Showing papers in "Chaos Solitons & Fractals in 2009"
TL;DR: In this article, a direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations, which provides a more systematical and convenient handling of the solution process of non-linear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, and the mapping method.
Abstract: A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F -expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3 + 1 dimensional Jimbo–Miwa equation is treated, together with a Backlund transformation.
510 citations
TL;DR: It has been detected that coupling emergent results in different areas, like those of PSO and complex dynamics, can improve the quality of results in some optimization problems.
Abstract: This paper proposes new particle swarm optimization (PSO) methods that use chaotic maps for parameter adaptation. This has been done by using of chaotic number generators each time a random number is needed by the classical PSO algorithm. Twelve chaos-embedded PSO methods have been proposed and eight chaotic maps have been analyzed in the benchmark functions. It has been detected that coupling emergent results in different areas, like those of PSO and complex dynamics, can improve the quality of results in some optimization problems. It has been also shown that, some of the proposed methods have somewhat increased the solution quality, that is in some cases they improved the global searching capability by escaping the local solutions.
451 citations
TL;DR: In this paper, the stability of linear integer-order circuits with one fractional element, two fractional elements of the same order or two fractions of different order is studied, and a general procedure for studying the system with many fractional components is also given.
Abstract: Linear integer-order circuits are a narrow subset of rational-order circuits which are in turn a subset of fractional-order. Here, we study the stability of circuits having one fractional element, two fractional elements of the same order or two fractional elements of different order. A general procedure for studying the stability of a system with many fractional elements is also given. It is worth noting that a fractional element is one whose impedance in the complex frequency s-domain is proportional to sα and α is a positive or negative fractional-order. Different transformations and methods will be illustrated via examples.
298 citations
TL;DR: In this paper, instead of Adomian polynomials, instead of the Adomians, He polynomial is introduced based on homotopy perturbation method and the solution procedure becomes easier, simpler and more straightforward.
Abstract: The Adomian decomposition method is widely used in approximate calculation. The main difficulty of the method is to calculate Adomian polynomials, the procedure is very complex. In order to overcome the demerit, this paper suggests an alternative approach to Adomian method, instead of Adomian polynomials, He polynomials are introduced based on homotopy perturbation method. The solution procedure becomes easier, simpler, and more straightforward.
285 citations
TL;DR: A chaos-based image encryption algorithm with variable control parameters that can effectively resist all known attacks against permutation–diffusion architectures is proposed.
Abstract: In recent years, a number of image encryption algorithms based on the permutation–diffusion structure have been proposed. However, the control parameters used in the permutation stage are usually fixed in the whole encryption process, which favors attacks. In this paper, a chaos-based image encryption algorithm with variable control parameters is proposed. The control parameters used in the permutation stage and the keystream employed in the diffusion stage are generated from two chaotic maps related to the plain-image. As a result, the algorithm can effectively resist all known attacks against permutation–diffusion architectures. Theoretical analyses and computer simulations both confirm that the new algorithm possesses high security and fast encryption speed for practical image encryption.
268 citations
TL;DR: The cause of potential flaws in the original algorithm is analyzed in detail, and then the corresponding enhancement measures are proposed, and it is indicated that the improved algorithm can overcome these flaws and maintain all the merits of the original one.
Abstract: The security of digital image attracts much attention recently. In Guan et al. [Guan Z, Huang F, Guan W. Chaos-based image encryption algorithm. Phys Lett A 2005; 346: 153–7.], a chaos-based image encryption algorithm has been proposed. In this paper, the cause of potential flaws in the original algorithm is analyzed in detail, and then the corresponding enhancement measures are proposed. Both theoretical analysis and computer simulation indicate that the improved algorithm can overcome these flaws and maintain all the merits of the original one.
247 citations
TL;DR: The results of several experimental, statistical analysis and key sensitivity tests show that the proposed image encryption scheme provides an efficient and secure way for real-time image encryption and transmission.
