Chaotic Operator Research Articles

Chaos for Cosine Operator Functions on Groups

Let1≤p<∞andGbe a locally compact group. We characterize chaotic cosine operator functions, generated by weighted translations on the Lebesgue spaceLp(G), in terms of the weight condition. In particular, chaotic cosine operator functions and chaotic weighted translations can only occur simultaneously. We also give a necessary and sufficient condition for the direct sum of a sequence of cosine operator functions to be chaotic.

Hybrid Artificial Bee Colony Algorithm and Particle Swarm Search for Global Optimization

Artificial bee colony (ABC) algorithm is one of the most recent swarm intelligence based algorithms, which has been shown to be competitive to other population-based algorithms. However, there is still an insufficiency in ABC regarding its solution search equation, which is good at exploration but poor at exploitation. To overcome this problem, we propose a novel artificial bee colony algorithm based on particle swarm search mechanism. In this algorithm, for improving the convergence speed, the initial population is generated by using good point set theory rather than random selection firstly. Secondly, in order to enhance the exploitation ability, the employed bee, onlookers, and scouts utilize the mechanism of PSO to search new candidate solutions. Finally, for further improving the searching ability, the chaotic search operator is adopted in the best solution of the current iteration. Our algorithm is tested on some well-known benchmark functions and compared with other algorithms. Results show that our algorithm has good performance.

A hybrid Univariate Marginal Distribution Algorithm for dynamic economic dispatch of units considering valve-point effects and ramp rates

Summary This paper presents a new approach for dynamic economic dispatch (DED) problem in power system by using a hybrid Univariate Marginal Distribution Algorithm (HUMDA). The DED problem with valve-point effects and ramp rate limits is a nonliner constrained optimization problem with non-convex and non-smooth characteristics. In the proposed method, a two-stage adaptive mechanism is devised to control parameters of the Univariate Marginal Distribution Algorithm in continuous domains (UMDAc) dynamically and lead the algorithm with better search efficiency; a chaotic local search operator is integrated with UMDAc to effectively avoid premature convergence. Moreover, a constraint handle according to the two-stage adaptive mechanism is proposed, and the results show that the strategy can handle constraints effectively. Finally, the efficiency of the proposed method is validated on two test systems consisting of 5, 10 and 30 thermal units. The results show the superiority of the proposed method while it is compared with other works in the area. Copyright © 2013 John Wiley & Sons, Ltd.

Restrictions of an invertible chaotic operator to its invariant subspaces

Let M be a closed subspace of a separable, infinite dimensional Hilbert space H with dim(H/M)=∞. We show that a bounded linear operator A:M→M has an invertible chaotic extension T:H→H if and only if A is bounded below. Motivated by our result, we further show that A:M→M has a chaotic Fredholm extension T:H→H if and only if A is left semi-Fredholm.

Solving Optimal Power Flow Problems Using Chaotic Self-adaptive Differential Harmony Search Algorithm

Optimal power flow is the basic tool that allows an electric utility to determine the economic and secure operating conditions of an electric power system. This article presents a chaotic self-adaptive differential harmony search algorithm to solve optimal power flow problems with non-smooth and non-convex cost functions. The searching capacity of the proposed chaotic self-adaptive differential harmony search algorithm has been improved by introducing a chaotic self-adaptive differential mutation operator instead of a pitch adjustment operator in the harmony search algorithm. The effectiveness of the proposed chaotic self-adaptive differential harmony search algorithm has been tested with IEEE 30-bus, IEEE 300-bus, and 66-bus Indian utility systems. The simulation results obtained using the proposed algorithm are compared with other variants of the improved harmony search algorithm, such as the differential harmony search algorithm, the chaotic differential harmony search algorithm, the interior point method, and other techniques reported in the literature, to show its effectiveness. The results obtained by the proposed algorithm are found to be better than the results obtained by the differential harmony search, chaotic differential harmony search, interior point method, and other algorithms reported in the literature in terms of solution quality and standard deviation of generation cost. In terms of speed of convergence and computational time, the proposed algorithm is better than the differential harmony search and chaotic differential harmony search algorithms.

