Eva Kaslik

2012 Special Issue: Nonlinear dynamics and chaos in fractional-order neural networks

Several topics related to the dynamics of fractional-order neural networks of Hopfield type are investigated, such as stability and multi-stability (coexistence of several different stable states), bifurcations and chaos. The stability domain of a steady state is completely characterized with respect to some characteristic parameters of the system, in the case of a neural network with ring or hub structure. These simplified connectivity structures play an important role in characterizing the networks dynamical behavior, allowing us to gain insight into the mechanisms underlying the behavior of recurrent networks. Based on the stability analysis, we are able to identify the critical values of the fractional order for which Hopf bifurcations may occur. Simulation results are presented to illustrate the theoretical findings and to show potential routes towards the onset of chaotic behavior when the fractional order of the system increases.

Impulsive hybrid discrete-time Hopfield neural networks with delays and multistability analysis

In this paper we investigate multistability of discrete-time Hopfield-type neural networks with distributed delays and impulses, by using Lyapunov functionals, stability theory and control by impulses. Example and simulation results are given to illustrate the effectiveness of the results.

Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions☆

Abstract Using the Mellin transform approach, it is shown that, in contrast with integer-order derivatives, the fractional-order derivative of a periodic function cannot be a function with the same period. The three most widely used definitions of fractional-order derivatives are taken into account, namely, the Caputo, Riemann–Liouville and Grunwald–Letnikov definitions. As a consequence, the non-existence of exact periodic solutions in a wide class of fractional-order dynamical systems is obtained. As an application, it is emphasized that the limit cycle observed in numerical simulations of a simple fractional-order neural network cannot be an exact periodic solution of the system.

Dynamics of fractional-order neural networks

In this paper we discuss the stability analysis for fractional-order neural networks of Hopfield type. The stability domain of a steady state is completely characterized with respect to some characteristic parameters of the system, in the case of a two-dimensional network and of a network of n ≥ 3 neurons with ring structure. The values of the characteristic parameters for which Hopf bifurcations occur are identified. Numerical simulations are given which substantiate the theoretical findings and suggest possible routes towards chaos when the fractional order of the system increases.

Complex and chaotic dynamics in a discrete-time-delayed Hopfield neural network with ring architecture

This paper is devoted to the analysis of a discrete-time-delayed Hopfield-type neural network of p neurons with ring architecture. The stability domain of the null solution is found, the values of the characteristic parameter for which bifurcations occur at the origin are identified and the existence of Fold/Cusp, Neimark-Sacker and Flip bifurcations is proved. These bifurcations are analyzed by applying the center manifold theorem and the normal form theory. It is proved that resonant 1:3 and 1:4 bifurcations may also be present. It is shown that the dynamics in a neighborhood of the null solution become more and more complex as the characteristic parameter grows in magnitude and passes through the bifurcation values. A theoretical proof is given for the occurrence of Marottos chaotic behavior, if the magnitudes of the interconnection coefficients are large enough and at least one of the activation functions has two simple real roots.

Analytical and numerical methods for the stability analysis of linear fractional delay differential equations

In this paper, several analytical and numerical approaches are presented for the stability analysis of linear fractional-order delay differential equations. The main focus of interest is asymptotic stability, but bounded-input bounded-output (BIBO) stability is also discussed. The applicability of the Laplace transform method for stability analysis is first investigated, jointly with the corresponding characteristic equation, which is broadly used in BIBO stability analysis. Moreover, it is shown that a different characteristic equation, involving the one-parameter Mittag-Leffler function, may be obtained using the well-known method of steps, which provides a necessary condition for asymptotic stability. Stability criteria based on the Argument Principle are also obtained. The stability regions obtained using the two methods are evaluated numerically and comparison results are presented. Several key problems are highlighted.

Multiple periodic solutions in impulsive hybrid neural networks with delays

Abstract The existence of multiple periodic solutions and their exponential stability are investigated for impulsive hybrid Hopfield-type neural networks with both time-dependent and distributed delays, using the Leray–Schauder fixed point theorem and Lyapunov functionals. The criteria given are easily verifiable, possess many adjustable parameters, and depend on impulses, providing flexibility for the analysis and design of delayed neural networks with impulse effects. Examples are given.

Stability Results for a Class of Difference Systems with Delay

Considering the linear delay difference system , where , is a real matrix, and is a positive integer, the stability domain of the null solution is completely characterized in terms of the eigenvalues of the matrix . It is also shown that the stability domain becomes smaller as the delay increases. These results may be successfully applied in the stability analysis of a large class of nonlinear difference systems, including discrete-time Hopfield neural networks.

Dynamics of complex-valued fractional-order neural networks

The dynamics of complex-valued fractional-order neuronal networks are investigated, focusing on stability, instability and Hopf bifurcations. Sufficient conditions for the asymptotic stability and instability of a steady state of the network are derived, based on the complex system parameters and the fractional order of the system, considering simplified neuronal connectivity structures (hub and ring). In some specific cases, it is possible to identify the critical values of the fractional order for which Hopf bifurcations may occur. Numerical simulations are presented to illustrate the theoretical findings and to investigate the stability of the limit cycles which appear due to Hopf bifurcations.

Dynamics of a discrete-time bidirectional ring of neurons with delay

This paper is devoted to the analysis of a discrete-time delayed Hopfield-type neural network of p ≥ 3 neurons with bidirectional ring architecture. The stability domain of the null solution is found, the values of the characteristic parameters for which bifurcations occur at the origin are identified and the existence of Fold/Cusp, Neimark-Sacker and higher codimension bifurcations is proved. The direction and stability of the Neimark-Sacker bifurcations are analyzed by applying the center manifold theorem and the normal form theory. Numerical simulations are given which substantiate the theoretical findings and suggest possible routes towards chaos when the absolute value of one of the characteristic parameters increases.