harmonic function with constant amplitude and random frequency and phase is called bounded noise. In this paper, the effect of bounded noise on the chaotic behavior of the Duffing oscillator under parametric excitation is studied in detail. The random Melnikov process is derived and a mean-square criterion is used to detect the chaotic dynamics in the system. It is found that the threshold of bounded noise amplitude for the onset of chaos in the system increases as the intensity of the noise in frequency increases. The threshold of bounded noise amplitude for the onset of chaos is also determined by the numerical calculation of the largest Lyapunov exponents. The effect of bounded noise on the Poincare map and power spectra is also investigated. The numerical results qualitatively confirm the conclusion drawn by using the random Melnikov process with mean-square criterion for larger noise intensity.

The existing results for multiperiodicity of threshold-linear networks (TLNs) are scale-free on time evolution and hence exhibit some restrictions. Due to the nature of the scale-limited activating set, it is interesting to study the dynamical properties of neurons on time scales. In this paper we analyze and obtain results concerning nondivergence, attractivity, and multiperiodic dynamics of TLNs on time scales. Using the notion of exponential functions on time scales, we obtain results for scale-limited type criteria for boundedness and global attractivity of TLNs. Moreover, by constructing simple algebraic inequalities over scale-limited activating sets, we achieve results regarding multiperiodicity of TLNs. This will show that each scale-limited activating set depends on scale-synchronous self-excitation, and the existence of inactive neurons will slow down convergence of TLNs. At the end of the paper, we perform computer simulations to illustrate the obtained new theories.

We investigate the number of periodic solutions of second-order asymptotically linear difference system. The main tools are Morse theory and twist number, and the discussion in this paper is divided into three cases. As the system is resonant at infinity, we use perturbation method to study the compactness condition of functional. We obtain some new results concerning the lower bounds of the nonconstant periodic solutions for discrete system.

Wilson-Cowan model of neuronal population with time-varying delays is considered in this paper. Some sufficient conditions for the existence and delay-based exponential stability of a unique almost periodic solution are established. The approaches are based on constructing Lyapunov functionals and the well-known Banach contraction mapping principle. The results are new, easily checkable, and complement existing periodic ones.

Two existence results are presented on boundary value problem for discrete Hamiltonian systems with forcing terms. Methods used here is based on the Saddle Point theorem and the Least action principle. Moreover, in order to apply variational methods, a new action functional is constructed which is different from the usual action functional for periodic boundary condition.

By using comparison theorem and constructing suitable Lyapunov functional, we study the following almost periodic nonlinear N-species competitive Lotka–Volterra model: x ˙ i ( t ) = x i ( t ) [ r i ( t ) − ∑ j = 1 N a i j ( t ) x j α i j ( t ) − ∑ j = 1 N b i j ( t ) x j α i j ( t − τ i j ( t ) ) − ∑ j = 1 N c i j ( t ) x i α i i ( t ) x j α i j ( t ) ] . A set of sufficient conditions is obtained for the existence and global attractivity of a unique positive almost periodic solution of the above model. As applications, some special competition models are studied again, our new results improve and generalize former results. Examples and their simulations show the feasibility of our main results.

Certain almost periodic forced perturbed systems with piecewise argument are considered in this paper. By using the contraction mapping principle and some new analysis technique, some sufficient conditions are obtained for the existence and uniqueness of almost periodic solution of these systems. Furthermore, we study the harmonic and subharmonic solutions of these systems. The obtained results generalize the previous known results such as [A.M. Fink, Almost Periodic Differential Equation, Lecture Notes in Math., vol. 377, Springer-Verlag, Berlin, 1974; C.Y. He, Almost Periodic Differential Equations, Higher Education Press, Beijing, 1992 (in Chinese); Z.S. Lin, The existence of almost periodic solution of linear system, Acta Math. Sinica 22 (5) (1979) 515–528 (in Chinese); C.Y. He, Existence of almost periodic solutions of perturbation systems, Ann. Differential Equations 9 (2) (1992) 173–181; Y.H. Xia, M. Lin, J. Cao, The existence of almost periodic solutions of certain perturbation system, J. Math. Anal. Appl. 310 (1) (2005) 81–96]. Finally, a tangible example and its numeric simulations show the feasibility of our results, the comparison between non-perturbed system and perturbed system, the relation between systems with and without piecewise argument.

In this article we employ Krasnoselskii’s fixed point theorem to obtain new biperiodicity criteria for neutral-type difference neural networks with delays. It is shown that the neutral-type term can leads to biperiodicity results. That is coexistence of a positive periodic sequence solution and its anti-sign periodic sequence solution. We illustrate our novel approach the biperiodicity dynamics of biperiodicity for neutral-type delay difference neural networks by two computer numerical examples. Mathematics Subject Classification 2010: 39A23; 39A10.