Abstract: Image encryption is somehow different from text encryption due to some inherent features of image such as bulk data capacity and high correlation among pixels, which are generally difficult to handle by conventional methods. The desirable cryptographic properties of the chaotic maps such as sensitivity to initial conditions and random-like behavior have attracted the attention of cryptographers to develop new encryption algorithms. Therefore, recent researches of image encryption algorithms have been increasingly based on chaotic systems, though the drawbacks of small key space and weak security in one-dimensional chaotic cryptosystems are obvious. This paper proposes a Coupled Nonlinear Chaotic Map, called CNCM, and a novel chaos-based image encryption algorithm to encrypt color images by using CNCM. The chaotic cryptography technique which used in this paper is a symmetric key cryptography with a stream cipher structure. In order to increase the security of the proposed algorithm, 240 bit-long secret key is used to generate the initial conditions and parameters of the chaotic map by making some algebraic transformations to the key. These transformations as well as the nonlinearity and coupling structure of the CNCM have enhanced the cryptosystem security. For getting higher security and higher complexity, the current paper employs the image size and color components to cryptosystem, thereby significantly increasing the resistance to known/chosen-plaintext attacks. The results of several experimental, statistical analysis and key sensitivity tests show that the proposed image encryption scheme provides an efficient and secure way for real-time image encryption and transmission.
216 citations
TL;DR: Two pseudorandom binary sequence generators, based on logistic chaotic maps intended for stream cipher applications, are proposed, which possess high linear complexity and very good statistical properties.
Abstract: Two pseudorandom binary sequence generators, based on logistic chaotic maps intended for stream cipher applications, are proposed. The first is based on a single one-dimensional logistic map which exhibits random, noise-like properties at given certain parameter values, and the second is based on a combination of two logistic maps. The encryption step proposed in both algorithms consists of a simple bitwise XOR operation of the plaintext binary sequence with the keystream binary sequence to produce the ciphertext binary sequence. A threshold function is applied to convert the floating-point iterates into binary form. Experimental results show that the produced sequences possess high linear complexity and very good statistical properties. The systems are put forward for security evaluation by the cryptographic committees.
204 citations
TL;DR: This paper proposes an OCML-based colour image encryption scheme with a stream cipher structure using a 192-bit-long external key to generate the initial conditions and the parameters of the OCML, which is modelled by one-way coupled-map lattices.
Abstract: The chaos-based cryptographic algorithms have suggested some new ways to develop efficient image-encryption schemes. While most of these schemes are based on low-dimensional chaotic maps, it has been proposed recently to use high-dimensional chaos namely spatiotemporal chaos, which is modelled by one-way coupled-map lattices (OCML). Owing to their hyperchaotic behaviour, such systems are assumed to enhance the cryptosystem security. In this paper, we propose an OCML-based colour image encryption scheme with a stream cipher structure. We use a 192-bit-long external key to generate the initial conditions and the parameters of the OCML. We have made several tests to check the security of the proposed cryptosystem namely, statistical tests including histogram analysis, calculus of the correlation coefficients of adjacent pixels, security test against differential attack including calculus of the number of pixel change rate (NPCR) and unified average changing intensity (UACI), and entropy calculus. The cryptosystem speed is analyzed and tested as well.
187 citations
TL;DR: In this paper, the authors extended fractional differential transform method (FDTM) to solve fractional integro-differential equations of Volterra type and proved new theorems for the transformation of integral terms having degenerate kernels.
Abstract: In this study, fractional differential transform method (FDTM), which is a semi analytical numerical technique, is extended to solve fractional integro-differential equations of Volterra type. New theorems for the transformation of integral terms having degenerate kernels that never existed before are introduced with their proofs. This implemented new technique is validated by solving and comparing four different examples that exist in the literature. It is observed that, FDTM can be utilized as a powerful and reliable tool for the solution of fractional integro-differential equations.
