Showing papers in "Chaos Solitons & Fractals in 2007"

New periodic solutions for nonlinear evolution equations using Exp-function method

Abstract: The Exp-function method is used to obtain generalized solitonary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics using symbolic computation. The method is straightforward and concise, and its applications are promising.

A fast image encryption system based on chaotic maps with finite precision representation

Abstract: In this paper, a fast chaos-based image encryption system with stream cipher structure is proposed. In order to achieve a fast throughput and facilitate hardware realization, 32-bit precision representation with fixed point arithmetic is assumed. The major core of the encryption system is a pseudo-random keystream generator based on a cascade of chaotic maps, serving the purpose of sequence generation and random mixing. Unlike the other existing chaos-based pseudo-random number generators, the proposed keystream generator not only achieves a very fast throughput, but also passes the statistical tests of up-to-date test suite even under quantization. The overall design of the image encryption system is to be explained while detail cryptanalysis is given and compared with some existing schemes.

Solution of fractional differential equations by using differential transform method

Abstract: In this study, we implement a well known transformation technique, Differential Transform Method (DTM), to the area of fractional differential equations. Theorems that never existed before are introduced with their proofs. Also numerical examples are carried out for various types of problems, including the Bagley–Torvik, Ricatti and composite fractional oscillation equations for the application of the method. The results obtained are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, accurate and easy to apply.

Variational approach for nonlinear oscillators

Abstract: We propose a novel variational approach for limit cycles of a kind of nonlinear oscillators Some examples are given to illustrate the effectiveness and convenience of the method The obtained results are valid for the whole solution domain with high accuracy

The extended F-expansion method and its application for a class of nonlinear evolution equations

Abstract: By means of a simple transformation technique, we have shown that the higher-order nonlinear Schrodinger equation in nonlinear optical fibers, a new Hamiltonian amplitude equation, generalized Hirota–Satsuma coupled system and generalized ZK-BBM equation can be reduced to the elliptic-like equation Then, the extended F-expansion method is used to obtain a series of solutions including the single and the combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions to the above mentioned class of nonlinear evolution equations

Numerical comparison of methods for solving linear differential equations of fractional order

Abstract: In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between the two methods and a conventional method such as the fractional difference method for solving linear differential equations of fractional order. The numerical results demonstrates that the new methods are quite accurate and readily implemented.

On the efficiency of chaos optimization algorithms for global optimization

Abstract: Chaos optimization algorithms as a novel method of global optimization have attracted much attention, which were all based on Logistic map. However, we have noticed that the probability density function of the chaotic sequences derived from Logistic map is a Chebyshev-type one, which may affect the global searching capacity and computational efficiency of chaos optimization algorithms considerably. Considering the statistical property of the chaotic sequences of Logistic map and Kent map, the improved hybrid chaos-BFGS optimization algorithm and the Kent map based hybrid chaos-BFGS algorithm are proposed. Five typical nonlinear functions with multimodal characteristic are tested to compare the performance of five hybrid optimization algorithms, which are the conventional Logistic map based chaos-BFGS algorithm, improved Logistic map based chaos-BFGS algorithm, Kent map based chaos-BFGS algorithm, Monte Carlo-BFGS algorithm, mesh-BFGS algorithm. The computational performance of the five algorithms is compared, and the numerical results make us question the high efficiency of the chaos optimization algorithms claimed in some references. It is concluded that the efficiency of the hybrid optimization algorithms is influenced by the statistical property of chaotic/stochastic sequences generated from chaotic/stochastic algorithms, and the location of the global optimum of nonlinear functions. In addition, it is inappropriate to advocate the high efficiency of the global optimization algorithms only depending on several numerical examples of low-dimensional functions.

New exact travelling wave solutions using modified extended tanh-function method

Abstract: Based on computerized symbolic computation and modified extended tanh-function method for constructing a new exact travelling wave solutions of nonlinear evolution equations (NEEs) is presented and implemented in a computer algebraic system. Applying this method, with the aid of Maple, we consider some (NEEs) with mathematical physics interests. As a results, we can successfully recover the previously known solitary wave solutions that had been found by the tanh-function method and other more sophisticated methods.

Complex dynamic behaviors of a discrete-time predator-prey system

Abstract: The dynamics of a discrete-time predator–prey system is investigated in the closed first quadrant R + 2 . It is shown that the system undergoes flip bifurcation and Hopf bifurcation in the interior of R + 2 by using center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-5, 6, 9, 10, 14, 18, 20, 25 orbits, cascade of period-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors.

