Prediction Of Chaotic System Research Articles

Complex network-based multistep forecasting model for hyperchaotic time series.

We present a method for predicting hyperchaotic time series using a complex network-based forecasting model. We first construct a network from a given time series, which serves as a coarse-grained representation of the underlying attractor. This network facilitates multistep forecasting by capturing the local nonlinearity of the dynamics and offers superior accuracy over more extended periods than traditional methods. The network is formed by converting the patterns of local oscillations into sequences of numerical symbols, which are then used to create nodes and edges in a network, capturing the system's dynamical behavior at a reduced resolution. The network allows predictions up to several steps ahead without the exponential error increase usually associated with linear first-order methods. The improved predictions result from the unique ability of the network to collect identical pattern transitions in the orbit dynamics into a system of neighborhoods in the network. The effectiveness of this approach is demonstrated through its application to several high-dimensional hyperchaotic systems, where it outperforms both the linear first-order and other network-based methods in terms of prediction accuracy and horizon. Besides enhancing the predictability of chaotic systems, this methodology also outlines a procedure to develop a discrete model flow within an attractor.

Using a snow ablation optimizer in an autonomous echo state network for the model-free prediction of chaotic systems

Using a snow ablation optimizer in an autonomous echo state network for the model-free prediction of chaotic systems

Comparative Study of Neural Network Models for Prediction of Chaotic System

Comparative Study of Neural Network Models for Prediction of Chaotic System

A Monte Carlo approach to understanding the impacts of initial-condition uncertainty, model uncertainty, and simulation variability on the predictability of chaotic systems: Perspectives from the one-dimensional logistic map.

The predictability of the logistic map is investigated for the joint impact of initial-condition (IC) and model uncertainty (bias + random variability) as well as simulation variability. To this end, Monte Carlo simulations are carried out where IC bias is varied in a wide range of 10-15-10-3, and, similarly, model bias is introduced in comparable range. It is found that while the predictability limit of the logistic map can be continuously extended by reducing IC bias, the introduction of the model bias imposes an upper limit to the predictability limit beyond which further reductions in IC bias do not lead to an extension in the predictability limit, effectively restricting the feasible joint space spanned by the IC-model biases. It is further observed that imposing a lower limit to the allowed variability among ensemble solutions (so as to prevent the ensemble variability from collapse) results in a similar constraint in the joint IC-model-bias space; but this correspondence breaks down when the imposed variability limit is too high (∼x>0.7 for the logistic map). Finally, although increasing the IC random variability in an ensemble is found to consistently extend the allowed predictability limit of the logistic map, the same is not observed for model parameter random variability. In contrast, while low levels of model parameter variability have no impact on the allowed predictability limit, there appears to be a threshold at which an abrupt transition occurs toward a distinctly lower predictability limit.

An elastic framework for ensemble-based large-scale data assimilation

Prediction of chaotic systems relies on a floating fusion of sensor data (observations) with a numerical model to decide on a good system trajectory and to compensate non-linear feedback effects. Ensemble-based data assimilation (DA) is a major method for this concern depending on propagating an ensemble of perturbed model realizations. In this paper, we develop an elastic, online, fault-tolerant and modular framework called Melissa-DA for large-scale ensemble-based DA. Melissa-DAallows elastic addition or removal of compute resources for state propagation at runtime. Dynamic load balancing based on list scheduling ensures efficient execution. Online processing of the data produced by ensemble members enables to avoid the I/O bottleneck of file-based approaches. Our implementation embeds the PDAF parallel DA engine, enabling the use of various DA methods. Melissa-DAcan support extra ensemble-based DA methods by implementing the transformation of member background states into analysis states. Experiments confirm the excellent scalability of Melissa-DA, propagating 16,384 members for a regional hydrological critical zone assimilation relying on the ParFlow model on a domain with about 4 M grid cells. The same use case was ported to the PDAF state-of-the-art DA framework relying on a MPI approach. A comparison with Melissa-DA at 2500 members on 20,000 cores shows our approach is about 50% faster per assimilation cycle.