177 citations
TL;DR: In this paper, the authors proposed a statistical interpretation for the variation of the b-value during the evolution of damage, based on a treatment originally proposed by Carpinteri A. The proposed model captures the transition from the condition of criticality, in which α = 3, to that of imminent failure, characterized by α = 2, in terms of damage localisation.
Abstract: Extensive research and studies on concrete fracture and failure by means of the acoustic emission (AE) technique have shown that fracture and damage growth can be characterized through a single synthetic parameter, namely the b-value, which changes systematically during the different stages of the failure process, as shown by several AE tests carried out from the specimen to the structural scale [Sammonds PR, Meredith PG, Murrel SAF, Main IG. Modelling the damage evolution in rock containing porefluid by acoustic emission. In: Proceedings of the Eurock’94; 1994; Colombo S, Main IG, Forde MC. Assessing damage of reinforced concrete beam using “b-value” analysis of acoustic emission signals. J Mater Civil Eng ASCE 2003;15:280–6; Carpinteri A, Lacidogna G, Niccolini G. Critical behaviour in concrete structures and damage localisation by Acoustic Emission. Key Eng Mater 2006;312:305–10]. This parameter can be linked to the value of the exponent α of the power-law distribution of the crack size in a damaged structure. In this paper, we propose a statistical interpretation for the variation of the b-value during the evolution of damage, based on a treatment originally proposed by [Carpinteri A. Mechanical damage and crack growth in concrete: plastic collapse to brittle fracture. Dordrecht: Martinus Nijhoff Publishers; 1986; Carpinteri A. Decrease of apparent tensile and bending strength with specimen size: two different explanations based on fracture mechanics. Int J Solid Struct 1989;25:407–29; Carpinteri A. Scaling laws and renormalization groups for strength and toughness of disordered materials. Int J Solid Struct 1994;31:291–302]. The proposed model captures the transition from the condition of criticality, in which α = 3, to that of imminent failure, characterized by α = 2, in terms of damage localisation.
TL;DR: A novel chaotic PSO combined with an implicit filtering (IF) local search method to solve economic dispatch problems using chaos mapping using Henon map sequences which increases its convergence rate and resulting precision.
Abstract: Particle swarm optimization (PSO) is a population-based swarm intelligence algorithm driven by the simulation of a social psychological metaphor instead of the survival of the fittest individual. Based on the chaotic systems theory, this paper proposed a novel chaotic PSO combined with an implicit filtering (IF) local search method to solve economic dispatch problems. Since chaotic mapping enjoys certainty, ergodicity and the stochastic property, the proposed PSO introduces chaos mapping using Henon map sequences which increases its convergence rate and resulting precision. The chaotic PSO approach is used to produce good potential solutions, and the IF is used to fine-tune of final solution of PSO. The hybrid methodology is validated for a test system consisting of 13 thermal units whose incremental fuel cost function takes into account the valve-point loading effects. Simulation results are promising and show the effectiveness of the proposed approach.
TL;DR: A tuning method for determining the parameters of PID control for an automatic regulator voltage (AVR) system using a chaotic optimization approach based on Lozi map is proposed, which introduces chaos mapping usingLozi map chaotic sequences which increases its convergence rate and resulting precision.
Abstract: Despite the popularity, the tuning aspect of proportional–integral-derivative (PID) controllers is a challenge for researchers and plant operators. Various controllers tuning methodologies have been proposed in the literature such as auto-tuning, self-tuning, pattern recognition, artificial intelligence, and optimization methods. Chaotic optimization algorithms as an emergent method of global optimization have attracted much attention in engineering applications. Chaotic optimization algorithms, which have the features of easy implementation, short execution time and robust mechanisms of escaping from local optimum, is a promising tool for engineering applications. In this paper, a tuning method for determining the parameters of PID control for an automatic regulator voltage (AVR) system using a chaotic optimization approach based on Lozi map is proposed. Since chaotic mapping enjoys certainty, ergodicity and the stochastic property, the proposed chaotic optimization introduces chaos mapping using Lozi map chaotic sequences which increases its convergence rate and resulting precision. Simulation results are promising and show the effectiveness of the proposed approach. Numerical simulations based on proposed PID control of an AVR system for nominal system parameters and step reference voltage input demonstrate the good performance of chaotic optimization.
