This paper presents several novel approaches of particle swarm
optimization (PSO) algorithm with new particle velocity equations
and three variants of inertia weight to solve the optimal control
problem of a class of hybrid systems, which are motivated by the
structure of manufacturing environments that integrate process and
optimal control. In the proposed PSO algorithm, the particle
velocities are conceptualized with the local best (or
<i>pbest</i>) and global best (or <i>gbest</i>) of the swarm,
which makes a quick decision to direct the search towards the
optimal (fitness) solution. The inertia weight of the proposed
methods is also described as a function of pbest and gbest, which
allows the PSO to converge faster with accuracy. A typical
numerical example of the optimal control problem is included to
analyse the efficacy and validity of the proposed algorithms.
Several statistical analyses including hypothesis test are done to
compare the validity of the proposed algorithms with the existing
PSO technique, which adopts linearly decreasing inertia weight.
The results clearly demonstrate that the proposed PSO approaches
not only improve the quality but also are more efficient in
converging to the optimal value faster.

M. Senthil Arumugam

M. V. C. Rao

On the optimal control of single-stage hybrid manufacturing systems via novel and different variants of particle swarm optimization algorithm

The paper investigates the conditions for full and partial
synchronization in systems of coupled chaotic maps that include
the presence of a major element, that is, an element that
interacts with all the other elements of the system. We consider a
system which consists of two globally coupled populations of
one-dimensional maps that interact via a major element. The
presence of this element can induce synchronization in both of the
globally coupled populations even though they operate in different
states. If a parameter mismatch is introduced between two
populations of uncoupled maps, the presence of a major element is
found to provide for the existence of states in which peripheral
elements with different parameter values display similar dynamics.

Iryna Omelchenko

Yuri Maistrenko

Erik Mosekilde

Synchronization in ensembles of coupled maps
       with a major element

The following map is studied:
<mml:math alttext="$(x,y)\rightarrow(1+a(|x|-y^2)+y,bx)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8594;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>a</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>x</mml:mi><mml:mo>|</mml:mo></mml:mrow><mml:mo>&#8722;</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>. It is proved numerically
that this model can   display two different chaotic
 attractors, one is new
and the other is a Lozi-type attractor. The new chaotic attractor
is allowed via a border-collision period-doubling scenario, which
is different from the classical period-doubling bifurcation.

Zeraoulia  Elhadj

A new chaotic attractor from 2D discrete mapping
via border-collision period-doubling scenario

A class of nonlinear delay difference equations are considered
and some sufficient conditions for global attractivity of
solutions of the equation are obtained. It is shown that the
stability properties, both local and global, of the equilibrium of
the delay equation can be derived from those of an associated
nondelay equation.

Binxiang Dai

Na Zhang

Stability and global attractivity for a class of
nonlinear delay difference equations

A sufficient condition to preserve the property of asymptotic
stability for a difference analogue of the linear mathematical
inverted pendulum is obtained.

L.  Shaikhet

Stability of difference analogue of linear mathematical inverted
pendulum

We consider the abstract Cauchy problem for differential equation of the hyperbolic type <mml:math alttext="$v''(t)+Av(t)=f(t)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>v</mml:mi><mml:mo>&#8243;</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mi>v</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> (<mml:math alttext="$ 0\leq t\leq T$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&#8804;</mml:mo><mml:mi>t</mml:mi><mml:mo>&#8804;</mml:mo><mml:mi>T</mml:mi></mml:math>), <mml:math alttext="$v(0)=v_{0}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>v</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>, <mml:math alttext="$v'(0)=v'_{0}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>v</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:msup><mml:mi>v</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mn>0</mml:mn></mml:msub></mml:math> in an arbitrary Hilbert space <mml:math alttext="$H$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math> with the selfadjoint positive definite operator <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by the Taylor decomposition on the three points for the numerical solutions of this problem are presented. The stability estimates for the solutions of these difference schemes are established. In applications, the stability estimates for the solutions of the high order of accuracy difference schemes of the mixed-type boundary value problems for hyperbolic equations are obtained.