A novel chaos danger model immune algorithm

Making use of ergodicity and randomness of chaos, a novel chaos danger model immune algorithm (CDMIA) is presented by combining the benefits of chaos and danger model immune algorithm (DMIA). To maintain the diversity of antibodies and ensure the performances of the algorithm, two chaotic operators are proposed. Chaotic disturbance is used for updating the danger antibody to exploit local solution space, and the chaotic regeneration is referred to the safe antibody for exploring the entire solution space. In addition, the performances of the algorithm are examined based upon several benchmark problems. The experimental results indicate that the diversity of the population is improved noticeably, and the CDMIA exhibits a higher efficiency than the danger model immune algorithm and other optimization algorithms.

Artificial bee colony algorithm based on chaos local search operator

In order to improve the ability of Artificial Bee Colony(ABC) algorithm at exploitation,a new Chaos Artificial Bee Colony(CH-ABC) algorithm was proposed for continuous function optimization problems.A new chaotic local search operator was embedded in the framework of the new algorithm.The new operator,whose search radius shrinks with the evolution generation,can do the local search around the best food source.The simulation results show that: compared with those of ABC algorithm,the solution quality and the convergence speed of the new algorithm are better for Rosenbrock and the convergence speed of the new algorithm is better for Griewank and Rastrigin.

Wind speed prediction by chaotic operator network based on Kalman Filter

A novel prediction network composed of some chaotic operators is proposed to predict the wind speed series. Training samples are constructed by the theory of phase space reconstruction. Genetic algorithm is adopted to optimize the control parameters of chaotic operators to change the dynamic characteristic of the network to approach to the predicted system. In this way, the dynamic prediction of wind speed series can be completed. The wind acceleration series can also be predicted by the same network. And the prediction results of both series can be fused by Kalman Filter to get the optimal estimation prediction result of the wind speed series, which is superior to the result obtained by each single method. Simulation results show that the prediction network has less computation cost than BP neural network, and it has better prediction performance than BP neural network and autoregressive integrated moving average model. Kalman Filter can improve the prediction performance further.

Chaotic self-adaptive differential harmony search algorithm based dynamic economic dispatch

Dynamic economic dispatch (DED) is one of the main optimization problems in electrical power system operation and control. DED problem is a non-smooth and non-convex problem when valve point effect, ramp-rate limits and prohibited operating zones of generation units are taken into account. This paper proposes an efficient chaotic self-adaptive differential harmony search (CSADHS) algorithm to solve the complicated DED problem in the presence of valve point effect, ramp-rate limits and prohibited operating zones constraints. In the proposed algorithm, chaotic self-adaptive differential mutation operator is used instead of pitch adjustment operator in the harmony search (HS) algorithm, to enhance the searching performance to find the quality solution. The effectiveness of the proposed algorithm is demonstrated on 10, 15 and 30 unit systems for a period of 24h. The simulation results obtained by the proposed algorithm are compared with the results obtained, using differential harmony search (DHS) algorithm, chaotic differential harmony search (CDHS) algorithm, and also with the results of other methods available in the literature. In terms of solution quality, the proposed algorithm is found to be better than other algorithms and in terms of speed of convergence, standard deviation of generation cost, and computational time, the proposed algorithm is better than DHS and CDHS algorithm.

An Adaptive Mutated Multi-objective Particle Swarm Optimization with an Entropy-based Density Assessment Scheme

In order to solve the problem of premature convergence in MOPSO and attain solutions with good diversity and distribution, a new algorithm has been proposed in this study. The algorithm adopts a new density assessment scheme on the basis of particles’ entropy information, which helps to obtain a Pareto set with uniformly distributed solutions. Also an adaptive chaotic mutation operator is designed to avoid premature convergence and help particles explore more efficiently in search space. In addition, the proposed algorithm is validated through comparisons with two existing state-of-the-art multi-objective algorithms using established benchmarks and metrics. Results show that the proposed algorithm shows better distribution performance than the compared algorithms while maintains a good convergence performance.

