Showing papers in "Chaos Solitons & Fractals in 2004"
TL;DR: The two-dimensional chaotic cat map is generalized to 3D for designing a real-time secure symmetric encryption scheme that uses the 3D cat map to shuffle the positions of image pixels and uses another chaotic map to confuse the relationship between the cipher-image and the plain-image, thereby significantly increasing the resistance to statistical and differential attacks.
Abstract: Encryption of images is different from that of texts due to some intrinsic features of images such as bulk data capacity and high redundancy, which are generally difficult to handle by traditional methods. Due to the exceptionally desirable properties of mixing and sensitivity to initial conditions and parameters of chaotic maps, chaos-based encryption has suggested a new and efficient way to deal with the intractable problem of fast and highly secure image encryption. In this paper, the two-dimensional chaotic cat map is generalized to 3D for designing a real-time secure symmetric encryption scheme. This new scheme employs the 3D cat map to shuffle the positions (and, if desired, grey values as well) of image pixels and uses another chaotic map to confuse the relationship between the cipher-image and the plain-image, thereby significantly increasing the resistance to statistical and differential attacks. Thorough experimental tests are carried out with detailed analysis, demonstrating the high security and fast encryption speed of the new scheme.
1,904 citations
TL;DR: In this paper, the basic conceptual framework of a new space-time theory with application to high energy particle physics is outlined, both achievements and limitations are discussed with direct reference to the mass spectrum problem.
Abstract: The essay outlines the basic conceptual framework of a new space–time theory with application to high energy particle physics. Both achievements and limitations are discussed with direct reference to the mass spectrum problem.
650 citations
TL;DR: In this article, the semi-inverse method was used to obtain variational principles for generalized Korteweg-de Vries equation and nonlinear Schrodinger's equation.
Abstract: Variational principles for generalized Korteweg–de Vries equation and nonlinear Schrodinger’s equation are obtained by the semi-inverse method. The most interesting features of the proposed method are its extreme simplicity and concise forms of variational functionals for a wide range of nonlinear problems. Comparison with the results obtained by the Noether’s theorem is made, revealing the present theorem is a straightforward and attracting mathematical tool.
601 citations
TL;DR: In this article, a new chaotic system is discussed and the forming mechanism of its compound structure obtained by merging together two simple attractors after performing one mirror operation has been investigated by detailed numerical as well as theoretical analysis.
Abstract: In this letter a new chaotic system is discussed. Some basic dynamical properties, such as Lyapunov exponents, Poincare mapping, fractal dimension, continuous spectrum and chaotic behaviors of this new butterfly attractor are studied. Furthermore, the forming mechanism of its compound structure obtained by merging together two simple attractors after performing one mirror operation has been investigated by detailed numerical as well as theoretical analysis.
539 citations
TL;DR: In this paper, the authors studied the chaotic behaviors in the fractional order Chen system and found that chaos exists in all the levels of the Chen system with order less than 3.1.
Abstract: In this letter, we study the chaotic behaviors in the fractional order Chen system. We found that chaos exists in the fractional order Chen system with order less than 3. The lowest order we found to have chaos in this system is 2.1. Linear feedback control of chaos in this system is also studied.
492 citations
TL;DR: In this paper, the authors used fractional calculus techniques to construct phase diagrams in Chen's system with fractional order and found that chaos does exist in the system with a fractional ordering, and some phase diagrams are constructed.
Abstract: In this paper, by utilizing the fractional calculus techniques, we found that chaos does exist in Chen's system with a fractional order, and some phase diagrams are constructed.
489 citations
TL;DR: Sgurev et al. as discussed by the authors defined the notion of intuitionistic fuzzy metric spaces and proved Baire's theorem and the Uniform limit theorem for intuitionistic metric spaces using fuzzy sets.
Abstract: Using the idea of intuitionistic fuzzy set due to Atanassov [Intuitionistic fuzzy sets. in: V. Sgurev (Ed.), VII ITKR's Session, Sofia June, 1983; Fuzzy Sets Syst. 20 (1986) 87], we define the notion of intuitionistic fuzzy metric spaces as a natural generalization of fuzzy metric spaces due to George and Veeramani [Fuzzy Sets Syst. 64 (1994) 395] and prove some known results of metric spaces including Baire's theorem and the Uniform limit theorem for intuitionistic fuzzy metric spaces.