TL;DR: In this article, a new analytic technique is proposed to solve non-linear Riccati differential equation with fractional order, which provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter.
Abstract: In this paper, based on the homotopy analysis method (HAM), a new analytic technique is proposed to solve non-linear Riccati differential equation with fractional order. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter ℏ . Besides, it is proved that well-known Adomian’s decomposition method is a special case of the homotopy analysis method when ℏ = −1. This work illustrates the validity and great potential of the homotopy analysis method for the non-linear differential equations with fractional order. The basic ideas of this approach can be widely employed to solve other strongly non-linear problems in fractional calculus.
TL;DR: In this paper, the existence and uniqueness of solutions for a class of fractional-order Lorenz chaotic systems are investigated theoretically, and the stability of the corresponding equilibria is also argued similarly to the integer-order counterpart.
Abstract: The dynamic behaviors of fractional-order differential systems have received increasing attention in recent decades. But many results about fractional-order chaotic systems are attained only by using analytic and numerical methods. Based on the qualitative theory, the existence and uniqueness of solutions for a class of fractional-order Lorenz chaotic systems are investigated theoretically in this paper. The stability of the corresponding equilibria is also argued similarly to the integer-order counterpart. According to the obtained results, the bifurcation conditions of these two systems are significantly different. Numerical solutions, together with simulations, finally verify the correctness of our analysis.
TL;DR: In this paper, the steady two-dimensional magnetohydrodynamic flow of an upper-convected Maxwell fluid near a stagnation point over a stretching surface is analyzed and the governing nonlinear partial differential equation for the flow are reduced to an ordinary differential equation by using similarity transformations.
Abstract: The present analysis comprises the steady two-dimensional magnetohydrodynamic flow of an upper-convected Maxwell fluid near a stagnation-point over a stretching surface. The governing non-linear partial differential equation for the flow are reduced to an ordinary differential equation by using similarity transformations. The analytic solution of nonlinear system is constructed in the series form using Homotopy analysis method. Convergence of the obtained series is discussed explicitly. The effects of the sundry parameters on the velocity profile is shown through graphs. The values of skin-friction coefficient for different parameters is tabulated.
TL;DR: In this paper, the He's homotopy perturbation method is applied to solve systems of Volterra integral equations of the second kind, and the results reveal that the method is very effective and simple.
Abstract: In this paper, the He’s homotopy perturbation method is applied to solve systems of Volterra integral equations of the second kind. Some examples are presented to illustrate the ability of the method for linear and non-linear such systems. The results reveal that the method is very effective and simple.
TL;DR: Theoretical analysis and experimental results show that the scheme has good cryptographic security and perceptual security, and it does not affect the compression efficiency apparently, which makes the scheme a suitable choice for practical applications.
Abstract: In this paper, an efficient image/video encryption scheme is constructed based on spatiotemporal chaos system. The chaotic lattices are used to generate pseudorandom sequences and then encrypt image blocks one by one. By iterating chaotic maps for certain times, the generated pseudorandom sequences obtain high initial-value sensitivity and good randomness. The pseudorandom-bits in each lattice are used to encrypt the Direct Current coefficient (DC) and the signs of the Alternating Current coefficients (ACs). Theoretical analysis and experimental results show that the scheme has good cryptographic security and perceptual security, and it does not affect the compression efficiency apparently. These properties make the scheme a suitable choice for practical applications.
TL;DR: In this paper, the homotopy analysis method (HAM) was applied to the case of two-dimensional and axisymmetric shrinking and the convergence of the obtained series solution was discussed explicitly.