A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm

In this paper, we proposed a hybrid system to predict corporate bankruptcy. The whole procedure consists of the following four stages: first, sequential forward selection was used to extract the most important features; second, a rule-based model was chosen to fit the given dataset since it can present physical meaning; third, a genetic ant colony algorithm (GACA) was introduced; the fitness scaling strategy and the chaotic operator were incorporated with GACA, forming a new algorithm—fitness-scaling chaotic GACA (FSCGACA), which was used to seek the optimal parameters of the rule-based model; and finally, the stratifiedK-fold cross-validation technique was used to enhance the generalization of the model. Simulation experiments of 1000 corporations’ data collected from 2006 to 2009 demonstrated that the proposed model was effective. It selected the 5 most important factors as “net income to stock broker’s equality,” “quick ratio,” “retained earnings to total assets,” “stockholders’ equity to total assets,” and “financial expenses to sales.” The total misclassification error of the proposed FSCGACA was only 7.9%, exceeding the results of genetic algorithm (GA), ant colony algorithm (ACA), and GACA. The average computation time of the model is 2.02 s.

A novel swarm optimiser for flexible flow shop scheduling

In this work, a particle swarm optimisation (PSO) algorithm with chaotic mutation operator is proposed to solve flexible flow shop scheduling problems. Mutation, a commonly used operator in genetic algorithm, has been introduced so that common problem of trapping of solutions at local minima in PSO can be avoided. Chaotic sequence using logistic mapping is used instead of random numbers to improve the diversity in solution space. The performance of schedules is evaluated in terms of total completion time or makespan (Cmax). The results are presented in terms of percentage deviation (PD) of the solution from the lower bound (LB). The results are compared with different versions of genetic algorithm (GA) used for the purpose from open literature. The results indicate that the proposed PSO algorithm is quite effective in reducing makespan because average percentage deviation is observed as 6.390 whereas GA produces an average percentage deviation of 9.657. Finally, influence of various PSO parameters on solution quality has been investigated.

Dense chaos and densely chaotic operators

Dense chaos and densely chaotic operators

A Hybrid Method to Search the Optimal Hamiltonian Circuit

The objective of the well-known travelling salesman problem (TSP) is to search the optimal Hamiltonian circuit (OHC) in a tourist map. Finding the OHC becomes hard once the number of the cities and routes in the tourist map are large. The four vertices and three lines inequality was introduced as the constraints of the local optimal Hamiltonian paths (LOHPs) included in the OHC. The chaotic depth-priority algorithm was designed by adding the computation process with the chaotic operator to verify the rationality of the LOHPs generated with the depth-priority algorithm under the inequality constraints. A lot of non-LOHPs are abandoned in the search process and the search space of the OHC is reduced greatly. The method was verified with an example and it can be applied to the network optimization, path planning, task scheduling, assembly sequence planning etc.

A chaotic non-dominated sorting genetic algorithm for the multi-objective automatic test task scheduling problem

Solving a task scheduling problem is a key challenge for automatic test technology to improve throughput, reduce test time, and operate the necessary instruments at their maximum capacity. Therefore, this paper attempts to solve the automatic test task scheduling problem (TTSP) with the objectives of minimizing the maximal test completion time (makespan) and the mean workload of the instruments. In this paper, the formal formulation and the constraints of the TTSP are established to describe this problem. Then, a new encoding method called the integrated encoding scheme (IES) is proposed. This encoding scheme is able to transform a combinatorial optimization problem into a continuous optimization problem, thus improving the encoding efficiency and reducing the complexity of the genetic manipulations. More importantly, because the TTSP has many local optima, a chaotic non-dominated sorting genetic algorithm (CNSGA) is presented to avoid becoming trapped in local optima and to obtain high quality solutions. This approach introduces a chaotic initial population, a crossover operator, and a mutation operator into the non-dominated sorting genetic algorithm II (NSGA-II) to enhance the local searching ability. Both the logistic map and the cat map are used to design the chaotic operators, and their performances are compared. To identify a good approach for hybridizing NSGA-II and chaos, and indicate the effectiveness of IES, several experiments are performed based on the following: (1) a small-scale TTSP and a large-scale TTSP in real-world applications and (2) a TTSP used in other research. Computational simulations and comparisons show that CNSGA improves the local searching ability and is suitable for solving the TTSP.