465 citations
TL;DR: In this paper, an adaptive sliding mode controller is presented for a class of master-slave chaotic synchronization systems with uncertainties, which can be implemented without the requirement that the bounds of the uncertainties and the disturbances should be known in advance.
Abstract: In this paper an adaptive sliding mode controller is presented for a class of master–slave chaotic synchronization systems with uncertainties. Using an adaptive technique to estimate the switching gain, an adaptive sliding mode controller is then proposed to ensure that the sliding condition is maintained in finite time. The proposed adaptive sliding mode control scheme can be implemented without the requirement that the bounds of the uncertainties and the disturbances should be known in advance. The concept of extended systems is used such that continuous control input is obtained using a sliding mode design scheme. By comparing with the results in the existed literatures, the results show that the master–slave chaotic systems with uncertainties can be synchronized accurately by this controller. Illustrative examples of chaos synchronization for uncertain Duffing–Holmes system are presented to demonstrate the superiority of the obtained results.
286 citations
TL;DR: In this paper, a sufficient condition for the existence, uniqueness, and global robust stability of equilibria for interval neural networks with time delays by constructing Lyapunov functional and using matrix-norm inequality is presented.
Abstract: This paper is concerned with the global robust stability of a class of delayed interval recurrent neural networks which contain time-invariant uncertain parameters whose values are unknown but bounded in given compact sets. A new sufficient condition is presented for the existence, uniqueness, and global robust stability of equilibria for interval neural networks with time delays by constructing Lyapunov functional and using matrix-norm inequality. An error is corrected in an earlier publication, and an example is given to show the effectiveness of the obtained results.
266 citations
TL;DR: A chaos-based watermarking algorithm is developed in the wavelet domain for still images that can gain high fidelity and high robustness, especially under the typical attack of geometric operations.
Abstract: In this paper, a chaos-based watermarking algorithm is developed in the wavelet domain for still images. The wavelet transform is commonly applied for watermarking, where the whole image is transformed in the frequency domain. In contrast to this conventional approach, we apply the wavelet transform only locally. We transform the subimage, which is extracted from the original image, in the frequency domain by using DWT and then embed the chaotic watermark into part of the subband coefficients. As usual, the watermark is detected by computing the correlation between the watermarked coefficients and the watermarking signal, where the watermarking threshold is chosen according to the Neyman–Pearson criterion based on some statistical assumptions. Watermark detection is accomplished without using the original image. Simulation results show that we can gain high fidelity and high robustness, especially under the typical attack of geometric operations.
234 citations
TL;DR: A new method, based on a recently defined centrality measure, allows to spot the critical components of a generic complex network.
Abstract: A new method, based on a recently defined centrality measure, allows to spot the critical components of a generic complex network.The identification and protection of the critical components of a given communication–transportation network should be the first concern in order to reduce the consequences of terrorist attacks.On the other hand, the critical components of a terrorist organization are the terrorists to target to disrupt the organization and reduce the possibility of terroristic attacks. 2003 Elsevier Ltd.All rights reserved.
TL;DR: In this article, a model for vegetation patterns in water limited systems is presented, which involves two variables, the vegetation biomass density and the soil water density, and takes into account positive feedback relations between the two.
Abstract: A continuum model for vegetation patterns in water limited systems is presented. The model involves two variables, the vegetation biomass density and the soil water density, and takes into account positive feedback relations between the two. The model predicts transitions from bare-soil at low precipitation to homogeneous vegetation at high precipitation through intermediate states of spot, stripe and gap patterns. It also predicts the appearance of ring-like shapes as transient forms toward asymptotic stripes. All these patterns have been identified in observations made on two types of perennial grasses in the Northern Negev. Another prediction of the model is the existence of wide precipitation ranges where different stable states coexist, e.g. a bare soil state and a spot pattern, a spot pattern and a stripe pattern, and so on. This result suggests the interpretation of desertification followed by recovery as an hysteresis loop and sheds light on the irreversibility of desertification. 2003 Elsevier Ltd. All rights reserved.
TL;DR: In this paper, the authors employ a rolling sample approach and calculate median Hurst exponents, R/S and modified Hurst statistics in order to assess relative efficiency of these equity markets.
