Parameter Estimation Of Chaotic Systems Research Articles

Model selection of chaotic systems from data with hidden variables using sparse data assimilation.

Many natural systems exhibit chaotic behavior, including the weather, hydrology, neuroscience, and population dynamics. Although many chaotic systems can be described by relatively simple dynamical equations, characterizing these systems can be challenging due to sensitivity to initial conditions and difficulties in differentiating chaotic behavior from noise. Ideally, one wishes to find a parsimonious set of equations that describe a dynamical system. However, model selection is more challenging when only a subset of the variables are experimentally accessible. Manifold learning methods using time-delay embeddings can successfully reconstruct the underlying structure of the system from data with hidden variables, but not the equations. Recent work in sparse-optimization based model selection has enabled model discovery given a library of possible terms, but regression-based methods require measurements of all state variables. We present a method combining variational annealing-a technique previously used for parameter estimation in chaotic systems with hidden variables-with sparse-optimization methods to perform model identification for chaotic systems with unmeasured variables. We applied the method to ground-truth time-series simulated from the classic Lorenz system and experimental data from an electrical circuit with Lorenz-system like behavior. In both cases, we successfully recover the expected equations with two measured and one hidden variable. Application to simulated data from the Colpitts oscillator demonstrates successful model selection of terms within nonlinear functions. We discuss the robustness of our method to varying noise.

A Novel Teaching-Learning-Based Optimization with Laplace Distribution and Experience Exchange

Teaching-Learning-Based Optimization (TLBO) algorithm is an evolutionary powerful algorithm that has better global searching capability. However, in the later period of evolution of the TLBO algorithm, the diversity of learners will be degraded with the increasing iteration of evolution and the smaller scope of solutions, which lead to a trap in local optima and premature convergence. This paper presents an improved version of the TLBO algorithm based on Laplace distribution and Experience exchange strategy (LETLBO). It uses Laplace distribution to expand exploration space. A new experience exchange strategy is applied to make good use of experience information to identify more promising solutions to make the algorithm converge faster. The experimental performances verify that the LETLBO algorithm enhances the solution accuracy and quality compared to original TLBO and various versions of TLBO and is very competitive with respect to other very popular and powerful evolutionary algorithms. Finally, the LETLBO algorithm is also applied to parameter estimation of chaotic systems, and the promising results show the applicability of the LETLBO algorithm for problem-solving.

An Improved Return Maps Method for Parameter Estimation of Chaotic Systems

Recently, an effective method called return maps is proposed for the parameter estimation of chaotic systems. However, high time-consumption limits practical applications. In this paper, we focus on this problem, and an improved return maps method is proposed. It combines the differential evolution algorithm with the return maps method, and simplifies the calculation process of Euclidean distance. Numerical simulations are carried out on two fractional-order chaotic systems, and the other five methods are used as the comparison. Results show that the improved method can accurately estimate the parameters of chaotic systems, and it saves much time than does the classical return maps method. Furthermore, the proposed method also exhibits good anti-noise performance.

Parameter estimation of chaotic systems based on extreme value points

Parameter estimation and synchronisation of chaotic systems are one of the hottest topics in the field of nonlinear science. In this paper, we addressed how to utilise the obtained experimental time series to estimate multiple parameters in chaotic systems. On the basis of relations of critical points and extreme value points, as well as the least squares estimation, we deduced a novel statistical parameter estimation corollary method to evaluate the unknown parameters in chaotic systems. In order to illustrate the feasibility and effectiveness of the proposed method, three numerical simulation results are presented, where the validity of the proposed method is verified in detail. Furthermore, we also investigated the effects of time-series noise and system disturbances for the proposed method, and the results showed that the proposed method is robust to uncertainties.