Robust Optimization and Validation of Echo State Networks for learning chaotic dynamics

An approach to the time-accurate prediction of chaotic solutions is by learning temporal patterns from data. Echo State Networks (ESNs), which are a class of Reservoir Computing, can accurately predict the chaotic dynamics well beyond the predictability time. Existing studies, however, also showed that small changes in the hyperparameters may markedly affect the network’s performance. The overarching aim of this paper is to improve the robustness in the selection of hyperparameters in Echo State Networks for the time-accurate prediction of chaotic solutions. We define the robustness of a validation strategy as its ability to select hyperparameters that perform consistently between validation and test sets. The goal is three-fold. First, we investigate routinely used validation strategies. Second, we propose the Recycle Validation, and the chaotic versions of existing validation strategies, to specifically tackle the forecasting of chaotic systems. Third, we compare Bayesian optimization with the traditional grid search for optimal hyperparameter selection. Numerical tests are performed on prototypical nonlinear systems that have chaotic and quasiperiodic solutions, such as the Lorenz and Lorenz-96 systems, and the Kuznetsov oscillator. Both model-free and model-informed Echo State Networks are analysed. By comparing the networks’ performance in learning chaotic (unpredictable) versus quasiperiodic (predictable) solutions, we highlight fundamental challenges in learning chaotic solutions. The proposed validation strategies, which are based on the dynamical systems properties of chaotic time series, are shown to outperform the state-of-the-art validation strategies. Because the strategies are principled – they are based on chaos theory such as the Lyapunov time – they can be applied to other Recurrent Neural Networks architectures with little modification. This work opens up new possibilities for the robust design and application of Echo State Networks, and Recurrent Neural Networks, to the time-accurate prediction of chaotic systems.

New Results for Prediction of Chaotic Systems Using Deep Recurrent Neural Networks

Prediction of nonlinear and dynamic systems is a challenging task, however with the aid of machine learning techniques, particularly neural networks, is now possible to accomplish this objective. Most common neural networks used are the multilayer perceptron (MLP) and recurrent neural networks (RNN) using long-short term memory units (LSTM-RNN). In recent years, deep learning neural network models have become more relevant due the improved results they show for various tasks. In this paper the authors compare these neural network models with deep learning neural network models such as long-short term memory deep recurrent neural network (LSTM-DRNN) and gate recurrent unit deep recurrent neural network (GRU-DRNN) when presented with the task of predicting three different chaotic systems such as the Lorenz system, Rabinovich–Fabrikant and the Rossler System. The results obtained show that the deep learning neural network model GRU-DRNN has better results when predicting these three chaotic systems in terms of loss and accuracy than the two other models using less neurons and layers. These results can be very helpful to solve much more complex problems such as the control and synchronization of these chaotic systems.

Accurate predictions of chaotic motion of a free fall disk

It is important to know the accurate trajectory of a free fall object in fluid (such as a spacecraft), whose motion might be chaotic in many cases. However, it is impossible to accurately predict its chaotic trajectory in a long enough duration by traditional numerical algorithms in double precision. In this paper, we give the accurate predictions of the same problem by a new strategy, namely, the Clean Numerical Simulation (CNS). Without loss of generality, a free-fall disk in water is considered, whose motion is governed by the Andersen–Pesavento–Wang model. We illustrate that convergent and reliable trajectories of a chaotic free-fall disk in a long enough interval of time can be obtained by means of the CNS, but different traditional algorithms in double precision give disparate trajectories. Besides, unlike the traditional algorithms in double precision, the CNS can predict the accurate posture of the free-fall disk near the vicinity of the bifurcation point of some physical parameters in a long duration. Therefore, the CNS can provide reliable prediction of chaotic systems in a long enough interval of time.

Predicting phase and sensing phase coherence in chaotic systems with machine learning.

Recent interest in exploiting machine learning for model-free prediction of chaotic systems focused on the time evolution of the dynamical variables of the system as a whole, which include both amplitude and phase. In particular, in the framework based on reservoir computing, the prediction horizon as determined by the largest Lyapunov exponent is often short, typically about five or six Lyapunov times that contain approximately equal number of oscillation cycles of the system. There are situations in the real world where the phase information is important, such as the ups and downs of species populations in ecology, the polarity of a voltage variable in an electronic circuit, and the concentration of certain chemical above or below the average. Using classic chaotic oscillators and a chaotic food-web system from ecology as examples, we demonstrate that reservoir computing can be exploited for long-term prediction of the phase of chaotic oscillators. The typical prediction horizon can be orders of magnitude longer than that with predicting the entire variable, for which we provide a physical understanding. We also demonstrate that a properly designed reservoir computing machine can reliably sense phase synchronization between a pair of coupled chaotic oscillators with implications to the design of the parallel reservoir scheme for predicting large chaotic systems.