On the Fibonacci k-numbers

Abstract: We introduce a general Fibonacci sequence that generalizes, between others, both the classic Fibonacci sequence and the Pell sequence. These general kth Fibonacci numbers { F k , n } n = 0 ∞ were found by studying the recursive application of two geometrical transformations used in the well-known four-triangle longest-edge (4TLE) partition. Many properties of these numbers are deduce directly from elementary matrix algebra.

Modified projective synchronization of chaotic system

Abstract: A modified projective synchronization is proposed to acquire a general kind of proportional relationships between the drive and response systems. From rigorously control theory, a sufficient condition is attained for the stability of the error dynamics, and is applied to guiding the design of the controllers. Finally, we take Lorenz system as an example for illustration and verification.

Fractional-order dynamical models of love

Abstract: This paper examines fractional-order dynamical models of love. It offers a generalization of a dynamical model recently reported in the literature. The generalization is obtained by permitting the state dynamics of the model to assume fractional orders. The fact that fractional systems possess memory justifies this generalization, as the time evolution of romantic relationships is naturally impacted by memory. We show that with appropriate model parameters, strange chaotic attractors may be obtained under different fractional orders, thus confirming previously reported results obtained from integer-order models, yet at an advantage of reduced system order. Furthermore, this work opens a new direction of research whereby fractional derivative applications might offer more insight into the modeling of dynamical systems in psychology and life sciences. Our results are validated by numerical simulations.

The k-Fibonacci sequence and the Pascal 2-triangle

Abstract: The general k-Fibonacci sequence { F k , n } n = 0 ∞ were found by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge (4TLE) partition. This sequence generalizes, between others, both the classical Fibonacci sequence and the Pell sequence. In this paper many properties of these numbers are deduced and related with the so-called Pascal 2-triangle.

A study on compactly supported orthogonal vector-valued wavelets and wavelet packets

Abstract: In this paper, vector-valued multiresolution analysis and orthogonal vector-valued wavelets are introduced. The definition for orthogonal vector-valued wavelet packets is proposed. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is derived by means of paraunitary vector filter bank theory. An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented. The properties of the vector-valued wavelet packets are investigated by using operator theory and algebra theory. In particular, it is shown how to construct various orthonormal bases of L2(R, Cs) from the orthogonal vector-valued wavelet packets.

An extended method for obtaining S-boxes based on three-dimensional chaotic Baker maps

Abstract: Tang et al. proposed a novel method for obtaining S-boxes based on the well-known two-dimensional chaotic Baker map. Unfortunately, some mistakes exist in their paper. The faults are corrected first in this paper and then an extended method is put forward for acquiring cryptographically strong S-boxes. The new scheme employs a three-dimensional chaotic Baker map, which has more intensive chaotic characters than the two-dimensional one. In addition, the cryptographic properties such as the bijective property, the nonlinearity, the strict avalanche criterion, the output bits independence criterion and the equiprobable input/output XOR distribution are analyzed in detail for our S-box and revised Tang et al.’s one, respectively. The results of numerical analysis show that both of the two boxes can resist several attacks effectively and the three-dimensional chaotic map, a stronger sense in chaotic characters, can perform more smartly and more efficiently in designing S-boxes.

Exponential stability of uncertain stochastic neural networks with mixed time-delays

Abstract: This paper is concerned with the global exponential stability analysis problem for a class of stochastic neural networks with mixed time-delays and parameter uncertainties. The mixed delays comprise discrete and distributed time-delays, the parameter uncertainties are norm-bounded, and the neural networks are subjected to stochastic disturbances described in terms of a Brownian motion. The purpose of the stability analysis problem is to derive easy-to-test criteria under which the delayed stochastic neural network is globally, robustly, exponentially stable in the mean square for all admissible parameter uncertainties. By resorting to the Lyapunov–Krasovskii stability theory and the stochastic analysis tools, sufficient stability conditions are established by using an efficient linear matrix inequality (LMI) approach. The proposed criteria can be checked readily by using recently developed numerical packages, where no tuning of parameters is required. An example is provided to demonstrate the usefulness of the proposed criteria.

Continuous-time random walk and parametric subordination in fractional diffusion

Abstract: The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit, we obtain a (generally non-Markovian) diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination , applied to a combination of a Markov process with a positively oriented Levy process, we generate and display sample paths for some special cases.

