Identification Of Chaotic Systems Research Articles

Sparse Identification of Fractional Chaotic Systems based on the time-domain data

The complexity inherent in defining and computing fractional derivatives renders the development of a data-driven modeling framework for fractional-order systems more formidable than for their integer-order counterparts. This study introduces a methodology for sparsely identifying fractional chaotic systems by leveraging time-domain data to ascertain the system’s governing equations. Initially, we establish a sparse identification framework specifically tailored for fractional order systems and introduce a joint iterative thresholding method designed to identify these systems’ fractional and integer order components concurrently. Moreover, this study conducts a comparative analysis of two error criteria for model selection. It introduces a strategy predicated on minimizing the distance to balance sparsity and model fitting error optimally. To validate the efficacy and precision of the proposed methodology, numerical simulations were conducted on fractional order simplified Lorenz and Chua’s circuits, affirming the robustness and accuracy of our approach.

Research on Information Identification of Chaotic Map with Multi-Stability

Influenced by the rapid development of artificial intelligence, the identification of chaotic systems with intelligent optimization algorithms has received widespread attention in recent years. This paper focuses on the intelligent information identification of chaotic maps with multi-stability properties, and an improved sparrow search algorithm is proposed as the identification algorithm. Numerical simulations show that different initial values can lead to the same dynamic behavior, making it impossible to stably and accurately identify the initial values of multi-stability chaotic maps. An identification scheme without considering the initial values is proposed for solving this problem, and simulations demonstrate that the proposed method has the highest identification precision among seven existing intelligent algorithms and a certain degree of noise resistance. In addition, the above research reveals that chaotic systems with multi-stability may have more potential applications in fields such as secure communication.

Optimized cascade chaotic fuzzy system (OCCFS) and its application to function approximation and chaotic systems identification

Optimized cascade chaotic fuzzy system (OCCFS) and its application to function approximation and chaotic systems identification

Bayesian system identification and chaotic prediction from data for stochastic Mathieu-van der Pol-Duffing energy harvester

• An improved approximate Bayesian method is proposed. • The unknown parameters of chaotic VEH system are identified by improved approximate Bayesian method. • The 0-1 test is used to judge the chaotic occur based on the correlation of the response data In this paper, the approximate Bayesian computation combines the particle swarm optimization and sequential Monte Carlo methods, which identify the parameters of the Mathieu-van der Pol-Duffing chaotic energy harvester system. Then the proposed method is applied to estimate the coefficients of the chaotic model and the response output paths of the identified coefficients compared with the observed, which verifies the effectiveness of the proposed method. Finally, a partial response sample of the regular and chaotic responses, determined by the maximum Lyapunov exponent, is applied to detect whether chaotic motion occurs in them by a 0-1 test. This paper can provide a reference for data-based parameter identification and chaotic prediction of chaotic vibration energy harvester systems.

Parameter Estimation of Fractional-Order Chaotic Power System Based on Lens Imaging Learning Strategy State Transition Algorithm

Parameter identification of fractional-order chaotic power systems is a multidimensional optimization problem that plays a decisive role in the synchronization and control of fractional-order chaotic power systems. In this paper, a state transition algorithm based on the lens imaging learning strategy is proposed for parameter identification of fractional-order chaotic power systems. Taking a fractional-order six-dimensional chaotic power system mathematical model as an example, the mathematical model and chaotic state are analyzed. First, the Tent chaotic mapping is used to initialize the population, thus increasing population diversity. The randomness and ergodicity of the Tent chaotic sequence are used to enhance the global searching ability of the algorithm. Second, a maturity index is employed to determine population maturity. The lens imaging learning strategy is used to suppress the premature convergence of the state transition algorithm effectively and help the population jump out of local optima. Finally, the improved state transition algorithm is used to identify the parameters of the fractional-order six-dimensional chaotic power system model. The proposed improved state transition algorithm shows high estimation accuracy and convergence speed, and is superior to the traditional state transition and particle swarm optimization algorithms. The simulation results show that the parameters of the fractional-order chaotic power system are identified accurately even in the presence of white noise, demonstrating the strong robustness and versatility of the proposed algorithm.

Finite-Time Generalized Synchronization and Parameter Identification of Chaotic Systems with Different Dimensions

Finite-time generalized synchronization and parameter identification of chaotic systems with different dimensions are studied in this paper. Based on finite-time stability theory, adaptive controllers and parameter identification rules are proposed to realize generalized synchronization of chaotic systems and identify accurately all their unknown parameters in finite time. At the same time, the obtained synchronization time formula can estimate the required longest time for realizing the generalized synchronization of chaotic systems, which solely depends on the initial values of chaotic systems and is easily calculated in numerical simulation. At the last, some numerical examples are presented to illustrate the effectiveness and correctness of the obtained conclusions.

