Complexity-entropy causality plane: A useful approach to quantify the stock market inefficiency

The complexity-entropy causality plane has been recently introduced as a powerful tool for discriminating Gaussian from non-Gaussian process and different degrees of correlations [O.A. Rosso, H.A. Larrondo, M.T. Martín, A. Plastino, M.A. Fuentes, Distinguishing noise from chaos, Phys. Rev. Lett. 99 (2007) 154102]. We propose to use this representation space to distinguish the stage of stock market development. Our empirical results demonstrate that this statistical physics approach is useful, allowing a more refined classification of stock market dynamics.     

Wavelet Jensen–Shannon divergence as a tool for studying the dynamics of frequency band components in EEG epileptic seizures

We develop a quantitative method of analysis of EEG records. The method is based on the wavelet analysis of the record and on the capability of the Jensen–Shannon divergence (JSD) to identify dynamical changes in a time series. The JSD is a measure of distance between probability distributions. Therefore for its evaluation it is necessary to define a (time dependent) probability distribution along the record. We define this probability distribution from the wavelet decomposition of the associated time series. The wavelet JSD provides information about dynamical changes in the scales and can be considered a complementary methodology reported earlier [O.A. Rosso, S. Blanco, A. Rabinowicz, Signal Processing 86 (2003) 1275; O.A. Rosso, S. Blanco, J. Yordanova, V. Kolev, A. Figliola, M. Schürmann, E. Ba?ar, J. Neurosci. Methods 105 (2001) 65; O.A. Rosso, M.T. Martin, A. Figliola, K. Keller, A. Plastino, J. Neurosci. Methods 153 (2006) 163]. In the present study we have demonstrated it by analyzing EEG signal of tonic–clonic epileptic seizures applying the JSD method. The display of the JSD curves enables easy comparison of frequency band component dynamics. This would, in turn, promise easy and successful comparison of the EEG records from various scalp locations of the brain.     

Link between the laws of geometrical optics and the radiative transfer equation in media with a spatially varying refractive index

We proposed in a previous paper [J.-M. Tualle, E. Tinet, Opt. Commun. 228 (2003) 33] a modified radiative transfer equation to describe radiative transfer in a medium with a spatially varying refractive index. The present paper is devoted to the demonstration that this equation perfectly works in the non-absorbing/non-scattering limit, what was contested by Martí-López and coworkers [L. Martí-López, J. Bouza-Domínguez, R.A. Martínez-Celorio, J.C. Hebden, Opt. Commun. 266 (2006) 44]. The assertion that this equation would imply a zero divergence of the rays is also commented.     

A Three-Dimensional Infinite Collapse Map with Image Encryption

Chaos is considered as a natural candidate for encryption systems owing to its sensitivity to initial values and unpredictability of its orbit. However, some encryption schemes based on low-dimensional chaotic systems exhibit various security defects due to their relatively simple dynamic characteristics. In order to enhance the dynamic behaviors of chaotic maps, a novel 3D infinite collapse map (3D-ICM) is proposed, and the performance of the chaotic system is analyzed from three aspects: a phase diagram, the Lyapunov exponent, and Sample Entropy. The results show that the chaotic system has complex chaotic behavior and high complexity. Furthermore, an image encryption scheme based on 3D-ICM is presented, whose security analysis indicates that the proposed image encryption scheme can resist violent attacks, correlation analysis, and differential attacks, so it has a higher security level.     

Qualitative analysis of the Rössler equations: Bifurcations of limit cycles and chaotic attractors

In this paper we study different aspects of the paradigmatic Rössler model. We perform a detailed study of the local and global bifurcations of codimension one and two of limit cycles. This provides us a global idea of the three-parametric evolution of the system. We also study the regions of parameters where we may expect a chaotic behavior by the use of different Chaos Indicators. The combination of the different techniques gives an idea of the different routes to chaos and the different kinds of chaotic attractors we may found in this system.     

Performance of encryption schemes in chaotic optical communication: A multifractal approach

From the estimation of the Hurst exponent and the multifractality degree we discriminate the security levels of two typical encoding schemes usually applied in chaos-based communication systems. We also analyze the effects that the sampling period and the message amplitude have on the goodness of these techniques. We compare our results with those obtained by considering an information theory approach [O.A. Rosso, R. Vicente, C.R. Mirasso, Phys. Lett. A 372 (2007) 1018]. The Hurst exponent seems to be a sensitive and powerful tool for discriminating the presence of a message embedded in a chaotic carrier.     

Lightweight secure image encryption scheme based on chaotic differential equation

The ultimate secure choice for block cryptosystem until now is advanced encryption standard (AES). It is very difficult to implement AES for the constrained situations such as sensor networks, image encryption and RFID tags. In this article, a chaotic oscillator generated by a second order differential equation is used to produce confusion and diffusion in the plaintext message to achieve the desired secrecy. The produced chaotic sequence of random numbers from dynamical system is utilized to scramble the pixels of an image to obtain an encrypted image. Chaos based encryption technique is found secure enough to tackle chosen plaintext attacks and brute force attacks. The specific attributes of chaotic system like, sensitivity to initial conditions, randomness and uncertainty make it suitable for the design of cryptosystem. The dominance of the proposed scheme is acknowledged due to the fact of better cryptographic properties when compared with the algorithms already developed in the literature.     