Allaberen  Ashyralyev

Pavel E. Sobolevskii

Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations

We study that the <mml:math alttext="$q$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math>-Bernoulli polynomials, which were constructed by Kim, are analytic continued to <mml:math alttext="$\beta_s(z)$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x3B2;</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. A new formula for the <mml:math alttext="$q$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math>-Riemann zeta function <mml:math alttext="$\zeta_q(s)$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x3B6;</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> due to Kim in terms of nested series of <mml:math alttext="$\zeta_q(n)$" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x3B6;</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> is derived. The new concept of dynamics of the zeros of analytic  continued polynomials is introduced, and an interesting phenomenon of &#x201C;scattering&#x201D; of the zeros of <mml:math alttext="$\beta_s(z)$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x3B2;</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> is observed. Following the idea of <mml:math alttext="$q$" id="E9" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math>-zeta function due to Kim, we are going to use &#x201C;Mathematica&#x201D; to explore a formula for <mml:math alttext="$\zeta_q(n)$" id="E10" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x3B6;</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>.

T. Kim

C. S. Ryoo

L. C. Jang

S. H. Rim

Exploring the <svg style="vertical-align:-3.50804pt" height="12.1125" version="1.1" viewBox="0 0 8.4750004 12.1125" width="8.4750004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
		<g transform="matrix(.017,-0,0,-.017,.062,7.675)">
			<path id="x1D45E" d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387&#xA;q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z"/>
		</g>
	</svg>-Riemann zeta function and <svg style="vertical-align:-3.50804pt" height="12.1125" version="1.1" viewBox="0 0 8.4750004 12.1125" width="8.4750004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
		<g transform="matrix(.017,-0,0,-.017,.062,7.675)">
			<use xlink:href="#x1D45E"/>
		</g>
	</svg>-Bernoulli polynomials



Modern theory of dynamical systems is mostly based on nonlinear differential equations and operations. At the same time, the theory of hypernumbers and extrafunctions, a novel approach in functional analysis, has been limited to linear systems. In this paper, nonlinear structures are introduced in spaces of real and complex hypernumbers by extending the concept of a hypernumber. In such a way, linear algebras of extended hypernumbers are built. A special topology of conical neighborhoods in these algebras is introduced and studied. It is proved that the space of all extended real hypernumbers is Hausdorff. This provides uniqueness for limits what is very important for analysis of dynamical systems. It is also proved that construction of extended real
hypernumbers is defined by a definite invariance principle: the space of all extended real hypernumbers is the biggest Hausdorff factorization of the sequential extension of the space of all real numbers with the topology of conical neighborhoods. In addition,
this topology turns the set of all bounded extended real hypernumbers into a topological algebra. Other topologies in spaces of extended hypernumbers are considered.

M. S. Burgin

Topology in nonlinear extensions of hypernumbers

We first give sufficient conditions for the permanence of nonautonomous discrete ratio-dependent predator-prey model. By linearization of the model at positive solutions and construction of Lyapunov function, we also obtain some conditions which ensure that a positive solution of the model is stable and attracts all positive solutions.

Hai-Feng Huo

Wan-Tong Li

Permanence and global stability of positive solutions of a nonautonomous discrete ratio-dependent predator-prey model

We study dynamical behavior of a class of cellular neural networks
system with distributed delays under dynamical thresholds. By using topological degree theory and Lyapunov functions, some new criteria ensuring the existence, uniqueness, global asymptotic stability, and global exponential stability of equilibrium point are derived. In particular, our criteria generalize and improve some known results in the literature.

Fei-Yu Zhang

Global stability of delayed Hopfield neural networks under
dynamical thresholds

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<p><em>Discrete Dynamics in Nature and Society</em> publishes research that links basic and applied research relating to discrete dynamics of complex systems encountered in the natural and social sciences.</p>


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<p>Chief Editor, Dr&nbsp;Renna, is an associate professor at the University of Basilicata, Italy. His research interests include manufacturing systems, production planning and enterprise networks.&nbsp;</p>


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Discrete Dynamics in Nature and Society