Showing papers in "Discrete Dynamics in Nature and Society in 2010"

A Clustering Approach Using Cooperative Artificial Bee Colony Algorithm

Abstract: Artificial Bee Colony (ABC) is one of the most recently introduced algorithms based on the intelligent foraging behavior of a honey bee swarm. This paper presents an extended ABC algorithm, namely, the Cooperative Article Bee Colony (CABC), which significantly improves the original ABC in solving complex optimization problems. Clustering is a popular data analysis and data mining technique; therefore, the CABC could be used for solving clustering problems. In this work, first the CABC algorithm is used for optimizing six widely used benchmark functions and the comparative results produced by ABC, Particle Swarm Optimization (PSO), and its cooperative version (CPSO) are studied. Second, the CABC algorithm is used for data clustering on several benchmark data sets. The performance of CABC algorithm is compared with PSO, CPSO, and ABC algorithms on clustering problems. The simulation results show that the proposed CABC outperforms the other three algorithms in terms of accuracy, robustness, and convergence speed.

Particle Swarm Optimization with Various Inertia Weight Variants for Optimal Power Flow Solution

Abstract: This paper proposes an efficient method to solve the optimal power flow problem in power systems using Particle Swarm Optimization (PSO). The objective of the proposed method is to find the steady-state operating point which minimizes the fuel cost, while maintaining an acceptable system performance in terms of limits on generator power, line flow, and voltage. Three different inertia weights, a constant inertia weight (CIW), a time-varying inertia weight (TVIW), and global-local best inertia weight (GLbestIW), are considered with the particle swarm optimization algorithm to analyze the impact of inertia weight on the performance of PSO algorithm. The PSO algorithm is simulated for each of the method individually. It is observed that the PSO algorithm with the proposed inertia weight yields better results, both in terms of optimal solution and faster convergence. The proposed method has been tested on the standard IEEE 30 bus test system to prove its efficacy. The algorithm is computationally faster, in terms of the number of load flows executed, and provides better results than other heuristic techniques.

Characterizing Growth and Form of Fractal Cities with Allometric Scaling Exponents

Abstract: Fractal growth is a kind of allometric growth, and the allometric scaling exponents can be employed to describe growing fractal phenomena such as cities. The spatial features of the regular fractals can be characterized by fractal dimension. However, for the real systems with statistical fractality, it is incomplete to measure the structure of scaling invariance only by fractal dimension. Sometimes, we need to know the ratio of different dimensions rather than the fractal dimensions themselves. A fractal-dimension ratio can make an allometric scaling exponent (ASE). As compared with fractal dimension, ASEs have three advantages. First, the values of ASEs are easy to be estimated in practice; second, ASEs can reflect the dynamical characters of system's evolution; third, the analysis of ASEs can be made through prefractal structure with limited scale. Therefore, the ASEs based on fractal dimensions are more functional than fractal dimensions for real fractal systems. In this paper, the definition and calculation method of ASEs are illustrated by starting from mathematical fractals, and, then, China's cities are taken as examples to show how to apply ASEs to depiction of growth and form of fractal cities.

Boundedness and Stability for Discrete-Time Delayed Neural Network with Complex-Valued Linear Threshold Neurons

Abstract: The discrete-time delayed neural network with complex-valued linear threshold neurons is considered. By constructing appropriate Lyapunov-Krasovskii functionals and employing linear matrix inequality technique and analysis method, several new delay-dependent criteria for checking the boundedness and global exponential stability are established. Illustrated examples are also given to show the effectiveness and less conservatism of the proposed criteria.

Complete Convergence for Weighted Sums of -Mixing Random Variables

Abstract: We obtain the complete convergence for weighted sums of -mixing random variables. Our result extends the result of Peligrad and Gut (1999) on unweighted average to a weighted average under a mild condition of weights. Our result also generalizes and sharpens the result of An and Yuan (2008).

