A Study on Multi-objective Chaotic Evolution Algorithms Using Multiple Chaotic Systems

This paper proposes a user-biometric-based image encryption using chaotic dynamical systems, SHA-256 hash function, and zigzag transformation. Three chaotic systems, namely, the Henon map, logistic map, and Lorenz system, are used to apply the chaotic properties to the encryption system. The hash value of the user biometric image is used to generate the initial value of the Henon map. The initial values for the logistic map and Lorenz system are generated from the hash value of the plain image. SHA-256 hash algorithm generates the hash values of the biometric and plain images. The plain image is scrambled using the two-dimensional chaotic values generated by the Henon map first. Next, the first scrambled image’s pixels are XORed with the one-dimensional chaotic values generated from the logistic map and produce the second scrambled image. Finally, take the second scrambled image’s pixels, and XOR them with the three-dimensional chaotic values generated from the Lorenz system and create the final encrypted image. The robustness and flexibility of the encryption system are analyzed using various security and performance analyses. The experiment results are compared with the existing algorithms to prove the efficiency of the proposed algorithm.

A minimum principle for chaotic dynamical systems

Large deviations in chaotic dynamics have potentially significant and dramatic consequences. We study large deviations of series of finite lengths N generated by chaotic maps. The distributions generally display an exponential decay with N, associated with large-deviation (rate) functions. We obtain the exact rate functions analytically for the doubling, tent, and logistic maps. For the latter two, the solution is given as a power series whose coefficients can be systematically calculated to any order. We also obtain the rate function for the cat map numerically, uncovering strong evidence for the existence of a remarkable singularity of it that we interpret as a second-order dynamical phase transition. Furthermore, we develop a numerical tool for efficiently simulating atypical realizations of sequences if the chaotic map is not invertible, and we apply it to the tent and logistic maps.

We have developed techniques to control the chaotic behavior in Fluidized Bed Systems (FBC) systems using recurrent neural networks. For the sake of comparison of the techniques we have developed with the traditional chaotic system control methods, in the past three months we have been investigating the most popular and first known chaotic system control technique known as the OGY method. This method was developed by Edward Ott, Celso Grebogi and James York in 1990. In the past few years this method was further developed and applied by many researchers in the field. It was shown that this method has potential applications to a large cross section of problems in many fields. The only remaining question is whether it will prove possible to move from laboratory demonstrations on model systems to real world situations of engineering importance. We have developed computer programs to compute the OGY parameters from a chaotic time series, to control a chaotic system to a desired periodic orbit, using small perturbations to an accessible system parameter. We have tested those programs on the logistic map and the Henon map. We were able to control the chaotic behavior in such typical chaotic systems to period 1, 2, 3, 5..., as shown in some sample results below. In the following sections a brief discussion for the OGY method will be introduced, followed by results for the logistic map and Henon map control.

The paper presents combining k-error approximate entropy with some tests in NIST's STS randomness test suite to analyze cryptographical properties of several chaotic pseudo- random sequences generated by diverse maps and quantified methods, including Logistic map, Cubic map, Henon map, Sine map, Tent map, Chebyshev map, Piecewise-linear map, Piece- wise-square-root map. Simulation results show that the k-error approximate entropy can distinguish the stability of different sequences; that the cryptological properties are influenced by quantified methods; that the sequences generated by Logistic map, Chebyshev map, Piecewise-linear map and Piecewise- square-root map have the best pseudorandom properties, while the sequences generated by Henon map and Tent map have the worst stability.

In recent years, the use of discrete memristors to enhance chaotic maps has received increasing attention. The introduction of memristors increases the complexity of chaotic maps, making them suitable for engineering applications based on chaotic systems. In this work, a fractional-order discrete memristor exhibiting local activity and controllable asymptotic stability points is constructed by using multiband nonlinear functions. The locally active property of this memristor is demonstrated by using the power-off plot and DC <i>v</i> - <i>i</i> plot. It is then introduced into the Henon map to construct a fractional-order memristive Henon map that can generate any number of coexisting attractors. Simulation results show that the number of fixed points in the system is controlled by the memristor parameters and related to the number of coexisting attractors, thus achieving controllable homogeneous multistability. The complex dynamical behaviors of this map are analyzed by using phase portraits, bifurcation diagrams, maximum Lyapunov exponent (MLE), and attractor basins. Numerical simulations show that the fractional-order map can generate various periodic orbits, chaotic attractors, and period-doubling bifurcations. The system is then implemented on an ARM digital platform. The experimental results are consistent with the simulation results, confirming the accuracy of the theoretical analysis and its physical feasibility. Finally, a parallel video encryption algorithm is designed by using the chaotic sequence iteratively generated by fraction-order memory Henon mapping, which mainly includes frame pixel scrambling and diffusion. Comprehensive security analyses are conducted, proving the robustness and reliability of the proposed encryption scheme. The results show that the encryption algorithm can effectively protect video information. In the future, we will explore other methods of constructing chaotic or hyperchaotic systems with controllable multistability and study their circuit implementation, synchronization control, and chaos-based engineering applications.