Formal equivalence between Tsallis and extended Boltzmann-Gibbs statistics

A formal correspondence between the q-distribution obtained from the Tsallis entropy and non-Maxwellian distributions obtained from the Boltzmann-Gibbs (BG) entropy is obtained. This formal correspondence is obtained by imposing an infinite number of constraints when one maximizes the BG entropy. Different from the approach of Tsallis, Prato and Plastino [C. Tsallis, D. Prato, A.R. Plastino, Astrophys. Space Sci., 290 (2004) 259-274], we relate the constraints to the central moments, providing a natural meaning to the q-parameter.     

Canonical equilibrium distribution derived from Helmholtz potential

Plastino and Curado [A. Plastino, E.M.F. Curado, Phys. Rev. E 72 (2005) 047103] recently determined the equilibrium probability distribution for the canonical ensemble using only phenomenological thermodynamical laws as an alternative to the entropy maximization procedure of Jaynes. In the current paper we present another alternative derivation of the canonical equilibrium probability distribution, which is based on the definition of the Helmholtz free energy (and its being constant at the equilibrium) and the assumption of the uniqueness of the equilibrium probability distribution. Noting that this particular derivation is applicable for all trace-form entropies, we also apply it to the Tsallis entropy, showing that the Tsallis entropy yields genuine inverse power laws.     

Synchronization Control of Two Different Chaotic Systems with Known and Unknown Parameters

Chaos synchronization of two different chaotic systems with known and unknown parameters is studied. Based on the Lyapunov stability theory, two different chaotic systems with known parameters realize global synchronization via the successfully designed nonlinear controller. By employing an adaptive synchronization scheme, the synchronization of two different chaotic systems with unknown parameters is achieved. Numerical simulations validate the effectiveness of the theoretical analysis.     

Energy distribution and energy fluctuation in Tsallis statistics

The energy distribution and the energy fluctuation in the Tsallis canonical ensemble are studied with the OLM formalism but following a new way. The resulting formula for the energy fluctuation is not the same as that in previous work [L.Y. Liu, J.L. Du, Physica A 387 (2008) 5417]. In discussing the application of an ideal gas, we find that the energy fluctuation can not be negligible in the thermodynamic limit, showing the ensemble nonequivalence for this case in Tsallis statistics. We investigate the energy fluctuation with a Tsallis generalized canonical distribution studied by Plastino and Plastino [A.R. Plastino, A. Plastino, Phys. Lett. A 193 (1994) 140] for describing a system in contact with a finite heat bath. For this situation, the two formulae for the energy fluctuation are shown to be equivalent, while the nonextensive parameter q$q$ plays a very important role.     

Adsorption of molecular gases on porous materials in the SAFT-VR approximation

A simple molecular thermodynamic approach is applied to the study of the adsorption of gases of chain molecules on solid surfaces. We use a model based on the Statistical Associating Fluid Theory for Variable Range (SAFT-VR) potentials [A. Gil-Villegas, A. Galindo, P.J. Whitehead, S.J. Mills, G. Jackson, A.N. Burgess, J. Chem. Phys. 106 (1997) 4168] that we extend by including quasi-two-dimensional approximation to describe the adsorption properties of these types of real gas [A. Martínez, M. Castro, C. McCabe, A. Gil-Villegas, J. Chem. Phys. 126 (2007) 074707]. The model is applied to ethane, ethylene, propane, and carbon dioxide adsorbed on activated carbon and silica gel, which are porous media of significant industrial interest. We show that the adsorption isotherms obtained by means of the present SAFT-VR modeling are in fair agreement with the experimental results provided in the literature.     

基于新的五维多环多翼超混沌系统的图像加密算法

本文提出了一种基于新的五维多环多翼超混沌系统的数字图像加密方法.首先,将明文图像矩阵和五条混沌序列分别通过QR分解法分解成一个正交矩阵和一个上三角矩阵,将混沌系统产生的五条混沌序列分别通过LU分解法分解成一个上三角矩阵和一个下三角矩阵,分别将两个上三角矩阵和一个下三角矩阵相加,得到五个离散后的混沌序列;其次,将明文图像矩阵分解出来的正交矩阵与五个混沌序列分解出来的五个正交矩阵相乘,同时把明文图像矩阵分解出来的上三角矩阵中的元素通过混沌序列进行位置乱,再将操作后的两个矩阵相乘;最后,将相乘后的矩阵通过混沌序列进行比特位位置乱,再用混沌序列与其进行按位“异或”运算,得到最终加密图像.理论分析和仿真实验结果表明该算法的密钥空间远大于10^200,密钥敏感性强,能够有效地抵御统计分析和灰度值分析的攻击,对数字图像的加密具有很好的加密效果.     