Long-term prediction of chaotic systems with machine learning

This paper develops a machine learning scheme to obtain long term predictions on chaotic systems, including high-dimensional, spatiotemporal chaotic systems, by using extremely rare updates

Chaotic System Prediction Using Data Assimilation and Machine Learning

Atmospheric systems are typically chaotic and their chaotic nature is an important limiting factor for weather forecasting and climate prediction. So far, there have been many studies on the simulation and prediction of chaotic systems using numerical simulation methods. However, there are many intractable problems in predicting chaotic systems using numerical simulation methods, such as initial value sensitivity, error accumulation, and unreasonable parameterization of physical processes, which often lead to forecast failure. With the continuous improvement of observational techniques, data assimilation has gradually become an effective method to improve the numerical simulation prediction. In addition, with the advent of big data and the enhancement of computing resources, machine learning has achieved great success. Studies have shown that deep neural networks are capable of mining and extracting the complex physical relationships behind large amounts of data to build very good forecasting models. Therefore, in this paper, we propose a prediction method for chaotic systems that combines deep neural networks and data assimilation. To test the effectiveness of the method, we use the model to perform forecasting experiments on the Lorenz96 model. The experimental results show that the prediction method that combines neural network and data assimilation is very effective in predicting the amount of state of Lorenz96. However, Lorenz96 is a relatively simple model, and our next step will be to continue the experiments on the complex system model to test the effectiveness of the proposed method in this paper and to further optimize and improve the proposed method.

On the existence of proper stochastic Markov models for statistical reconstruction and prediction of chaotic time series

Abstract In this paper, the problem of statistical reconstruction and prediction of chaotic systems with unknown governing equations using stochastic Markov models is investigated. Using the time series of only one measurable state, an algorithm is proposed to design any orders of Markov models and the approach is state transition matrix extraction. Using this modeling, two goals are followed: first, using the time series, statistical reconstruction is performed through which the probability density and conditional probability density functions are reconstructed; and second, prediction is performed. For this problem, some estimators are required and here the maximum likelihood and the conditional expected value are used. The efficiency of this algorithm has been investigated by applying it to some typical data. To illustrate the advantages of this model, a deterministic model based on the nearest neighbors in the delayed phase space is used. It will be seen that using the appropriate Markov model order and sufficient number of meshes, the proposed algorithm, unlike the deterministic model, is capable of reconstructing and predicting the chaotic data in a many steps.

Synchronization of chaotic systems and their machine-learning models.

Recent advances have demonstrated the effectiveness of a machine-learning approach known as "reservoir computing" for model-free prediction of chaotic systems. We find that a well-trained reservoir computer can synchronize with its learned chaotic systems by linking them with a common signal. A necessary condition for achieving this synchronization is the negative values of the sub-Lyapunov exponents. Remarkably, we show that by sending just a scalar signal, one can achieve synchronism in trained reservoir computers and a cascading synchronization among chaotic systems and their fitted reservoir computers. Moreover, we demonstrate that this synchronization is maintained even in the presence of a parameter mismatch. Our findings possibly provide a path for accurate production of all expected signals in unknown chaotic systems using just one observational measure.

Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system

For an n-dimensional chaotic system, we extend the definition of the nonlinear local Lyapunov exponent (NLLE) from one- to n-dimensional spectra, and present a method for computing the NLLE spectrum. The method is tested on three chaotic systems with different complexity. The results indicate that the NLLE spectrum realistically characterizes the growth rates of initial error vectors along different directions from the linear to nonlinear phases of error growth. This represents an improvement over the traditional Lyapunov exponent spectrum, which only characterizes the error growth rates during the linear phase of error growth. In addition, because the NLLE spectrum can effectively separate the slowly and rapidly growing perturbations, it is shown to be more suitable for estimating the predictability of chaotic systems, as compared to the traditional Lyapunov exponent spectrum.

Predictive uncertainty of chaotic daily streamflow using ensemble wavelet networks approach