A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method

Abstract: In this paper, Adomian’s decomposition method is proposed to solve the well-known Blasius equation. Comparison with homotopy perturbation method and Howarth’s numerical solution reveals that the Adomian’s decomposition method is of high accuracy.

Parameter estimation for chaotic systems by particle swarm optimization

Abstract: Parameter estimation for chaotic systems is an important issue in nonlinear science and has attracted increasing interests from various research fields, which could be essentially formulated as a multi-dimensional optimization problem. As a novel evolutionary computation technique, particle swarm optimization (PSO) has attracted much attention and wide applications, owing to its simple concept, easy implementation and quick convergence. However, to the best of our knowledge, there is no published work on PSO for estimating parameters of chaotic systems. In this paper, a PSO approach is applied to estimate the parameters of Lorenz system. Numerical simulation and the comparisons demonstrate the effectiveness and robustness of PSO. Moreover, the effect of population size on the optimization performances is investigated as well.

The one-dimensional heat equation subject to a boundary integral specification

Abstract: Various processes in the natural sciences and engineering lead to the nonclassical parabolic initial boundary value problems which involve nonlocal integral terms over the spatial domain. The integral term may appear in the boundary conditions. It is the reason for which such problems gained much attention in recent years, not only in engineering but also in the mathematics community. In this paper the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered. Several approaches for the numerical solution of this boundary value problem which have been considered in the literature, are reported. New finite difference techniques are proposed for the numerical solution of the one-dimensional heat equation subject to the specification of mass. Numerical examples are given at the end of this paper to compare the efficiency of the new techniques. Some specific applications in various engineering models are introduced.

Variational iteration method for one dimensional nonlinear thermoelasticity

Abstract: This paper applies the variational iteration method to solve the Cauchy problem arising in one dimensional nonlinear thermoelasticity. The advantage of this method is to overcome the difficulty of calculation of Adomian’s polynomials in the Adomian’s decomposition method. The numerical results of this method are compared with the exact solution of an artificial model to show the efficiency of the method. The approximate solutions show that the variational iteration method is a powerful mathematical tool for solving nonlinear problems.

Topology of the Italian airport network: A scale-free small-world network with a fractal structure?

Abstract: In this paper, for the first time we analyze the structure of the Italian Airport Network (IAN) looking at it as a mathematical graph and investigate its topological properties. We find that it has very remarkable features, being like a scale-free network, since both the degree and the “betweenness centrality” distributions follow a typical power-law known in literature as a Double Pareto Law. From a careful analysis of the data, the Italian Airport Network turns out to have a self-similar structure. In short, it is characterized by a fractal nature, whose typical dimensions can be easily determined from the values of the power-law scaling exponents. Moreover, we show that, according to the period examined, these distributions exhibit a number of interesting features, such as the existence of some “hubs”, i.e. in the graph theory’s jargon, nodes with a very large number of links, and others most probably associated with geographical constraints. Also, we find that the IAN can be classified as a small-world network because the average distance between reachable pairs of airports grows at most as the logarithm of the number of airports. The IAN does not show evidence of “communities” and this result could be the underlying reason behind the smallness of the value of the clustering coefficient, which is related to the probability that two nearest neighbors of a randomly chosen airport are connected.

Application of He’s homotopy perturbation method to functional integral equations

Abstract: In this paper, an application of He’s homotopy perturbation method is applied to solve functional integral equations. Comparisons are made between expansion method based on Lagrange interpolation formulae and the homotopy perturbation method. The results reveal that the He’s homotopy perturbation method is very effective and simple and gives the exact solution.

Auxiliary equation method and new solutions of Klein-Gordon equations

Abstract: Many new types of exact solutions of an auxiliary ordinary differential equation are introduced. They are used to generate new exact travelling wave solutions of the quadratic and the cubic nonlinear Klein–Gordon equations. This approach is also applicable to a large variety of nonlinear partial differential equations.

Complex dynamics of a Holling type II prey–predator system with state feedback control

Abstract: The complex dynamics of a Holling type II prey–predator system with impulsive state feedback control is studied in both theoretical and numerical ways. The sufficient conditions for the existence and stability of semi-trivial and positive periodic solutions are obtained by using the Poincare map and the analogue of the Poincare criterion. The qualitative analysis shows that the positive periodic solution bifurcates from the semi-trivial solution through a fold bifurcation. The bifurcation diagrams, Lyapunov exponents, and phase portraits are illustrated by an example, in which the chaotic solutions appear via a cascade of period-doubling bifurcations. The superiority of the state feedback control strategy is also discussed.