Abstract: In this paper we test for long-range dependence and efficiency in stock indices for 11 emerging markets and also for the US and Japan. We employ a “rolling sample” approach and calculate median Hurst exponents, R/S and modified R/S statistics in order to assess relative efficiency of these equity markets. Our results suggest that Asian equity markets show greater inefficiency than those of Latin America (with the exception of Chile), and that developed markets rank first in terms of efficiency.
TL;DR: The feasibility of the communication scheme in high-dimensional chaotic systems, such as the hyperchaotic Rossler system, is demonstrated and the unpredictability of the scaling factor in projective synchronization can additionally enhance the security of communication.
Abstract: Most secure communication schemes using chaotic dynamics are based on identical synchronization. In this paper, we show the possibility of secure communication using projective synchronization (PS). The unpredictability of the scaling factor in projective synchronization can additionally enhance the security of communication. It is also showed that the scaling factor can be employed to improve the robustness against noise contamination. The feasibility of the communication scheme in high-dimensional chaotic systems, such as the hyperchaotic Rossler system, is demonstrated. Numerical results show the success in transmitting a sound signal through chaotic systems.
TL;DR: Simulation results of typical complex function optimization show that CSA improves the convergence and is efficient, applicable and easy to implement.
Abstract: Simulated annealing (SA) has been applied with success to many numerical and combinatorial optimization problems in recent years. SA has a rather slow convergence rate, however, on some function optimization problems. In this paper, by introducing chaotic systems to simulated annealing, we propose a optimization algorithm named chaos simulated annealing (CSA). The distinctions between CSA and SA are chaotic initialization and chaotic sequences replacing the Gaussian distribution. Simulation results of typical complex function optimization show that CSA improves the convergence and is efficient, applicable and easy to implement. In addition, we discuss the advantages of CSA, and show the reasons why CSA performs better than SA.
TL;DR: In this article, an adaptive backstepping design is proposed to synchronize two uncertain chaos systems, which can be applied to a variety of chaos systems and can be transformed into the so-called general strict feedback form no matter whether it contains external excitation or not.
Abstract: In this paper, an adaptive backstepping design is proposed to synchronize two uncertain chaos systems. This method can be applied to a variety of chaos systems which can be transformed into the so-called general strict-feedback form no matter whether it contains external excitation or not. Rossler system and Duffing oscillator are used as examples for detailed description. Numerical simulations show the effectiveness and feasibility of the method.
TL;DR: In this article, the anti-control of chaos for a rigid body has been studied, where basic dynamical behaviors, such as symmetry, invariance, dissipativity and existence of attractor, are discussed.
Abstract: Anti-control of chaos for a rigid body has been studied in the paper. For certain feedback gains, a rigid body can easily generate chaotic motion. Basic dynamical behaviors, such as symmetry, invariance, dissipativity and existence of attractor, are also discussed. The transient behaviors of the chaotic system have also been presented as the feedback gain changed. Of particular interesting is the fact that the chaotic system can generate a complex multi-scroll chaotic attractor under the appropriate feedback gains. Finally, it was shown that the system could be related to the famous Lorenz equations and Chen system. In other words, the system can easily display all the dynamical behaviors of the famous Lorenz equations and Chen system.
TL;DR: In this paper, a direct algebraic method is described to construct the exact travelling wave solutions for nonlinear evolution equations, and the modified Kawahara equations are investigated and new exact traveling wave solutions are explicitly obtained with the aid of symbolic computation.
Abstract: By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact travelling wave solutions for nonlinear evolution equations. By this method the Kawahara and the modified Kawahara equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.
TL;DR: In this article, the authors take the Burgers equation and Sharma-Tasso-Olver equation as two concrete examples to show the fission and fusion of the solitary wave and the soliton solutions respectively which are studied by means of Hirota's direct method and the Backlund transformation.
Abstract: Fission and fusion phenomena can happen for solitons (sometimes solitary waves may be more accurate) which have been recently discovered both theoretically and experimentally. In this paper, taking the Burgers equation and the Sharma–Tasso–Olver equation as two concrete examples to show the fission and fusion of the solitary wave and the soliton solutions respectively which are studied by means of the Hirota’s direct method and the Backlund transformation. Furthermore, the amplitude and velocity relations between solitons and/or solitary waves before and after interactions are given and a possible general condition for fission and/or fusion is proposed.