Hierarchical Swarm Model: A New Approach to Optimization

Abstract: This paper presents a novel optimization model called hierarchical swarm optimization (HSO), which simulates the natural hierarchical complex system from where more complex intelligence can emerge for complex problems solving. This proposed model is intended to suggest ways that the performance of HSO-based algorithms on complex optimization problems can be significantly improved. This performance improvement is obtained by constructing the HSO hierarchies, which means that an agent in a higher level swarm can be composed of swarms of other agents from lower level and different swarms of different levels evolve on different spatiotemporal scale. A novel optimization algorithm (named PS2O), based on the HSO model, is instantiated and tested to illustrate the ideas of HSO model clearly. Experiments were conducted on a set of 17 benchmark optimization problems including both continuous and discrete cases. The results demonstrate remarkable performance of the PS2O algorithm on all chosen benchmark functions when compared to several successful swarm intelligence and evolutionary algorithms.

Exploring the Fractal Parameters of Urban Growth and Form with Wave-Spectrum Analysis

Abstract: The Fourier transform and spectral analysis are employed to estimate the fractal dimension and explore the fractal parameter relations of urban growth and form using mathematical experiments and empirical analyses. Based on the models of urban density, two kinds of fractal dimensions of urban form can be evaluated with the scaling relations between the wave number and the spectral density. One is the radial dimension of self-similar distribution indicating the macro-urban patterns, and the other, the profile dimension of self-affine tracks indicating the micro-urban evolution. If a city's growth follows the power law, the summation of the two dimension values may be a constant under certain condition. The estimated results of the radial dimension suggest a new fractal dimension, which can be termed “image dimension”. A dual-structure model named particle-ripple model (PRM) is proposed to explain the connections and differences between the macro and micro levels of urban form.

Neimark-Sacker Bifurcation in a Discrete-Time Financial System

Abstract: A discrete-time financial system is proposed by using forward Euler scheme. Based on explicit Neimark-Sacker bifurcation (also called Hopf bifurcation for map) criterion, normal form method and center manifold theory, the system's existence, stability and direction of Neimark-Sacker bifurcation are studied. Numerical simulations are employed to validate the main results of this work. Some comparison of bifurcation between the discrete-time financial system and its continuous-time system is given.

Analysis of a Simple Vector-Host Epidemic Model with Direct Transmission

Abstract: Vector-host epidemic models with direct transmission are proposed and analyzed. It is shown that the stability of the equilibria in the proposed models can be controlled by the basic reproduction number of the disease transmission. One model considers that the dynamics of human hosts and vectors are described by SIS and SI model, respectively, where the global asymptotical stability for the equilibria of the model is analyzed by constructing Lyapunov function, respectively. The other model considers that the dynamics of the human hosts and vectors are described by SIRS and SI model, respectively, where the global stability of the disease-free equilibrium and the persistence of the disease in the model are also analyzed, respectively.

A Fixed Point Approach to the Stability of a General Mixed AQCQ-Functional Equation in Non-Archimedean Normed Spaces

Abstract: Using the fixed point methods, we prove the generalized Hyers-Ulam stability of the general mixed additive-quadratic-cubic-quartic functional equation 𝑓(𝑥

Convergence Properties for Asymptotically almost Negatively Associated Sequence

Abstract: We get the strong law of large numbers, strong growth rate, and the integrability of supremum for the partial sums of asymptotically almost negatively associated sequence. In addition, the complete convergence for weighted sums of asymptotically almost negatively associated sequences is also studied.

Existence and Uniqueness of Solutions for the Cauchy-Type Problems of Fractional Differential Equations

Abstract: By using the Banach fixed point theorem and step method, we study the existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equations. Meanwhile, by citing some counterexamples, it is pointed out that there exist a few defects in the proofs of the known results.

Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

Abstract: Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalue of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

Generalized Hyers-Ulam-Rassias Theorem in Menger Probabilistic Normed Spaces

Abstract: We introduce two reasonable versions of approximately additive functions in a Serstnev probabilistic normed space endowed with Π 𝑀 triangle function. More precisely, we show under some suitable conditions that an approximately additive function can be approximated by an additive mapping in above mentioned spaces.