Perfect or even mediocre weather predictions over a long period are almost impossible because of the ultimate growth of a small initial error into a significant one. Even though the sensitivity of initial conditions limits the predictability in chaotic systems, an ensemble of prediction from different possible initial conditions and also a prediction algorithm capable of resolving the fine structure of the chaotic attractor can reduce the prediction uncertainty to some extent. All of the traditional chaotic prediction methods in hydrology are based on single optimum initial condition local models which can model the sudden divergence of the trajectories with different local functions. Conceptually, global models are ineffective in modeling the highly unstable structure of the chaotic attractor. This paper focuses on an ensemble prediction approach by reconstructing the phase space using different combinations of chaotic parameters, i.e., embedding dimension and delay time to quantify the uncertainty in initial conditions. The ensemble approach is implemented through a local learning wavelet network model with a global feed‐forward neural network structure for the phase space prediction of chaotic streamflow series. Quantification of uncertainties in future predictions are done by creating an ensemble of predictions with wavelet network using a range of plausible embedding dimensions and delay times. The ensemble approach is proved to be 50% more efficient than the single prediction for both local approximation and wavelet network approaches. The wavelet network approach has proved to be 30%–50% more superior to the local approximation approach. Compared to the traditional local approximation approach with single initial condition, the total predictive uncertainty in the streamflow is reduced when modeled with ensemble wavelet networks for different lead times. Localization property of wavelets, utilizing different dilation and translation parameters, helps in capturing most of the statistical properties of the observed data. The need for taking into account all plausible initial conditions and also bringing together the characteristics of both local and global approaches to model the unstable yet ordered chaotic attractor of a hydrologic series is clearly demonstrated.

Route to hyperchaos in a system of coupled oscillators with multistability

This work presents the results of a detailed experimental study into the transition between synchronized, low-dimensional, and unsynchronized, high-dimensional dynamics using a system of coupled electronic chaotic oscillators. Novel data analysis techniques have been employed to reveal that a hyperchaotic attractor can arise from the amalgamation of two nonattracting sets. These originate from initially multistable low-dimensional attractors which experience a smooth transition from low- to high-dimensional chaotic behavior, losing stability through a bubbling bifurcation. Numerical techniques were also employed to verify and expand on the experimental results, giving evidence on the locally unstable invariant sets contained within the globally stable hyperchaotic attractor. This particular route to hyperchaos also results in the possibility of phenomena (such as unstable dimension variability) that can be a major obstruction to shadowing and predictability in chaotic systems.

Modeling and prediction of chaotic systems with artificial neural networks

AbstractThis 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.

Prediction of chaotic systems with multidimensional recurrent least squares support vector machines

In this paper, we propose a multidimensional version of recurrent least squares support vector machines (MDRLS-SVM) to solve the problem about the prediction of chaotic system. To acquire better prediction performance, the high-dimensional space, which provides more information on the system than the scalar time series, is first reconstructed utilizing Takens's embedding theorem. Then the MDRLS-SVM instead of traditional RLS-SVM is used in the high-dimensional space and the prediction performance can be improved from the point of view of reconstructed embedding phase space. In addition, the MDRLS-SVM algorithm is analysed in the context of noise and we also find that the MDRLS-SVM has lower sensitivity to noise than the RLS-SVM.

Periodicity and Predictability in Chaotic Systems

Most of the good mathematical models available for natural phenomena are non linear, and these are often chaotic. Sensitive dependence on initial conditions can foil any attempt at prediction. In fact, measurement errors in collecting the initial data, intrinsic to any observational method, will propagate under the computational use of the model. With this, the final results obtained from the model may be useless when compared with the real future of the phenomena. Even in the strictly computational context, the sensitive dependence on initial conditions often makes a chaotic system S unpredictable because of representation errors. In fact, most of the orbits obtained in computers are not even good approximations to the actual ones. Real number representations by computers fall under two headings: floating-point (or numerical) and algebraic (or symbolic). In the former case, given the finite amount of memory, only numbers with finite decimal representations are treated without (roundoff) errors. In the symbolic setting, exact representation extends to some alge braic irrationals, to some transcendentals, and to various algebraic combinations of these. Therefore, the development of a dynamical system inside a computer is prone to successive errors, which is usually fatal to predictions concerning chaotic systems. In this article, we look at the question of computational predictability more closely and show that unpredictability in chaotic systems is also related to the scheme in which the system is represented. For simplicity, we restrict our discussions to discrete-time dynamical systems (i.e., to systems S defined by maps / : V -> V for metric spaces V). The orbit of S corresponding to the initial condition x0 is defined as the sequence y : N -+ V given by y(n) ? xn ? fn(x0), where N = {0, 1, 2, ...}. The time variable of S is n, and V is its state space. The space V is usually a subset of M.d for some natural number d, the dimension of the system. The orbits of S may be open, eventually periodic, or periodic. We write (eventually) periodic when referring to either of the last two categories. We call S periodic if all its orbits are (eventually) periodic.

Periodicity and Predictability in Chaotic Systems

Periodicity and Predictability in Chaotic Systems