The identification of fractional order systems by multiscale multivariate analysis

Abstract The recognition of complex systems plays a vital role in system dynamics. To distinguish chaotic systems and stochastic processes, the complex-entropy causality plane (CECP) was proposed and received considerable attention. However, few methods have been applied to the identification of fractional order chaotic systems. In this paper, we propose multiscale multivariate fractional dispersion entropy (MMFDE) and MMFDE plane to study the structural complexity of multivariate nonlinear systems. Unlike the ordinal pattern in CECP, the dispersion pattern is used in MMFDE plane to solve the computational burden in fractional order systems. Moreover, the fractional order α in MMFDE can be used to adjust the ability of MMFDE plane to distinguish between different systems. Through simulation data, we find that the MMFDE plane can classify not only chaotic systems and stochastic processes but also different fractional order chaotic systems even if adding noise to the signals. It is then applied to financial time series and epileptic EEG recordings. The results show that the MMFDE plane can identify developed and emerging markets. For epileptic EEG recordings, both the EEG subbands and the original data without filtering can be used to distinguish different states of healthy subjects and patients with epilepsy. The coarse-graining process is also discussed to demonstrate the consequences further.

Whale optimization and sine–cosine optimization algorithms with cellular topology for parameter identification of chaotic systems and Schottky barrier diode models

This research study aims to enhance the optimization accuracy of the two recently emerged metaheuristics of whale and sine–cosine optimizers by means of the balanced improvements in intensification and diversification phases of the algorithms provided by cellular automata (CA). Stagnation at the early phases of the iterations, which leads to entrapment in local optimum points in the search space, is one of the inherent drawbacks of the metaheuristic algorithms. As a favorable solution alternative to this problem, different types of cellular topologies are implemented into these two algorithms with a view to ameliorating their search mechanisms. Exploitation of the fertile areas in the search domain is maintained by the interaction between the topological neighbors, whereas the improved exploration is resulted from the smooth diffusion of the available population information among the structured neighbors. Numerical experiments have been carried out to assess the optimization performance of the proposed cellular-based algorithms. Optimization benchmark problems comprised of unimodal and multimodal test functions have been applied and numerical results have been compared with those found by some of the state-of-the-art literature optimizers including particle swarm optimization, differential evolution, artificial cooperative search and differential search. Cellular variants have been outperformed by the base algorithms for multimodal benchmark problems of Levy and Penalized1 functions. Then, the proposed cellular algorithms have been applied to two different parameter identification cases in order to test their efficiencies on real-world optimization problems. Extensive performance evaluations on different parameter optimization cases reveal that incorporating the CA concepts on these algorithms not only improves the optimization accuracy but also provides considerable robustness to acquired solutions.

Secure Communication Method of Metallurgical System Data Network based on Chaotic System Sequence Parameter Identification

In the research process of chaotic system sequence parameter identification, we find that there is only one chaotic attractor in chaotic direct spread spectrum signals, which cannot satisfy the confidentiality of communications. Based on the thought of generalized synchronization of chaotic system, we proposed a chaotic secure communication method, in other words, the partial sequences of chaotic system were used for the parameter identification of chaotic system. The chaotic ant colony optimization algorithm was used for the parameter identification of partial sequences of chaotic system, so as to achieve the purpose of understanding all the information of chaotic system. The parameter space and the ant colony space were introduced into the parameter identification process. We used the space transformation function to transform the parameter space and the ant colony space. Experimental results show that using the Lorenz system numerical method can verify the feasibility of chaotic system partial sequence parameter identification and chaotic secure communication.

Recursive second-order Volterra filter based on Dawson function for chaotic memristor system identification

In this paper, we propose a modified version of exponential cost function to improve the stability of adaptive algorithm, where the recursive algorithm is based on the Dawson function. The second-order Volterra (SOV) filter is incorporated into the proposed recursive algorithm, resulting the SOV-ExRLS algorithm, to achieve the improved performance in both $$\alpha $$-stable and Gaussian environments. Moreover, the mean and mean-square behavior of the SOV-ExRLS algorithm is analyzed. In particular, the proposed cExRLS-IDLMS method is convexly combined with the functional link artificial neural network filter to flexibly model the chaotic memristor system. Simulation studies verify the analytical findings and reveal enhanced identification performance of the proposed algorithms over the existing nonlinear algorithms.