TL;DR: In this paper, an algebraic method is devised to construct new exact wave soliton solutions for two generalized nonlinear Hirota-Satsuma coupled KdV systems of partial differential equations using symbolic software like Mathematica or Maple in terms of sn(x)−cn(x)- Jacobic elliptic functions and sec q − tanh q q -deformed hyperbolic functions based on the idea of the homogeneous balance method.
Abstract: In this paper an algebraic method is devised to construct new exact wave soliton solutions for two generalized nonlinear Hirota–Satsuma coupled KdV systems of partial differential equations using symbolic software like Mathematica or Maple in terms of sn(x)−cn(x) Jacobic elliptic functions and sec q − tanh q q -deformed hyperbolic functions based on the idea of the homogeneous balance method.
TL;DR: A systematic review of chaos theory in geophysics, covering a wide spectrum of geophysical phenomena studied (e.g., rainfall, river flow, sediment transport, temperature, pressure, tree ring series, etc.), is presented in this paper.
Abstract: The past two decades of research on chaos theory in geophysics has brought about a significant shift in the way we view geophysical phenomena. Research on chaos theory in geophysics continues to grow at a much faster pace, with applications to a wide variety of geophysical phenomena and geophysical problems. In spite of our success in understanding geophysical phenomena also from a different (i.e. chaotic) perspective, there still seems to be lingering suspicions on the scope of chaos theory in geophysics. The goal of this paper is to present a comprehensive account of the achievements and status of chaos theory in geophysics, and to disseminate the hope and scope for the future. A systematic review of chaos theory in geophysics, covering a wide spectrum of geophysical phenomena studied (e.g. rainfall, river flow, sediment transport, temperature, pressure, tree ring series, etc.), is presented to narrate our past achievements not only in understanding and predicting geophysical phenomena but also in improving the chaos identification and prediction techniques. The present state of chaos research in geophysics (in terms of geophysical phenomena, problems, and chaos methods) and potential for future improvements (in terms of where, why and possibly how) are also highlighted. Our popular views of nature (i.e. stochastic and deterministic), and of geophysical phenomena in particular, are discussed, and the usefulness of chaos theory as a bridge between such views is also put forth.
TL;DR: In this article, the authors investigated the effect of delay on the dynamics of Chen's system with delayed feedback and showed that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.
Abstract: Time-delayed feedback has been introduced as a powerful tool for control of unstable periodic orbits or control of unstable steady states. In the present paper, regarding the delay as parameter, we investigate the effect of delay on the dynamics of Chen's system with delayed feedback. We first consider the effect of delay on the steady states, and then investigate the existence of local Hopf bifurcations. By using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.
TL;DR: The main concepts of E Infinity theory and some of the principle results have been explained in this article, and the problem of the mass spectrum of the standard model from various viewpoints, in particular, from that of gravitational instanton.
Abstract: The paper is an elementary introduction to the concepts of E Infinity of quantum physics. In the first two paragraphs the main concepts of E Infinity theory and some of the principle results have been explained. Subsequently we address the problem of the mass spectrum of the standard model from various viewpoints, in particular, from that of gravitational instanton. Particular attention is paid throughout the paper to giving an intuitive grasp for an extended theory of vacuum fluctuation based on Dirac's hole theory and E Infinity theory. It is further shown how the golden mean and logarithmic scalings can be used to understand quantum gravity and how the new transfinite Dirac's theory can explain certain anomalous positron production which was observed by several experimental groups worldwide.
TL;DR: In this paper, an adaptive complementary variable structure control is proposed for chaotic synchronization, based on Lyapunov's stability theory and Babalat's lemma, which has been shown to render the synchronous error to zero.
Abstract: A novel adaptive complementary variable structure control is proposed in this paper for chaotic synchronization. The bounded parameters of the model approximation error and the external disturbance are all regarded as unknown constants in this paper. Based on Lyapunov’s stability theory and the Babalat’s lemma the proposed controller has been shown to render the synchronous error to zero. The Duffing–Holmes oscillator was used as an illustrative example. Simulation results validated that the proposed scheme in the application of secure communication.
TL;DR: In this paper, a recursive procedure for constructing rational solutions to the Toda lattice equation through the Casoratian formulation is presented, allowing us to compute a broad class of rational solutions directly, without computing long wave limits in soliton solutions.