A Hybrid Method for a Countable Family of Multivalued Maps, Equilibrium Problems, and Variational Inequality Problems

Abstract: We introduce a new monotone hybrid iterative scheme for finding a common element of the set of common fixed points of a countable family of nonexpansive multivalued maps, the set of solutions of variational inequality problem, and the set of the solutions of the equilibrium problem in a Hilbert space. Strong convergence theorems of the purposed iteration are established.

The Fixed Point Method for Fuzzy Approximation of a Functional Equation Associated with Inner Product Spaces

Abstract: Th. M. Rassias (1984) proved that the norm defined over a real vector space 𝑋 is induced by an inner product if and only if for a fixed integer 𝑛≥2,∑𝑛𝑖=1‖𝑥𝑖∑−(1/𝑛)𝑛𝑗=1𝑥𝑗‖2=∑𝑛𝑖=1‖𝑥𝑖‖2∑−𝑛‖(1/𝑛)𝑛𝑖=1𝑥𝑖‖2 holds for all 𝑥1,…,𝑥𝑛∈𝑋. The aim of this paper is to extend the applications of the fixed point alternative method to provide a fuzzy stability for the functional equation ∑𝑛𝑖=1𝑓(𝑥𝑖∑−(1/𝑛)𝑛𝑗=1𝑥𝑗∑)=𝑛𝑖=1𝑓(𝑥𝑖∑)−𝑛𝑓((1/𝑛)𝑛𝑖=1𝑥𝑖) which is said to be a functional equation associated with inner product spaces.

Furstenberg Families and Sensitivity

Abstract: We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system is -sensitive if there exists a positive such that for every and every open neighborhood of there exists such that the pair is not --asymptotic; that is, the time set belongs to , where is a Furstenberg family. A dynamical system is (, )-sensitive if there is a positive such that every is a limit of points such that the pair is -proximal but not --asymptotic; that is, the time set belongs to for any positive but the time set belongs to , where and are Furstenberg families.

Adaptive Pinning Synchronization of Complex Networks with Stochastic Perturbations

Abstract: The adaptive pinning synchronization is investigated for complex networks with nondelayed and delayed couplings and vector-form stochastic perturbations. Two kinds of adaptive pinning controllers are designed. Based on an Lyapunov-Krasovskii functional and the stochastic stability analysis theory, several sufficient conditions are developed to guarantee the synchronization of the proposed complex networks even if partial states of the nodes are coupled. Furthermore, three examples with their numerical simulations are employed to show the effectiveness of the theoretical results.

A Note on the Modified -Bernstein Polynomials

Abstract: We propose the modified -Bernstein polynomials of degree which are different -Bernstein polynomials of Phillips (1997). From these modified -Bernstein polynomials of degree , we derive some recurrence formulae for the modified -Bernstein polynomials.

Nonlinear Modelling and Qualitative Analysis of a Real Chemostat with Pulse Feeding

Abstract: The control of substrate concentration in the bioreactor medium should be due to the substrate inhibition phenomenon Moreover, the oxygen demand in a bioreactor should be lower than the dissolved oxygen content The biomass concentration is one of the most important factors which affect the oxygen demand In order to maintain the dissolved oxygen content in an appropriate range, the biomass concentration should not exceed a critical level Based on the design ideas, a mathematical model of a chemostat with Monod-type kinetics and impulsive state feedback control for microorganisms of any biomass yield is proposed in this paper By the existence criteria of periodic solution of a general planar impulsive autonomous system, the conditions for the existence of period-1 solution of the system are obtained The results simplify the choice of suitable operating conditions for continuous culture systems It also points out that the system is not chaotic according to the analysis on the existence of period-2 solution The results and numerical simulations show that the chemostat system with state impulsive control tends to a stable state or a period solution

Improved Results on Fuzzy Filter Design for T-S Fuzzy Systems

Abstract: The fuzzy filter design problem for T-S fuzzy systems with interval time-varying delay is investigated. The delay is considered as the time-varying delay being either differentiable uniformly bounded with delay derivative in bounded interval or fast varying (with no restrictions on the delay derivative). A novel Lyapunov-Krasovskii functional is employed and a tighter upper bound of its derivative is obtained. The resulting criterion thus has advantages over the existing ones since we estimate the upper bound of the derivative of Lyapunov-Krasovskii functional without ignoring some useful terms. A fuzzy filter is designed to ensure that the filter error system is asymptotically stable and has a prescribed performance level. An improved delay-derivative-dependent condition for the existence of such a filter is derived in the form of linear matrix inequalities (LMIs). Finally, numerical examples are given to show the effectiveness of the proposed method.

