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- Further Results on the Stability of Neural Network for Solving Variational
Inequalities

pp. 341–353 | DOI: 10.5890/DNC.2016.12.001

Mi Zhou1†, Xiaolan Liu2,3‡

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Abstract

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.

- How the Minimal Poincar´e Return Time Depends on the Size of a Return Region in
a Linear Circle Map

pp. 335–364 | DOI: 10.5890/DNC.2016.12.002

N. Semenova, E. Rybalova, V. Anishchenko†

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

- Reversible Mixed Dynamics: A Concept and Examples

pp. 365–374 | DOI: 10.5890/DNC.2016.12.003

S.V. Gonchenko†

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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 Speak Up the Time, and Time Bespeaks Us

pp. 375–395 | DOI: 10.5890/DNC.2016.12.004

Dimitri Volchenkov†, Anna Cabigiosu, MassimoWarglien

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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.

- On Quasi-periodic Perturbations of Duffing Equation

pp. 397–406 | DOI: 10.5890/DNC.2016.12.005

A.D. Morozov†, T.N. Dragunov

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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.

- A Study of the Dynamics of the Family f
_{ λ ,μ} = λsinz+μ/(z−kπ) where λ ,μ ∈ R\{0} and k ∈ Z\{0}

pp. 407–414 | DOI: 10.5890/DNC.2016.12.006

Patricia Dom´ınguez†, Josu´e V´azquez, Marco A. Montes de Oca

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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.

- New Results on Exponential Stability of Fractional Order Nonlinear Dynamic
Systems

pp. 415–425 | DOI: 10.5890/DNC.2016.12.007

Tianzeng Li1,2†, Yu Wang1,3, Yong Yang

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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.

- Robust Exponential Stability of Impulsive Stochastic Neural Networks with
Markovian Switching and Mixed Time-varying Delays

pp. 427–446 | DOI: 10.5890/DNC.2016.12.008

Haoru Li1, Yang Fang2, Kelin Li2†

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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.

- Slowing Down of So-called Chaotic States: “Freezing” the Initial State

pp. 447–455 | DOI: 10.5890/DNC.2016.12.009

M. Belger1, S. De Nigris†2, X. Leoncini‡1,3

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Abstract

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 →∞. 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.

- Relaxation Oscillations and Chaos in a Duffing Type Equation: A Case Study

pp. 457–474 | DOI: 10.5890/DNC.2016.12.0010

L. Lerman1†, A. Kazakov2,1, N.Kulagin3

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Abstract

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.