Nano-effects, quantum-like properties in electrospun nanofibers

Abstract: Electrospun nanofiber technology bridges the gap between deterministic laws (Newton mechanics) and probabilistic laws (quantum mechanics). Our research reveals that fascinating phenomena arise when the diameter of the electrospun nanofibers is less than 100 nm. The nano-effect has been demonstrated for unusual strength, high surface energy, surface reactivity, high thermal and electric conductivity. Dragline silk is made of many nano-fibers with diameter of about 20 nm, thus it can make full use of nano-effects. It is a challenge to developing technologies capable of preparing for nanofibers within 100 nm. Vibration-melt-electrospinning is uniquely qualified to address this challenge. The flexibility and adaptation provided by the method have made the method a strong candidate for producing nanofibers on such a scale. The application of Sirofil technology to strengthen nanofibers is also addressed, E-infinity theory is emphasized as a challenging theory for nano-scale technology and science.

Effects of small-world connectivity on noise-induced temporal and spatial order in neural media

Abstract: We present an overview of possible effects of small-world connectivity on noise-induced temporal and spatial order in a two-dimensional network of excitable neural media with FitzHugh–Nagumo local dynamics. Small-world networks are characterized by a given fraction of so-called long-range couplings or shortcut links that connect distant units of the system, while all other units are coupled in a diffusive-like manner. Interestingly, already a small fraction of these long-range couplings can have wide-ranging effects on the temporal as well as spatial noise-induced dynamics of the system. Here we present two main effects. First, we show that the temporal order, characterized by the autocorrelation of a firing-rate function, can be greatly enhanced by the introduction of small-world connectivity, whereby the effect increases with the increasing fraction of introduced shortcut links. Second, we show that the introduction of long-range couplings induces disorder of otherwise ordered, spiral-wave-like, noise-induced patterns that can be observed by exclusive diffusive connectivity of spatial units. Thereby, already a small fraction of shortcut links is sufficient to destroy coherent pattern formation in the media. Although the two results seem contradictive, we provide an explanation considering the inherent scale-free nature of small-world networks, which on one hand, facilitates signal transduction and thus temporal order in the system, whilst on the other hand, disrupts the internal spatial scale of the media thereby hindering the existence of coherent wave-like patterns. Additionally, the importance of spatially versus temporally ordered neural network functioning is discussed.

Some remarks on an experiment suggesting quantum-like behavior of cognitive entities and formulation of an abstract quantum mechanical formalism to describe cognitive entity and its dynamics

Abstract: We have executed for the first time an experiment on mental observables concluding that there exists equivalence (that is to say, quantum-like behavior) between quantum and cognitive entities. Such result has enabled us to formulate an abstract quantum mechanical formalism that is able to describe cognitive entities and their time dynamics.

Solution of the fully fuzzy linear systems using iterative techniques

Abstract: This paper mainly intends to discuss the iterative solution of fully fuzzy linear systems which we call FFLS. We employ Dubois and Prade’s approximate arithmetic operators on LR fuzzy numbers for finding a positive fuzzy vector x ˜ which satisfies A ∼ x ˜ = b ∼ , where A ∼ and b ∼ are a fuzzy matrix and a fuzzy vector, respectively. Please note that the positivity assumption is not so restrictive in applied problems. We transform FFLS and propose iterative techniques such as Richardson, Jacobi, Jacobi overrelaxation (JOR), Gauss–Seidel, successive overrelaxation (SOR), accelerated overrelaxation (AOR), symmetric and unsymmetric SOR (SSOR and USSOR) and extrapolated modified Aitken (EMA) for solving FFLS. In addition, the methods of Newton, quasi-Newton and conjugate gradient are proposed from nonlinear programming for solving a fully fuzzy linear system. Various numerical examples are also given to show the efficiency of the proposed schemes.

Controllability of impulsive functional differential systems with infinite delay in Banach spaces

Abstract: The paper establishes a sufficient condition for the controllability of the first-order impulsive functional differential systems with infinite delay in Banach spaces. We use Schauder’s fixed point theorem combined with a strongly continuous operator semigroup. An example is given to illustrate our results.