The influence of samples on meta-heuristic algorithm for parameter estimation of chaotic system

Parameter estimation of chaotic system is an important issue in nonlinear science. The meta-heuristic algorithm is one of the effective estimation approaches, and has received increasing attention. However, few people have noticed the influence of the samples on parameter estimation. In fact, when using meta-heuristic algorithm for parameter estimation, the number of samples will affect the efficiency greatly. In this paper, this problem is investigated. The relationship between sample and step size, and the factors that affect the difficulty of parameter estimation are also considered. Experimental results show that it is more efficient to set the samples to a small number, and the number of estimated parameters is the most important factor affecting the difficulty of parameter estimation. Finally, to improve the precision further, a new hybrid chaotic particle swarm optimization algorithm which combines a special inertia weight with chaotic search is proposed. Results demonstrate that the new hybrid algorithm is more effective than the other four meta-heuristic algorithms.

Fractional calculus-based firefly algorithm applied to parameter estimation of chaotic systems

This paper deals with the parameters estimation problem of chaotic systems using firefly algorithm (FA). The main contribution of the present work is to introduce a modified version of FA by incorporating fractional calculus during the search process, namely fractional-order FA (FOFA). FOFA simulates the behavior of each firefly with more historical memory, leading to enhance the performance of the basic FA by controlling its convergence speed. A simple structure and straightforward to implement are the main aspects of the proposed FOFA. First, the capability of FOFA is evaluated on the well-known test functions adopted from CEC'2015. Then, FOFA is applied for parameter estimation of chaotic systems. Results reveal that the incorporation of a memory term into the FA is an extremely significant development in comparison with other types of FAs.

Cost function based on hidden Markov models for parameter estimation of chaotic systems

In this note, we deal with parameter estimation methods of chaotic systems. The parameter estimation of the chaotic systems has some significant issues due to their butterfly effects. It can be formulated as an optimization problem and needs a suitable cost function. In this paper, we propose a new cost function based on a hidden Markov model which is a statistical tool for modeling of time series data. It can model dynamical characteristics of the chaotic systems. Moreover, the use of dynamical features of their strange attractors is investigated to achieve a better cost function in the procedure of parameter estimation. Our experimental results indicate the success of the proposed cost function in the one-dimensional parameter estimation of a new four-dimensional chaotic system and Lorenz system as a well-known three-dimensional chaotic system.

An Effective Hybrid Cuckoo Search Algorithm for Unknown Parameters and Time Delays Estimation of Chaotic Systems

Parameter estimation is an important issue in nonlinear science, which can be formulated as a multi-dimensional problem. Numbers of nature-inspired meta-heuristic algorithms have been applied for parameter estimation of chaotic systems; however, many of them are not able to achieve an appropriate trade-off between exploration and exploitation. Therefore, this paper proposes an effective hybrid cuckoo search (HCS) algorithm to obtain higher quality solutions and convergence speed. Inspired by the powerful efficiency of differential evolution, the proposed HCS provides two novel mutation strategies to fully exploit the neighborhood among the current population. Furthermore, a crossover operator under self-adaptive parameters control is introduced to balance the exploration and exploitation ability of the proposed two mutation strategies. Besides, the opposition-based learning is incorporated into HCS for initializing population and producing new candidate solutions during the evolutionary process. HCS is further employed to estimate the unknown parameters and time delays of chaotic systems. Numerical simulations and comparisons with some other optimization methods are conducted on three chaotic systems with and without time delays to demonstrate the performance of HCS. The experimental results show a superiority of HCS in parameter estimation of chaotic systems, and can be regarded as a promising method in terms of its high calculation accuracy, fast convergence speed, and strong robustness.

Parameter estimation for chaotic systems using improved bird swarm algorithm

Parameter estimation of chaotic systems is an important problem in nonlinear science and has aroused increasing interest of many research fields, which can be basically reduced to a multidimensional optimization problem. In this paper, an improved boundary bird swarm algorithm is used to estimate the parameters of chaotic systems. This algorithm can combine the good global convergence and robustness of the bird swarm algorithm and the exploitation capability of improved boundary learning strategy. Experiments are conducted on the Lorenz system and the coupling motor system. Numerical simulation results reveal the effectiveness and with desirable performance of IBBSA for parameter estimation of chaotic systems.