Randomly chosen chaotic maps can give rise to nearly ordered behavior

Parrondo’s paradox [J.M.R. Parrondo, G.P. Harmer, D. Abbott, New paradoxical games based on Brownian ratchets, Phys. Rev. Lett. 85 (2000), 5226–5229] (see also [O.E. Percus, J.K. Percus, Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox, Math. Intelligencer 24 (3) (2002) 68–72]) states that two losing gambling games when combined one after the other (either deterministically or randomly) can result in a winning game: that is, a losing game followed by a losing game = a winning game. Inspired by this paradox, a recent study [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124–132] asked an analogous question in discrete time dynamical system: can two chaotic systems give rise to order, namely can they be combined into another dynamical system which does not behave chaotically? Numerical evidence is provided in [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124–132] that two chaotic quadratic maps, when composed with each other, create a new dynamical system which has a stable period orbit. The question of what happens in the case of random composition of maps is posed in [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124–132] but left unanswered. In this note we present an example of a dynamical system where, at each iteration, a map is chosen in a probabilistic manner from a collection of chaotic maps. The resulting random map is proved to have an infinite absolutely continuous invariant measure (acim) with spikes at two points. From this we show that the dynamics behaves in a nearly ordered manner. When the foregoing maps are applied one after the other, deterministically as in [O.E. Percus, J.K. Percus, Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox, Math. Intelligencer 24 (3) (2002) 68–72], the resulting composed map has a periodic orbit which is stable.     

Cryptanalyzing a novel image fusion encryption algorithm based on DNA sequence operation and hyper-chaotic system

Recently, an image fusion encryption algorithm [Optik 124 (18) (2013) 3596–3600] was proposed based on DNA sequence operation and hyper-chaotic system. The security of this algorithm depends mainly on both five keys as the initial conditions of hyper-chaotic system and a key image. In this paper, we cryptanalyze the algorithm and find that two chaotic keystream determined by the five keys keep unchanged for different image encryption processes. The two chaotic keystream can be revealed and the computational complexity is approximately O(mn), where m and n represent the height and width, respectively. In addition, the key image needs to be changed for each encryption. Finally, experimental results also verify our idea.     

Cryptanalysis of a cryptosystem based on discretized two-dimensional chaotic maps

Recently, an encryption algorithm based on two-dimensional discretized chaotic maps was proposed [Xiang et al., Phys. Lett. A 364 (2007) 252]. In this Letter, we analyze the security weaknesses of the proposal. Using the algebraic dependencies among system parameters, we show that its effective key space can be shrunk. We demonstrate a chosen-ciphertext attack that reveals a portion of the key.     

Image Encryption Scheme Based on Mixed Chaotic Bernoulli Measurement Matrix Block Compressive Sensing

Many image encryption schemes based on compressive sensing have poor reconstructed image quality when the compression ratio is low, as well as difficulty in hardware implementation. To address these problems, we propose an image encryption algorithm based on the mixed chaotic Bernoulli measurement matrix block compressive sensing. A new chaotic measurement matrix was designed using the Chebyshev map and logistic map; the image was compressed in blocks to obtain the measurement values. Still, using the Chebyshev map and logistic map to generate encrypted sequences, the measurement values were encrypted by no repetitive scrambling as well as a two-way diffusion algorithm based on GF(257) for the measurement value matrix. The security of the encryption system was further improved by generating the Secure Hash Algorithm-256 of the original image to calculate the initial values of the chaotic mappings for the encryption process. The scheme uses two one-dimensional maps and is easier to implement in hardware. Simulation and performance analysis showed that the proposed image compression–encryption scheme can improve the peak signal-to-noise ratio of the reconstructed image with a low compression ratio and has good encryption against various attacks.     

A Novel Hyperchaotic 2D-SFCF with Simple Structure and Its Application in Image Encryption

In this paper, a novel image encryption algorithm is proposed based on hyperchaotic two-dimensional sin-fractional-cos-fractional (2D-SFCF), called sin-fractional-cos-fractional image-encryption (SFCF-IE). The 2D-SFCF is constructed from two one-dimensional cosine fractional (1-DCFs), and it has a more complex chaotic behavior with a larger parameter space than one-dimensional chaotic systems. Compared with the two-dimensional (2D) chaotic system, the 2D-SFCF has a simple structure, and the parameter space in the chaotic state is continuous, which is beneficial to generating the keystream in the cryptosystem. Therefore, in the novel image encryption algorithm, we use the 2D-SFCF to generate the keystream of the cryptosystem. The encryption algorithm is a process of scrambling and diffusion. Different from common diffusion methods, the diffusion starting position of the SFCF-IE is randomly generated, enhancing the algorithm’s security. Simulation experiments show that the image encrypted by this algorithm has better distribution characteristics and can resist common attack methods.