Vol. 5, No. 4 (2016): Regular Issue

This paper analyzes and proves the global Lyapunov stability of the neural network proposed by Yashtini and Malek when the mapping is continuously differentiable and the Jacobian matrix of the mapping is positive semi-definite. Furthermore, the neural network is shown to be exponentially stable under stronger conditions. In particular, the stability results can be applied to the stability analysis of variational inequalities with linear constraints and bounded constraints. Some examples show that the proposed neural network can be used to solve the various nonlinear optimization problems. The new results improve the existing ones in the literature.

It is found that the step function of dependence of the minimal Poincaré return time on the size of a return region τinf(ε) for the linear circle map with an arbitrary rotation number can be approximated analytically. All analytical results are confirmed by numerical simulation.

We observe some recent results related to the new type of dynamical chaos, the so-called, “mixed dynamics” which can be considered as an intermediate link between “strange attractor“ and “conservative chaos”. We propose a mathematical concept of mixed dynamics for two-dimensional reversible maps and consider several examples.

We have presented the first study integrating the analysis of temporal patterns of interaction, interaction preferences and the local vs. global structure of communication in networks of agents. We analyzed face-to-face interactions in two organizations over a period of three weeks. Data on interactions among ca 140 individuals have been collected through a wearable sensors study carried on two start-up organizations in the North-East of Italy. Our results suggest that simple principles reflecting interaction propensities, time budget and institutional constraints underlie the distribution of interaction events. Both data on interaction duration and those on intervals between interactions respond to a common logic, based on the propensities of individuals to interact with each other, the cost of interrupting other activities to interact, and the institutional constraints over behavior. These factors affect the decision to interact with someone else. Our data suggest that there are three regimes of interaction arising from the organizational context of our observations: casual, spontaneous (or deliberate) and institutional interaction. Such regimes can be naturally expressed by different parameterizations of our models.

Quasi-periodic two-frequency perturbations are studied in a system which is close to a nonlinear two-dimensional Hamiltonian one. The example of Duffing equation with a saddle and two separatix loops is considered. Several problems are studied: dynamical behavior in a neighborhood of a resonance level of the unperturbed system, conditions for the existence of resonance quasi-periodic solutions (two-dimensional resonance tori), global behavior of solutions inside domains separated from the unperturbed separatrix. In a neighborhood of the unperturbed separatrix the problem of relative position of stable an unstable separatrix manifolds is studied, conditions for the existence of doubly asymptotic solutions are found.

In this article we investigate the dynamics of the meromorphic family f λ ,μ (z) = λ sin z+ μ/( z−kπ) , λ ,μ ∈ R \ {0} and k ∈ Z \ {0}. We show that for some parameters λ ,μ the Stable set contains an attracting component which is multiply connected and completely invariant. We give a definition of a cut of the space of parameters, with μ and kπ fixed, and show examples of a cut and the Stable and Chaotic sets related to the cut, for some λ given.

In this letter stability analysis of fractional order nonlinear systems is studied. An extension of Lyapunov direct method for fractional order systems is proposed by using the properties of Mittag-Leffler function and Laplace transform. Some new sufficient conditions which ensure local exponential stability of fractional order nonlinear systems are proposed firstly. And we apply these conditions to the Riemann-Liouville fractional order systems by using fractional comparison principle. Finally, three examples are provided to illustrate the validity of the proposed approach.

This paper is concerned with the robust exponential stability problem for a class of impulsive stochastic neural networks with Markovian switching, mixed time-varying delays and parametric uncertainties. By construct a novel Lyapunov-Krasovskii functional, and using linear matrix inequality (LMI) technique, Jensen integral inequality and free-weight matrix method, several novel sufficient conditions in the form of LMIs are derived to ensure the robust exponential stability in mean square of the trivial solution of the considered system. The results obtained in this paper improve many known results, since the parametric uncertainties have been taken into account, and the derivatives of discrete and distributed time-varying delays need not to be 0 or smaller than 1. Finally, three illustrative examples are given to show the effectiveness of the proposed method.

The so-called chaotic states that emerge on the model of XY interacting on regular critical range networks are analyzed. Typical time scales are extracted from the time series analysis of the global magnetization. The large spectrum confirms the chaotic nature of the observable, anyhow different peaks in the spectrum allows for typical characteristic time-scales to emerge. We find that these time scales τ (N) display a critical slowing down, i.e they diverge as N →ꝏ. The scaling law is analyzed for different energy densities and the behavior τ (N) ∼ √ N is exhibited. This behavior is furthermore explained analytically using the formalism of thermodynamicequations of the motion and analyzing the eigenvalues of the adjacency matrix.

Results of numerical simulations of a Duffing type Hamiltonian system with a slow periodically varying parameter are presented. Using theory of adiabatic invariants, reversibility of the system and theory of symplectic maps, along with thorough numerical experiments, we present many details of the orbit behavior for the system. In particular, we found many symmetric mixed mode periodic orbits, both being hyperbolic and elliptic, the regions with a perpetual adiabatic invariant and chaotic regions. For the latter region we present details of chaotic behavior: calculation of homoclinic tangles and Lyapunov exponents.