The observer-based synchronization and parameter estimation of a class of chaotic system via a single output

Observer-based synchronization and parameter estimation of chaotic systems has been an interesting and important issue in theory and various fields of application. In this paper first we investigate the observer-based synchronization of a class of chaotic systems, and then discuss its parameter estimation via a single output. We assume that only the sum of the first and second state variables is available. By constructing a proper observer, some novel criteria for observer-based synchronization and parameter estimation are proposed via a scalar input. The Lu chaotic system is taken as an example to demonstrate the efficiency of the proposed approach.

Parameter estimation for chaotic systems via a hybrid flower pollination algorithm

Parameter estimation is a fundamental research issue which has attracted great concern in the control and synchronization of chaotic systems. This problem can be mathematically described as a multi-dimensional continuous optimization problem through constructing an appropriate fitness function, and then solved via meta-heuristic algorithms. A hybrid flower pollination algorithm is proposed in this paper for solving this problem more efficiently. The proposed algorithm well combines the good global exploration ability of the original flower pollination algorithm and the powerful local exploitation ability of the Nelder–Mead simplex method together by integrating these two methods in a very simple way. The experimental results tested on three typical chaotic system parameter estimation problems with three unknown system parameters, including the Lorenz system, the Rossler system, and the Lorenz system under the noise condition, demonstrate that the algorithm can estimate the unknown parameters efficiently and accurately. The comparisons with the basic flower pollination algorithm and some other four reported methods including the quantum particle swarm optimization, the hybrid adaptive cuckoo search optimization algorithm, the oppositional backtracking search optimization algorithm, and the hybrid algorithm combining differential evolution with artificial bee colony suggest that the proposed algorithm is energy efficient and superior. The proposed algorithm can be used as a new and effective choice for parameter estimation of chaotic systems.

Rising Above Chaotic Likelihoods

Berliner [J. Amer. Statist. Assoc., 86 (1991), pp. 983--952] identified a number of difficulties in using the likelihood function within the Bayesian paradigm which arise both for state estimation and for parameter estimation of chaotic systems. Even when the equations of the system are given, he demonstrated “chaotic likelihood functions” both of initial conditions and of parameter values in the logistic map. Chaotic likelihood functions, while ultimately smooth, have such complicated small scale structure as to cast doubt on the possibility of identifying high likelihood states in practice. In this paper, the challenge of chaotic likelihoods is overcome by embedding the observations in a higher dimensional sequence space; this allows good state estimation with finite computational power. An importance sampling approach is introduced, where pseudo-orbit data assimilation is employed in the sequence space, first to identify relevant pseudo-orbits and then relevant trajectories. Estimates are identified with likelihoods orders of magnitude higher than those previously identified in the examples given by Berliner. The pseudo-orbit data assimilation importance sampler exploits the information both from the model dynamics and from the observations. While sampling from the relevant prior (here, the natural measure) will, of course, eventually yield an accountable sample, given the realistic computational resource this traditional approach would provide no high likelihood points at all. While one of the challenges Berliner posed is overcome, his central conclusion is supported. Chaotic likelihood functions for parameter estimation still pose a challenge; this fact helps clarify why physical scientists maintain a strong distinction between the initial condition uncertainty and parameter uncertainty.

Parameter Identification of a Chaotic Circuit with a Hidden Attractor Using Krill Herd Optimization

Parameter estimation plays an important role in modeling and system identification. However, parameter estimation of chaotic systems has some basic differences with other dynamical systems due to butterfly effect. In this paper, we apply a new cost function for parameter estimation in a very interesting chaotic system, a system with a plane of equilibrium which belongs to a newly introduced category of dynamical systems: systems with hidden attractor. The nonlinear dynamics of this system is described in terms of equilibria and its stability, phase portraits, bifurcation diagram and Lyapunov exponents. In order to minimize the proposed cost function and obtain the correct parameters, we use a new efficient optimization method, Krill Herd algorithm. The results show the success of proposed procedures.