Abstract: A recursive procedure is presented for constructing rational solutions to the Toda lattice equation through the Casoratian formulation. It allows us to compute a broad class of rational solutions directly, without computing long wave limits in soliton solutions. All rational solutions arising from the Taylor expansions of the generating functions of soliton solutions are special ones of the general class, but only a Taylor expansion containing even or odd powers leads to non-constant rational solutions. A few rational solutions of lower order are worked out.
TL;DR: In this article, the authors applied EMD analysis to decompose the heartbeat intervals series, derived from one electrocardiographic (ECG) signal of 13 subjects, into their components in order to identify the modes associated with breathing.
Abstract: Heart rate variability (HRV) is a well-known phenomenon whose characteristics are of great clinical relevance in pathophysiologic investigations. In particular, respiration is a powerful modulator of HRV contributing to the oscillations at highest frequency. Like almost all natural phenomena, HRV is the result of many nonlinearly interacting processes; therefore any linear analysis has the potential risk of underestimating, or even missing, a great amount of information content. Recently the technique of empirical mode decomposition (EMD) has been proposed as a new tool for the analysis of nonlinear and nonstationary data. We applied EMD analysis to decompose the heartbeat intervals series, derived from one electrocardiographic (ECG) signal of 13 subjects, into their components in order to identify the modes associated with breathing. After each decomposition the mode showing the highest frequency and the corresponding respiratory signal were Hilbert transformed and the instantaneous phases extracted were then compared. The results obtained indicate a synchronization of order 1:1 between the two series proving the existence of phase and frequency coupling between the component associated with breathing and the respiratory signal itself in all subjects.
TL;DR: In this paper, the convergence of continuous-time BAM neural networks is studied and sufficient conditions are obtained for the networks to converge exponentially toward the equilibrium associated with the constant input sources.
Abstract: First, convergence of continuous-time Bidirectional Associative Memory (BAM) neural networks are studied. By using Lyapunov functionals and some analysis technique, the delay-independent sufficient conditions are obtained for the networks to converge exponentially toward the equilibrium associated with the constant input sources. Second, discrete-time analogues of the continuous-time BAM networks are formulated and studied. It is shown that the convergence characteristics of the continuous-time systems are preserved by the discrete-time analogues without any restriction imposed on the uniform discretionary step size. An illustrative example is given to demonstrate the effectiveness of the obtained results.
TL;DR: In this paper, a new concept of robust periodicity is introduced, and problem of robust stability and robust periodsicity is discussed for delayed neural networks, and several sufficient conditions are derived for globally exponentially robust stability of delayed neural network based Lyapunov method.
Abstract: In this paper, a new concept of robust periodicity is introduced, and problem of robust stability and robust periodicity is discussed for delayed neural networks. Several sufficient conditions are derived for globally exponentially robust stability and robust periodicity of delayed neural networks based Lyapunov method. These results improve and extend those given in the earlier references.
TL;DR: In this article, the extended F-expansion method is applied to some different kinds of nonlinear PDEs. And more Jacobi elliptic function solutions are obtained including the single function solutions and the combined function solutions.
Abstract: The extended F-expansion method is proposed and applied to some different kinds of nonlinear PDEs. More Jacobi elliptic function solutions are obtained including the single function solutions and the combined function solutions. When the modulus m of Jacobi elliptic function is driven to the limit 1 and 0, hyperbolic function solutions and trigonometric function solutions can also be obtained respectively.
TL;DR: In this paper, the authors investigated the traveling-wave solutions of the Degasperis-Procesi equation, which are characterized by two parameters: the distance to the source and the distance from the sink.
Abstract: Travelling-wave solutions of the Degasperis-Procesi equation are investigated. The solutions are characterized by two parameters. For propagation in the positive x-direction, hump-like, inverted loop-like and coshoidal periodic-wave solutions are found; hump-like, inverted loop-like and peakon solitary-wave solutions are obtained as well. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. A transformed version of the Degasperis-Procesi equation, which is a generalization of the Vakhnenko equation, is also considered. For propagation in the positive x-direction, hump-like, loop-like, inverted loop-like, bell-like and coshoidal periodic-wave solutions are found; loop-like, inverted loop-like and kink-like solitary-wave solutions are obtained as well. For propagation in the negative x-direction, well-like and inverted coshoidal periodic-wave solutions are found; well-like and inverted peakon solitary-wave solutions are obtained as well. In an appropriate limit, the previously known solutions of the Vakhnenko equation are recovered.