Frequent hypercyclicity, chaos, and unconditional Schauder decompositions

We prove that if X is any complex separable infinite-dimensional Banach space with an unconditional Schauder decomposition, X supports an operator T which is chaotic and frequently hypercyclic. In contrast with the complex case, we observe that there are real Banach spaces with an unconditional basis which support no chaotic operator.

CHAOTIC PROPERTY OF WEIGHTED COMPOSITION OPERATORS

Abstract. In the present paper, we study the chaotic property of wei-ghted composition operators acting on the holomorphic function space H (U). 1. IntroductionA continuous linear operator T acting on a separable Frechet space X iscalled hypercyclic provided there exists a vector x 2 X such that its orbitorb( T;x ) = fT n x : n = 0 ; 1 ;:::g is dense in X . A periodic point for T is avector x2 X such that T n x = x for some n2 N. Finally, T is said to bechaotic if it is hypercyclic and its set of periodic points is dense in X .In 1929, ff [1] showed that the translation operator T a : H (C) ! H (C)de ned by ( T a f )( z ) = f ( z + a ), a= 0, is hypercyclic on the Frechet space H (C)of entire functions. This result was generalized by Godefroy and Shapiro [7]who proved that each operator on H (C N ) which commutes with all translationsand is not a scalar multiple of the identity, is chaotic. Other classical examplesof hypercyclic and chaotic operators are weighted shifts on l p spaces [8, 10] andadjoints of multiplication operators on Hilbert spaces of holomorphic functions[7].Let U stand for the open unit disk in C. Each

On the equivalence of four chaotic operators

In this paper, we study chaos for bounded operators on Banach spaces. First, it is proved that, for a bounded operator T defined on a Banach space, Li–Yorke chaos, Li–Yorke sensitivity, spatio-temporal chaos, and distributional chaos in a sequence are equivalent, and they are all strictly stronger than sensitivity. Next, we show that T is sensitive dependence iff sup { ‖ T n ‖ : n ∈ N } = ∞ . Finally, the following results are obtained: (1) T is chaotic iff T n is chaotic for each n ∈ N . (2) The product operator T n ∗ = ∏ i = 1 n T i is chaotic iff T k is chaotic for some k ∈ { 1 , 2 , … , n } .

Improving Clustering-based Differential Evolution with Chaotic Sequences and New Mutation Operator

Differential evolution (DE) algorithms compose an efficient type of evolutionary algorithm (EA) for the global optimization domain. But DE is not completely free from the problems of slow and/or premature convergence. In this paper, an improving clustering-based differential evolution with chaotic sequences and new mutation operator (CCDE) is proposed for the unconstrained global optimization problems. In CCDE, the population is partitioned into k subsets by a clustering algorithm, and each subset is considered to be the cluster neighborhood. And these cluster centers and chaotic sequences using logistic equation are used to design the new differential evolution mutation operators. This method utilizes the concept of the cluster neighborhood of each population member. The CEC2005 benchmark functions are employed for experimental verification. Experimental results indicate that CCDE is highly competitive compared to the state-of-the-art DE algorithms.

Chaotic extensions of operators on Hilbert subspaces

If M is a closed subspace of a separable, infinite dimensional Hilbert space H with dim (H/M) = ∞, we show that every bounded linear operator A: M → M can be extended to a chaotic operator T: H → H that satisfies the hypercyclicity criterion in the strongest possible sense.