A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation

In this paper, we construct a novel four dimensional fractional-order chaotic system. Compared with all the proposed chaotic systems until now, the biggest difference and most attractive place is that there exists no equilibrium point in this system. Those rigorous approaches, i.e., Melnikov??s and Shilnikov??s methods, fail to mathematically prove the existence of chaos in this kind of system under some parameters. To reconcile this awkward situation, we resort to circuit simulation experiment to accomplish this task. Before this, we use improved version of the Adams?CBashforth?CMoulton numerical algorithm to calculate this fractional-order chaotic system and show that the proposed fractional-order system with the order as low as 3.28 exhibits a chaotic attractor. Then an electronic circuit is designed for order q=0.9, from which we can observe that chaotic attractor does exist in this fractional-order system. Furthermore, based on the final value theorem of the Laplace transformation, synchronization of two novel fractional-order chaotic systems with the help of one-way coupling method is realized for order q=0.9. An electronic circuit is designed for hardware implementation to synchronize two novel fractional-order chaotic systems for the same order. The results for numerical simulations and circuit experiments are in very good agreement with each other, thus proving that chaos exists indeed in the proposed fractional-order system and the one-way coupling synchronization method is very effective to this system.     

Parameter identification of chaotic systems using improved differential evolution algorithm

In this paper, an improved differential evolution algorithm, named the Taguchi-sliding-based differential evolution algorithm (TSBDEA), is proposed to solve the problem of parameter identification for Chen, Lü and Rossler chaotic systems. The TSBDEA, a powerful global numerical optimization method, combines the differential evolution algorithm (DEA) with the Taguchi-sliding-level method (TSLM). The TSLM is used as the crossover operation of the DEA. Then, the systematic reasoning ability of the TSLM is provided to select the better offspring to achieve the crossover, and consequently enhance the DEA. Therefore, the TSBDEA can be more robust, statistically sound, and quickly convergent. Three illustrative examples of parameter identification for Chen, Lü and Rossler chaotic systems are given to demonstrate the applicability of the proposed TSBDEA, and the computational experimental results show that the proposed TSBDEA not only can find optimal or close-to-optimal solutions but also can obtain both better and more robust results than the DEA.     

Parameter estimation for chaotic systems by hybrid differential evolution algorithm and artificial bee colony algorithm

This paper is concerned with the parameter estimation of nonlinear chaotic system, which could be essentially formulated as a multi-dimensional optimization problem. In this paper, a hybrid algorithm by combining differential evolution with artificial bee colony is implemented to solve parameter estimation for chaotic systems. Hybrid algorithm combines the exploration of differential evolution with the exploitation of the artificial bee colony effectively. Experiments have been conducted on Lorenz system and Chen system. The proposed algorithm is applied to estimate the parameters of two chaotic systems. Simulation results and comparisons demonstrate that the proposed algorithm is better or at least comparable to differential evolution, artificial bee colony, particle swarm optimization, and genetic algorithm from literature when considering the quality of the solutions obtained.     

Chaos synchronization of discrete-time dynamic systems with a limited capacity communication channel

This paper offers a new control strategy for discrete-time chaos synchronization where the drive system and the response system are coupled via a limited capacity communication channel (LCCC for short). One simple condition is presented to ensure synchronization between the two chaotic systems coupled by a LCCC. Based on this condition, an explicit coder–decoder pair for the coding algorithm is designed and the synchronization error between the two chaotic systems decays to zero exponentially based on this coding algorithm. Finally, the proposed control strategy is applied to the well-known H\′{e}non system, and numerical simulations illustrate the validity of the obtained result.     

Adaptive output feedback control of uncertain nonlinear chaotic systems based on dynamic surface control technique

In this paper, an adaptive output feedback control algorithm based on the dynamic surface control (DSC) is proposed for a class of uncertain chaotic systems. Because the system states are assumed to be unavailable, an observer is designed to estimate those unavailable states. The main advantage of this algorithm can overcome the problem of “explosion of complexity” inherent in the backstepping design. Thus, the proposed control approach is simpler than the traditional backstepping control for the uncertain chaotic systems. The stability analysis shows that the system is stable in the sense that all signals in the closed-loop system are uniformly ultimately bounded (UUB) and the system output can track the reference signal to a bounded compact set. Finally, an example is provided to illustrate the effectiveness of the proposed control system.     

Chaos detection and parameter identification in fractional-order chaotic systems with delay

The paper first applies the 0–1 test for chaos to detecting chaos exhibited by fractional-order delayed systems. The results of the test reveal that there exists chaos in some fractional-order delayed systems with specific parameter values, which coincides with previous reports based on the phase portrait. In addition, it is very important to identify exactly the unknown specific parameters of fractional-order chaotic delayed systems in chaos control and synchronization. Thus, a method for parameter identification of fractional-order chaotic delayed systems based on particle swarm optimization (PSO) is presented. By treating the orders as parameters, the parameters and orders are identified through minimizing an objective function. PSO can efficiently find the optimal feasible solution of the objective function. Finally, numerical simulations on fractional-order chaotic logistic delayed system and fractional-order chaotic Chen delayed system show that the proposed method has effective performance of parameter identification.     