Artificial Cooperative Search algorithm for parameter identification of chaotic systems

Parameter estimation of chaotic systems is a challenging and critical topic in nonlinear science. Problem at hand is multi-dimensional and highly nonlinear thereof conventional optimization methods generally fail to extract the unknown parameters of chaotic system. In this study, Artificial Cooperative Search algorithm is put into practice for successful parameter estimation of chaotic systems and compared the parameter estimation performance of Artificial Cooperative Search with Bat, Artificial Bee Colony, Quantum behaved Particle Swarm Optimization algorithms. Parameter identification performance of each algorithm is outlined and benchmarked with several numerical simulations including Lorenz system, Duffing equation and Josephson junction. Results show that Artificial Cooperative Search algorithm outperforms other algorithms in terms of robustness and effectiveness.

An Alternate Iterative Differential Evolution Algorithm for Parameter Identification of Chaotic Systems

Parameter estimation of chaotic systems plays a key role for control and synchronization of chaotic systems. At first, the parameter estimation of chaotic systems is mathematically formulated as a global continuous optimization problem. Then through integrating two differential mutation strategies, an improved greedy selection mechanism and a population diversity balance scheme, an alternate iterative differential algorithm, called AIDE, is presented to solve the problem in this paper. Subsequently, experiments are tested on a set of cases of parameter estimation of chaotic systems and the results show that AIDE is better than or at least equal to DE/rand/1/bin, DE/best/1/bin, and other four well-known algorithms in all cases.

A New Cost Function for Parameter Estimation of Chaotic Systems Using Return Maps as Fingerprints

Estimating parameters of a model system using observed chaotic scalar time series data is a topic of active interest. To estimate these parameters requires a suitable similarity indicator between the observed and model systems. Many works have considered a similarity measure in the time domain, which has limitations because of sensitive dependence on initial conditions. On the other hand, there are features of chaotic systems that are not sensitive to initial conditions such as the topology of the strange attractor. We have used this feature to propose a new cost function for parameter estimation of chaotic models, and we show its efficacy for several simple chaotic systems.

Parameter estimation for chaotic systems using a hybrid adaptive cuckoo search with simulated annealing algorithm

This paper introduces a novel hybrid optimization algorithm to establish the parameters of chaotic systems. In order to deal with the weaknesses of the traditional cuckoo search algorithm, the proposed adaptive cuckoo search with simulated annealing algorithm is presented, which incorporates the adaptive parameters adjusting operation and the simulated annealing operation in the cuckoo search algorithm. Normally, the parameters of the cuckoo search algorithm are kept constant that may result in decreasing the efficiency of the algorithm. For the purpose of balancing and enhancing the accuracy and convergence rate of the cuckoo search algorithm, the adaptive operation is presented to tune the parameters properly. Besides, the local search capability of cuckoo search algorithm is relatively weak that may decrease the quality of optimization. So the simulated annealing operation is merged into the cuckoo search algorithm to enhance the local search ability and improve the accuracy and reliability of the results. The functionality of the proposed hybrid algorithm is investigated through the Lorenz chaotic system under the noiseless and noise condition, respectively. The numerical results demonstrate that the method can estimate parameters efficiently and accurately in the noiseless and noise condition. Finally, the results are compared with the traditional cuckoo search algorithm, genetic algorithm, and particle swarm optimization algorithm. Simulation results demonstrate the effectiveness and superior performance of the proposed algorithm.

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.

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.

An Improved Biogeography Based Optimization for Parameter Estimation of Chaotic Systems

An Improved Biogeography Based Optimization for Parameter Estimation of Chaotic Systems