Parameter Identification of Fractional Order Chaotic System via Opposition Based Learning Bare-Bones Imperialist Competition Algorithm

Parameter Identification of Fractional Order Chaotic System via Opposition Based Learning Bare-Bones Imperialist Competition Algorithm

Parameter identification of engineering problems using a differential shuffled complex evolution

An accurate mathematical model has a vital role in controlling and synchronization of different systems. But generally in real-world problems, parameters are mixed with mismatches and distortions. In this paper, an improved shuffled complex evolution (SCE) is proposed for parameter identification of engineering problems. The SCE by employing parallel search efficiently finds neighborhoods of the optimal point. So it carries out exploration in a proper way. But its drawback is due to exploitation stages. The SCE cannot converge accurately to an optimal point, in many cases. The current study focuses to overcome this drawback by inserting a shrinkage stage to an original version of SCE and presents a powerful global numerical optimization method, named the differential SCE. The efficacy of the proposed algorithm is first tested on some benchmark problems. After achieving satisfactory performance on the test problems, to demonstrate the applicability of the proposed algorithm, it is applied to ten identification problems includes parameter identification of ordinary differential equations and chaotic systems. Practical experiences show that the proposed algorithm is very effective and robust so that it produces similar and promising results over repeated runs. Also, a comparison against other evolutionary algorithms reported in the literature demonstrates a significantly better performance of our proposed algorithm.

Enhanced global flower pollination algorithm for parameter identification of chaotic and hyper-chaotic system

The problem of system parameter identification is a fundamental problem in the field of nonlinear science, which can be described as a multidimensional optimization problem. In this paper, an enhanced global flower pollination algorithm (GFPA) is proposed for parameter identification of chaotic and hyper-chaotic systems. The motion trajectory of the flower pollination algorithm is analyzed for the first time, and the equation of the algorithm exploration phase is improved by the chaotic mapping method to ensure the convergence of the algorithm in the exploration phase. In addition, in order to improve the convergence speed of the algorithm, the update method of the exploitation phase is reset by using the best information to guide the searching. Through analysis, the proposed new algorithm can guarantee the convergence of the algorithm without increasing the time complexity. Finally, we identify and validate the system of the Lorenz, Rossler, Chen and the system of the Rossler hyper-chaotic, Chen hyper-chaotic. The experimental results show that GFPA has better identification effect.

On the Identification and Generation of Discrete-Time Chaotic Systems with Recurrent Neural Networks

We address the identification and generation of the discrete-time chaotic system (DTCS) with a two-layered recurrent neural network (RNN). First, we propose an identification procedure of the DTCS in which the RNN is required to have less layers than in the conventional procedures. Next, based on Li–Yorke theorem, we propose a generation procedure which enables us to predict a range of chaotic behavior of the DTCS in advance. Simulation results demonstrate that the proposed identification procedure, employing the Levenberg–Marquardt algorithm and a two-layered RNN, requires lower computational complexity than the conventional identification procedures at comparable performance.

Parameter Identification of Chaotic Systems Using a Modified Cost Function Including Static and Dynamic Information of Attractors in the State Space

Parameter identification (PI) is very important in system analysis and controller design and has wide applications in industry. PI is an essential step in designing mathematical models of the dynamical systems based on the measured data. Identification of chaotic systems is very challenging due to the butterfly effect. Recently, a state-space-based cost function based on Gaussian mixture model (GMM) has been proposed for the PI of the chaotic systems. In there, a GMM as a statistical model of the measured data modeled static features of the chaotic attractors in the state space. In this paper, we propose a new method which incorporates static and dynamic features in the GMM modeling in order to achieve a better cost function for the PI problem. The proposed method can extract suitable information from the trajectory of the chaotic attractor. We conduct some experiments for one-dimensional PI. Using measured data from 4D and 3D chaotic systems, empirical results indicate success of the proposed method.

Parameter identification of fractional-order chaotic systems without or with noise: Reply to comments

Parameter identification of chaotic systems without or with noise are two different cases of ideal and real situations. These two cases have been widely studied. This paper is a reply to the comments by Jafari et al. (2013) regarding the objective function in the process of parameter identification. The following scripts are to address and discuss the comments. Positive and satisfactory responses are provided to all the ambiguous questions. Furthermore, we have also analyzed parameter identification in fractional-order chaotic systems with noise and published the related key codes. Experimental results show that the proposed technique and adopted objective function is reliable, even with the presence of noise.