Abstract: This work is concerned with the magnetohydrodynamic (MHD) viscous flow due to a shrinking sheet. The cases of two dimensional and axisymmetric shrinking have been discussed. Exact series solution is obtained using the homotopy analysis method (HAM). The convergence of the obtained series solution is discussed explicitly. The obtained HAM solution is valid for all values of the suction parameter and Hartman number.
TL;DR: This paper proposes a more efficient diffusion mechanism using simple table lookup and swapping techniques as a light-weight replacement of the 1D chaotic map iteration and shows that at a similar security level, the proposed cryptosystem needs about one-third the encryption time of a similar cryptos system.
Abstract: One of the existing chaos-based image cryptosystems is composed of alternative substitution and diffusion stages. A multi-dimensional chaotic map is usually employed in the substitution stage for image pixel permutation while a one-dimensional (1D) chaotic map is used for diffusion purpose. As the latter usually involves real number arithmetic operations, the overall encryption speed is limited by the diffusion stage. In this paper, we propose a more efficient diffusion mechanism using simple table lookup and swapping techniques as a light-weight replacement of the 1D chaotic map iteration. Simulation results show that at a similar security level, the proposed cryptosystem needs about one-third the encryption time of a similar cryptosystem. The effective acceleration of chaos-based image cryptosystems is thus achieved.
TL;DR: This paper studies the concept of statistically convergent and statistically Cauchy double sequences in intuitionistic fuzzy normed spaces and constructs an example of a double sequence to show that in IFNS statistical convergence does not imply convergence and the method of convergence even for double sequences is stronger than the usual convergence in intuitionism fuzzy norming space.
Abstract: Recently, the concept of intuitionistic fuzzy normed spaces was introduced by Saadati and Park [Saadati R, Park JH. Chaos, Solitons & Fractals 2006;27:331–44]. Karakus et al. [Karakus S, Demirci K, Duman O. Chaos, Solitons & Fractals 2008;35:763–69] have quite recently studied the notion of statistical convergence for single sequences in intuitionistic fuzzy normed spaces. In this paper, we study the concept of statistically convergent and statistically Cauchy double sequences in intuitionistic fuzzy normed spaces. Furthermore, we construct an example of a double sequence to show that in IFNS statistical convergence does not imply convergence and our method of convergence even for double sequences is stronger than the usual convergence in intuitionistic fuzzy normed space.
TL;DR: In this paper, the derivatives of k-Fibonacci polynomials are presented in the form of convolution of KF-FBNs and their properties admit a straightforward proof.
Abstract: The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of their properties admit a straightforward proof. Here in particular, we present the derivatives of these polynomials in the form of convolution of k-Fibonacci polynomials. This fact allows us to present in an easy form a family of integer sequences in a new and direct way. Many relations for the derivatives of Fibonacci polynomials are proven. � 2007 Elsevier Ltd. All rights reserved.
TL;DR: In this article, the properties of δ-β-continuous functions are investigated and characterizations and relationships with related functions are discussed, and a decomposition of continuity and complete continuity is given.
Abstract: In Hatir and Noiri [Hatir E, Noiri T. Decompositions of continuity and complete continuity. Acta Math Hungary 113(4);2006:281–287], δ–β-continuity has given to obtain a decomposition of continuity. In this paper, we investigate the properties of δ–β-continuous functions and discuss characterizations and the relationships with related functions.
TL;DR: In this article, B-spline functions were used to develop a numerical method for computing approximations to the solution of non-linear singular boundary value problems associated with physiology science.
Abstract: We use B-spline functions to develop a numerical method for computing approximations to the solution of non-linear singular boundary value problems associated with physiology science. The original differential equation is modified at singular point then the boundary value problem is treated by using B-spline approximation. The numerical method is tested for its efficiency by considering three model problems from physiology.
TL;DR: In this paper, an SIRS epidemic model with a nonlinear incidence rate and a time delay is investigated, and the local stability of an endemic equilibrium and a disease-free equilibrium is discussed.