A novel numerical algorithm based on Galerkin–Petrov time-discretization method for solving chaotic nonlinear dynamical systems

Nonlinear chaotic systems yield many interesting features related to different physical phenomena and practical applications. These systems are very sensitive to initial conditions at each time-iteration level in a numerical algorithm. In this article, we study the behavior of some nonlinear chaotic systems by a new numerical approach based on the concept of Galerkin–Petrov time-discretization formulation. Computational algorithms are derived to calculate dynamical behavior of nonlinear chaotic systems. Dynamical systems representing weather prediction model and finance model are chosen as test cases for simulation using the derived algorithms. The obtained results are compared with classical RK-4 and RK-5 methods, and an excellent agreement is achieved. The accuracy and convergence of the method are shown by comparing numerically computed results with the exact solution for two test problems derived from another nonlinear dynamical system in two-dimensional space. It is shown that the derived numerical algorithms have a great potential in dealing with the solution of nonlinear chaotic systems and thus can be utilized to delineate different features and characteristics of their solutions.     

Adaptive intelligence learning for nonlinear chaotic systems

In this paper, a reinforcement learning algorithm is proposed for a class of nonlinear differential chaotic systems. The nonlinear function of the chaotic systems is assumed to be bounded but the bounds are unknown. The unknown bounds need to be on-line adjusted. An adaptive optimal (or near optimal) control input with the reinforcement signal can be obtained compared with the current adaptive control for chaotic systems. The reinforcement signal is approximated by the neural networks. Based on Lyapunov analysis theory and by using Young’s inequalities, the closed-loop system is guaranteed to be stable. Finally, the simulation results are given to illustrate the effectiveness of the approach.     

Modeling and prediction of chaotic systems with artificial neural networks

This study of chaotic systems and their prediction is motivated by the fact that many phenomena, both natural and man‐made, are of a chaotic nature. Such phenomena include but are not limited to earthquakes, laser systems, epileptic seizures, combustion, and weather patterns. These phenomena have previously been thought to be unpredictable. However, it is indeed possible to predict time series generated by chaotic systems. The primary objective of this study is to develop a system that would train the artificial neural network (ANN) and then predict the future data of the process. In the present application, the chosen chaotic data set was obtained by solving Lorenz's equations. To predict the future data, the concept of a multilayer feed‐forward ANN with nonlinear auto‐regressive moving averages with exogenous input is used. A Backpropagation algorithm is used to train the network for the chaotic data. The final updated weights from the trained network were then used for the prediction of the future values of the system. Lyapunov exponents, phase diagrams and statistical analyses were used to evaluate the neural network output. A correlation of 94% and a negative Lyapunov exponent indicate that the results obtained from ANN are in good agreement with the actual values. Copyright © 2009 John Wiley & Sons, Ltd.     

Emergence of chaotic regimes in the generalized Lorenz canonical form: a competitive modes analysis

The recently-developed technique of competitive modes analysis is applied to determine parameter regimes for which the generalized Lorenz canonical form, a system constructed by Celikovsky and Chen, which holds many other chaotic systems (such as the Lorenz system, the Lü system, the Chen system, and the Shimizu?CMorioka system), may exhibit chaotic behavior. We verify that the generalized Lorenz canonical form exhibits interesting behaviors in the many parameter regimes thus obtained, thereby demonstrating the great utility of the competitive modes approach in delineating chaotic regimes in multi-parameter systems, where their identification can otherwise involve tedious numerical searches.     

Directly adaptive fuzzy control of discrete-time chaotic systems by least squares algorithm with dead-zone

A new design scheme of directly adaptive fuzzy control for a class of discrete-time chaotic systems is proposed in this paper. The T-S fuzzy model is employed to represent the discrete-time chaotic systems. Then a fuzzy controller is designed and the unknown coefficients of the controller are identified by least squares algorithm with dead-zone. By Lyapunov method, all the signals involved in the closed-loop systems are shown to be bounded and the error between the system output and the reference output is proved to converge to a small neighborhood of zero. Simulation results demonstrate the effectiveness of the theoretical results.     

Parameter estimation of chaotic systems by an oppositional seeker optimization algorithm

The parameter estimation can be formulated as a multi-dimensional optimization problem. By combining the seeker optimization algorithm with the opposition-based learning method, an oppositional seeker optimization algorithm is proposed in this work, and is applied to the parameter estimation of chaotic systems. The seeker optimization algorithm provides a new alternative for population-based heuristic search. By considering an estimate and its opposite of current solutions at the same time, the opposition-based learning method is employed for population initialization and also for generation jumping in seeker optimization algorithm. Numerical simulations on two typical chaotic systems are conducted to show the effectiveness and robustness of the proposed scheme.     

Designing a multi-scroll chaotic system by operating Logistic map with fractal process

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.     

Chaos suppression via periodic change of variables in a class of discontinuous dynamical systems of fractional order

By introducing a suitable change of variable theorem for a class of fractional discontinuous equations, we study the possibility to use a periodic perturbation algorithm to stabilize chaotic trajectories. For this purpose, some new issues of fractional differential inclusions and results on Filippov systems are used. The algorithm, which periodically changes the system variables, has been used so far to stabilize discrete, continuous and discontinuous systems of integer order. As an example, a piece-wise continuous variant of the Chen system is utilized.     