Non-linear Fractional-Order Chaotic Systems Identification with Approximated Fractional-Order Derivative based on a Hybrid Particle Swarm Optimization-Genetic Algorithm Method

Although many mathematicians have searched on the fractional calculus since many years ago, but its application in engineering, especially in modeling and control, does not have many antecedents. Since there are much freedom in choosing the order of differentiator and integrator in fractional calculus, it is possible to model the physical systems accurately. This paper deals with time-domain identification fractional-order chaotic systems where conventional derivation is replaced by a fractional one with the help of a non-integer derivation. This operator is itself approximated by a N-dimensional system composed of an integrator and a phase-lead filter. A hybrid particle swarm optimization (PSO) and genetic algorithm (GA) method has been applied to estimate the parameters of approximated nonlinear fractional-order chaotic system that modeled by a state-space representation. The feasibility of this approach is demonstrated through identifying the parameters of approximated fractional-order Lorenz chaotic system. The performance of the proposed algorithm is compared with the genetic algorithm (GA) and standard particle swarm optimization (SPSO) in terms of parameter accuracy and cost function. To evaluate the identification accuracy, the time-domain output error is designed as the fitness function for parameter optimization. Simulation results show that the proposed method is more successful than other algorithms for parameter identification of fractional order chaotic systems.

Parametric identification of fractional-order nonlinear systems

This work presents a new method for the identification of fractional-order nonlinear systems from time domain data. A parametric identification technique for integer-order systems using multiple trials is generalized and adapted for identification of fractional-order nonlinear systems. The time response of the system to be identified by two different harmonic excitations are considered. An initial value of the fractional-order term is assumed. Other system parameters are identified for each of the trial data by finding the pseudo-inverse of an equivalent algebraic system. A termination criterion is specified in terms of the error between identified values of system parameters from the two trials. If the error in system parameters does not fall in the specified tolerance, the value of the fractional-order term is varied using gradient descent method and identification is carried out until convergence. The validity of the proposed algorithm is demonstrated by applying it to fractional-order Duffing, van der Pol and van der Pol–Duffing oscillators. Numerical simulations show that the exact and identified parameters are in very close agreement. Comparison of the time taken for identification by the proposed method and those which define error in terms of time response shows the effectiveness of the method. Furthermore, the validity of the proposed method for identification of chaotic fractional-order systems is demonstrated by showing that the identified values are independent of signal time length used.

Online identification of large-scale chaotic system

The ensemble prediction system (EPS) is an approach employed in meteorology to estimate forecast uncertainty of dynamical systems. In EPS, an ensemble of auxiliary simulations is launched along with the main prediction. Recently, an application with the EPS framework was proposed as a method that enables algorithmic tuning of parameters of large-scale models, in cases where high-CPU demands make usual iterative optimization impractical. The approach was aimed and tested for operational numerical weather prediction models, with a relatively small number of parameters and well-tuned initial values. Here, we present a new version of the approach as a general-purpose parameter estimation method for situations where effective parallel computing is available, but high-CPU requirements exclude the use of standard sequential approaches. We treat the problem as a stochastic optimization task and employ an evolutionary approach, the differential evolution as the optimizer. We demonstrate improved convergence properties, especially for strongly biased initial values or higher number of parameters. For parametric uncertainty quantification, the approach can be considered as a heuristic sampler of the parameter distributions.

Identification of Uncertain Incommensurate Fractional-Order Chaotic Systems Using an Improved Quantum-Behaved Particle Swarm Optimization Algorithm

This paper is concerned with a significant issue in the research of nonlinear science, i.e., parameter identification of uncertain incommensurate fractional-order chaotic systems, which can be essentially formulated as a multidimensional optimization problem. Motivated by the basic particle swarm optimization and quantum mechanics theories, an improved quantum-behaved particle swarm optimization (IQPSO) algorithm is proposed to tackle this complex optimization problem. In this work, both systematic parameters and fractional derivative orders are regarded as independent unknown parameters to be identified. Numerical simulations are conducted to identify two typical incommensurate fractional-order chaotic systems. Simulation results and comparisons analyses demonstrate that the proposed method is suitable for parameter identification with advantages of high effectiveness and efficiency. Moreover, we also, respectively, investigate the effect of systematic parameters, fractional derivative orders, and additional noise on the optimization performances. The corresponding results further validate the superior searching capabilities of the proposed algorithm.