Abstract: In this paper, an SIRS epidemic model with a nonlinear incidence rate and a time delay is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. By comparison arguments, it is proved that if the basic reproductive number R 0 1 , the disease-free equilibrium is globally asymptotically stable. If R 0 > 1 , by means of an iteration technique, sufficient conditions are derived for the global asymptotic stability of the endemic equilibrium.
TL;DR: In this paper, a multiple delayed state-feedback control design for exponential H ∞ synchronization problem of a class of delayed neural networks with multiple time-varying discrete delays is presented.
Abstract: This article aims to present a multiple delayed state-feedback control design for exponential H ∞ synchronization problem of a class of delayed neural networks with multiple time-varying discrete delays. On the basis of the drive–response concept and by introducing a descriptor technique and using Lyapunov–Krasovskii functional, new delay-range-dependent sufficient conditions for exponential H ∞ synchronization of the drive–response structure of neural networks are driven in terms of linear matrix inequalities (LMIs). The explicit expression of the controller gain matrices are parameterized based on the solvability conditions such that the drive system and the response system can be exponentially synchronized. A numerical example is included to illustrate the applicability of the proposed design method.
TL;DR: In this article, the main concepts and ideas upon which the theory of fractal-cantorian spacetime is based are reviewed and a rather detailed introduction and up-to-date review is given.
Abstract: The paper gives a rather detailed introduction and up to date review of the main concepts and ideas upon which the theory of fractal-Cantorian spacetime is based.
TL;DR: In this article, the authors presented fractional Taylor type formulae with fractional integral remainder and fractional differential forms for the right Caputo fractional derivative, the right generalized fractional derivatives of Canavati type [Canavati JA. Nieuw Archief Voor Wiskunde 1987;5(1):53-75] and their corresponding right fractional integrals.
Abstract: Here are presented fractional Taylor type formulae with fractional integral remainder and fractional differential formulae, regarding the right Caputo fractional derivative, the right generalized fractional derivative of Canavati type [Canavati JA. The Riemann–Liouville integral. Nieuw Archief Voor Wiskunde 1987;5(1):53–75] and their corresponding right fractional integrals. Then are given representation formulae of functions as fractional integrals of their above fractional derivatives, as well as of their right and left Weyl fractional derivatives. At the end, we mention some far reaching implications of our theory to mathematical analysis computational methods. Also we compare the right Caputo fractional derivative to right Riemann–Liouville fractional derivative.
TL;DR: Numerical results reveal that the proposed IHS method is a powerful search and controller design optimization tool for synchronization of chaotic systems.
Abstract: The harmony search (HS) algorithm is a recently developed meta-heuristic algorithm, and has been very successful in a wide variety of optimization problems. HS was conceptualized using an analogy with music improvisation process where music players improvise the pitches of their instruments to obtain better harmony. The HS algorithm does not require initial values and uses a random search instead of a gradient search, so derivative information is unnecessary. Furthermore, the HS algorithm is simple in concept, few in parameters, easy in implementation, imposes fewer mathematical requirements, and does not require initial value settings of the decision variables. In recent years, the investigation of synchronization and control problem for discrete chaotic systems has attracted much attention, and many possible applications. The tuning of a proportional–integral–derivative (PID) controller based on an improved HS (IHS) algorithm for synchronization of two identical discrete chaotic systems subject the different initial conditions is investigated in this paper. Simulation results of the IHS to determine the PID parameters to synchronization of two Henon chaotic systems are compared with other HS approaches including classical HS and global-best HS. Numerical results reveal that the proposed IHS method is a powerful search and controller design optimization tool for synchronization of chaotic systems.
TL;DR: In this article, the authors proposed a reliable algorithm to develop exact and approximate solutions for the linear and nonlinear Schrodinger equations based mainly on two-dimensional differential transform method which is one of the approximate methods.
Abstract: In this paper, we propose a reliable algorithm to develop exact and approximate solutions for the linear and nonlinear Schrodinger equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.