To obtain chaotic systems with good properties to help chaos based image encryption, this paper first study the condition under which quadratic functions can be used as pseudo-random sequence generator. We give a necessary and sufficient condition for quadratic functions topologically conjugate with the most frequently used Logistic map and Tent map. Since topological conjugacy maintains chaotic characteristics, more quadratic functions can be used as pseudo-random sequence generators in the digital image encryption. Then, it analyzes a diffusion algorithm with a good effect but poor in resisting the chosen plaintext attack. It is also verified by simulation. The quadratic function chaotic system is applied in the improvement of the diffusion algorithm. Furthermore, the improved algorithm contains plaintext related and cyclic shift operations. Finally, the performance analysis shows that the improved scheme has good security and high efficiency.

The Internet advances in the past decade have been upping at a tremendous pace; especially in the fields like medicine, military, and technology, where lot of internet threats have also been raising, at least at the same rate, if not higher. The communication takes place in the form of text, image, voice, and video signals etc. And we deal with more image-based applications. There are many encryption techniques existing-DES, AES, RSA, etc. but they provide less security and are vulnerable to brute force attacks due to factors like small key space, to avoid which, chaos-based encryption was introduced. Chaotic systems are very sensitive to initial conditions and distort the image to beyond reversible state, except with a protection key with which it is encrypted, which is hard to find in such a large key space. In this paper we will discuss image encryption using combined chaos by combining various chaotic maps like logistics map, singer map, tent map, sine map, piecewise map etc. and attempt to compare the results for better combination for encryption.

This paper experimentally investigates the effect of nine chaotic maps on the performance of two Particle Swarm Optimization (PSO) variants, namely, Random Inertia Weight PSO (RIW‐PSO) and Linear Decreasing Inertia Weight PSO (LDIW‐PSO) algorithms. The applications of logistic chaotic map by researchers to these variants have led to Chaotic Random Inertia Weight PSO (CRIW‐PSO) and Chaotic Linear Decreasing Inertia Weight PSO (CDIW‐PSO) with improved optimizing capability due to better global search mobility. However, there are many other chaotic maps in literature which could perhaps enhance the performances of RIW‐PSO and LDIW‐PSO more than logistic map. Some benchmark mathematical problems well‐studied in literature were used to verify the performances of RIW‐PSO and LDIW‐PSO variants using the nine chaotic maps in comparison with logistic chaotic map. Results show that the performances of these two variants were improved more by many of the chaotic maps than by logistic map in many of the test problems. The best performance, in terms of function evaluations, was obtained by the two variants using Intermittency chaotic map. Results in this paper provide a platform for informative decision making when selecting chaotic maps to be used in the inertia weight formula of LDIW‐PSO and RIW‐PSO.

Recently, chaotic systems have been widely investigated in several engineering applications. This paper presents a new chaotic system based on Julia’s fractal process, chaotic attractors and Logistic map in a complex set. Complex dynamic characteristics were analyzed, such as equilibrium points, bifurcation, Lyapunov exponents and chaotic behavior of the proposed chaotic system. As we know, one positive Lyapunov exponent proved the chaotic state. Numerical simulation shows a plethora of complex dynamic behaviors, which coexist with an antagonist form mixed of bifurcation and attractor. Then, we introduce an algorithm for image encryption based on chaotic system. The algorithm consists of two main stages: confusion and diffusion. Experimental results have proved that the proposed maps used are more complicated and they have a key space sufficiently large. The proposed image encryption algorithm is compared to other recent image encryption schemes by using different security analysis factors including differential attacks analysis, statistical tests, key space analysis, information entropy test and running time. The results demonstrated that the proposed image encryption scheme has better results in the level of security and speed.