On the Max-Type Difference Equation X_(n+1) = max {A/X_(n ,) X_(n-3)}

Abstract: The study of max-type difference equations attracted recently a considerable attention, see, for example, 1–27 , and the references listed therein. This type of difference equations stems from, for example, certain models in automatic control theory see 28 . In the beginning of the study of these equations experts have been focused on the investigation of the behavior of some particular cases of the following general difference equation of order k ∈ N:

Asymptotic Stability for a Class of Nonlinear Difference Equations

Abstract: We study the global asymptotic stability of the equilibrium point for the fractional difference equation 𝑥𝑛

Almost Periodic Solutions of a Discrete Mutualism Model with Feedback Controls

Abstract: We consider a discrete mutualism model with feedback controls. Assuming that the coefficients in the system are almost periodic sequences, we obtain the existence and uniqueness of the almost periodic solution which is uniformly asymptotically stable.

Global Dissipativity on Uncertain Discrete-Time Neural Networks with Time-Varying Delays

Abstract: The problems on global dissipativity and global exponential dissipativity are investigated for uncertain discrete-time neural networks with time-varying delays and general activation functions. By constructing appropriate Lyapunov-Krasovskii functionals and employing linear matrix inequality technique, several new delay-dependent criteria for checking the global dissipativity and global exponential dissipativity of the addressed neural networks are established in linear matrix inequality (LMI), which can be checked numerically using the effective LMI toolbox in MATLAB. Illustrated examples are given to show the effectiveness of the proposed criteria. It is noteworthy that because neither model transformation nor free-weighting matrices are employed to deal with cross terms in the derivation of the dissipativity criteria, the obtained results are less conservative and more computationally efficient.

Asymptotic Properties of a Hepatitis B Virus Infection Model with Time Delay

Abstract: A hepatitis B virus infection model with time delay is discussed. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the model is studied. By using comparison arguments, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable. If the basic reproduction ratio is greater than unity, by means of an iteration technique, sufficient conditions are derived for the global asymptotic stability of the virus-infected equilibrium. Numerical simulations are carried out to illustrate the theoretical results.

Oscillation Behavior of a Class of Second-Order Dynamic Equations with Damping on Time Scales

Abstract: By using a Riccati transformation and inequality, we present some new oscillation theorems for the second-order nonlinear dynamic equation with damping on time scales. An example illustrating the importance of our results is also included.

Stability and Stabilization of Impulsive Stochastic Delay Difference Equations

Abstract: When an impulsive control is adopted for a stochastic delay difference system (SDDS), there are at least two situations that should be contemplated. If the SDDS is stable, then what kind of impulse can the original system tolerate to keep stable? If the SDDS is unstable, then what kind of impulsive strategy should be taken to make the system stable? Using the Lyapunov-Razumikhin technique, we establish criteria for the stability of impulsive stochastic delay difference equations and these criteria answer those questions. As for applications, we consider a kind of impulsive stochastic delay difference equation and present some corollaries to our main results.

Extended and Unscented Kalman Filtering Applied to a Flexible-Joint Robot with Jerk Estimation

Abstract: Robust nonlinear control of flexible-joint robots requires that the link position, velocity, acceleration, and jerk be available. In this paper, we derive the dynamic model of a nonlinear flexible-joint robot based on the governing Euler-Lagrange equations and propose extended and unscented Kalman filters to estimate the link acceleration and jerk from position and velocity measurements. Both observers are designed for the same model and run with the same covariance matrices under the same initial conditions. A five-bar linkage robot with revolute flexible joints is considered as a case study. Simulation results verify the effectiveness of the proposed filters.