Implementation of FPGA-based real time novel chaotic oscillator

Currently, chaotic systems and chaos-based applications are commonly used in the engineering fields. One of the main structures used in these applications is the chaos-based signal generators. Chaotic signal generators have an important role, particularly in chaotic communication and cryptology. In this study, the Pehlivan-Wei chaotic system, which is a recently developed chaotic system, has been implemented with FPGA using three distinct algorithms (the Euler, Heun, and RK4) for the first time in literature. Numerical and HDL approaches are implemented by these three algorithms to compare the performance of each model for use in chaotic generators. In addition, the Lyapunov exponents and phase portraits of the system have been extracted for chaos analysis. RMSE analysis has been conducted on the chaotic generators, which are modeled using the Euler, Heun, and RK4 algorithms in order to observe error rates of each numerical algorithm in a comparative aspect. The performance of new chaotic system with various data sets has been analyzed. The operation frequency of the chaotic oscillators synthesized and tested for the Virtex-6 FPGA chip has been able to reach up to 463.688 MHz and the chaotic system has been able to calculate 300,000 data sets in 0.0284 s. However, PC-based algorithm having highest performance score can calculate 300,000 data sets in a period of 75.363 s. A comparison study has been performed on the performance of the FPGA-based and PC-based solutions to evaluate each approach.     

The state space reconstruction technology of different kinds of chaotic data obtained from dynamical system

Certain deterministic nonlinear systems may show chaotic behavior. We consider the motion of qualitative information and the practicalities of extracting a part from chaotic experimental data. Our approach based on a theorem of Takens draws on the ideas from the generalized theory of information known as singular system analysis. We illustrate this technique by numerical data from the chaotic region of the chaotic experimental data. The method of the singular-value decomposition is used to calculate the eigenvalues of embedding space matrix. The corresponding concrete algorithm to calculate eigenvectors and to obtain the basis of embedding vector space is proposed in this paper. The projection on the orthogonal basis generated by eigenvectors of timeseries data and concrete paradigm are also provided here. Meanwhile the state space reconstruction technology of different kinds of chaotic data obtained from dynamical system has also been discussed in detail. The project supported by the National Natural Science Foundation of China (19672043)     

Parametric Identification of a Base-Excited Single Pendulum

A harmonic balance based identification algorithm was applied to the simulated single pendulum with horizontal base-excitation. The purpose of this simulation was to examine the applicability of the algorithm on parametrically excited, whirling chaotic systems. Modifications were adopted to adapt to the whirling systems. The system was supposed to be unknown except only the excitation frequency. Linear interpolation functions and the Fourier series functions were tested to approximate unknown nonlinear functions in the governing differential equation. After extracting unstable periodic orbits, all of the parameters were simultaneously identified. By direct comparison, Poincaré section plots and reconstructed phase portrait techniques, it was shown that the identified system had similar dynamical characteristics to the original simulated pendulum, which implies the effectiveness of the examined algorithm.     

A novel chaotic Jaya algorithm for unconstrained numerical optimization

Nonlinear Dynamics - Jaya algorithm is one of the recent algorithms developed to solve optimization problems. The basic concept of this algorithm consists in moving the obtained solution, for a...     

Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control

In this paper, we apply the nonsingular terminal sliding mode control technique to realize the novel combination-combination synchronization between combination of two chaotic systems as drive system and combination of two chaotic systems as response system with unknown parameters in a finite time. On the basic of the adaptive laws and finite-time stability theory, an adaptive combination sliding mode controller is proposed to ensure the occurrence of the sliding motion in a given finite time for four different chaotic systems. In theory, it is proved that the sliding mode technique can realize fast convergence for four different chaotic systems in the finite time. Some criteria and corollaries are derived for finite-time combination-combination synchronization of four different chaotic systems. Numerical simulation results are shown to verify the effectiveness and correctness of the combination-combination synchronization.     

Complexity of chaotic binary sequence and precision of its numerical simulation

Numerical simulation is one of primary methods in which people study the property of chaotic systems. However, there is the effect of finite precision in all processors which can cause chaos to degenerate into a periodic function or a fixed point. If it is neglected the precision of a computer processor for the binary numerical calculations, the numerical simulation results may not be accurate due to the chaotic nature of the system under study. New and more accurate methods must be found. A quantitative computable method of sequence complexity evaluation is introduced in this paper. The effect of finite precision is evaluated from the viewpoint of sequence complexity. The simulation results show that the correlation function based on information entropy can effectively reflect the complexity of pseudorandom sequences generated by a chaotic system, and it is superior to the other measure methods based on entropy. The finite calculation precision of the processor has significant effect on the complexity of chaotic binary sequences generated by the Lorenz equation. The pseudorandom binary sequences with high complexity can be generated by a chaotic system as long as the suitable computational precision and quantification algorithm are selected and behave correctly. The new methodology helps to gain insight into systems that may exist in various application domains such as secure communications and spectrum management.