Medical images include confidential and sensitive information about patients. Hence, ensuring the security of these images is a crucial requirement. This paper proposes an efficient and secure medical image encryption‐decryption scheme based on deoxyribonucleic acid (DNA), one‐dimensional chaotic maps (tent and logistic maps), and hash functions (SHA‐256 and MD5). The first part of the proposed scheme is the key generation based on the hash functions of the image and its metadata. The key then is highly related and intensely sensitive to the original image. The second part is the rotation and permutation of the first two MSB bit‐plans of the medical image to reduce its black background that produces redundant DNA encoded sequences. The third part is the DNA encoding‐decoding using dynamically chosen DNA rules for every 2‐bit pixel value through the logistic map. Meanwhile, the confusion‐diffusion is performed using the tent map and XOR operation. Simulation results and security analysis prove the good encryption effects of the proposed scheme compared to the state‐of‐art methods with a correlation of 6.66617e‐7 and a very large key space of 2 624 . Furthermore, the proposed system has a strong ability to resist various common attacks such as chosen/known‐plaintext attacks and cropping/noise attacks.

This paper proposed a secure and fast image encryption algorithm which is composed of permutation and substitution of pixels. In order to eliminate the relation between adjacent pixels in the image sufficiently, firstly, the sequence which derives from Logistic map is sorted as an ascending order, and the transformation function of element's index can be obtained in this process. Secondly, perform XOR operation between the scrambled pixel sequence and another chaotic sequence which derives from Henon map. Experiment simulations show that the proposed algorithm is secure against different types of attacks and can be applied to the real time encryption.

Since chaos systems generally have the intrinsic properties of sensitivity to initial conditions, topological mixing and density of periodic orbits, they may tactfully use the chaotic ergodic orbits to achieve the global optimum or their better approximation to given cost functions with high probability. During the past decade, they have increasingly received much attention from academic community and industry society throughout the world. To improve the performance of particle swarm optimization (PSO), we herein propose a chaotic proportional integral derivative (PID) controlling PSO algorithm by the hybridization of chaotic logistic dynamics and hierarchical inertia weight. The hierarchical inertia weight coefficients are determined in accordance with the present fitness values of the local best positions so as to adaptively expand the particles’ search space. Moreover, the chaotic logistic map is not only used in the substitution of the two random parameters affecting the convergence behavior, but also used in the chaotic local search for the global best position so as to easily avoid the particles’ premature behaviors via the whole search space. Thereafter, the convergent analysis of chaotic PID controlling PSO is under deep investigation. Empirical simulation results demonstrate that compared with other several chaotic PSO algorithms like chaotic PSO with the logistic map, chaotic PSO with the tent map and chaotic catfish PSO with the logistic map, chaotic PID controlling PSO exhibits much better search efficiency and quality when solving the optimization problems. Additionally, the parameter estimation of a nonlinear dynamic system also further clarifies its superiority to chaotic catfish PSO, genetic algorithm (GA) and PSO.

The logistic map is one of the most important but common examples of chaotic dynamics. The object shows the crucial belief of the deterministic chaos theory that brings a new procedural structure and apparatus for exploring and understanding complex behavior in dynamical systems. We put an importance on report of the Verhulst logistic map which is one of the potential models and methods for researching dynamical systems that could develop to chaotic. Chaotic signals present a special difficulty in parameter estimation. The difficulty arises from the definition of a chaotic system because of sensitive dependence on initial conditions. It is seen that very slight changes in the initial conditions cause significant effects in the evolution. In general the chaotic systems are nonlinear and apparently random but they are deterministic. The main objective of this paper is how can find the logistic map equation and investigated the chaotic behavior for the logistic equation by varying the control parameters and finally discover Lyaponov exponent, Bifurcation diagrams etc.

In this paper, we aim to solve the optimal tracking control problem for the Henon Mapping chaotic system using the direct heuristic dynamic programming (DHDP) setting with filtered tracking error. The fuzzy logic system is used to approximate the long-term utility function. Compared with the results for chaotic discrete-time system, the cost of the controller is reduced. The Lyapunov analysis approach is utilized to prove the stability of the chaotic system. It is shown that the tracking error, the adaptation law and the control input retain the property of uniformly ultimate boundedness. A simulation example is given to demonstrate the effectiveness of the proposed approach.