Fitting ordinary differential equations to chaotic data

Deep-learning reconstruction of complex dynamical networks from incomplete data

Reconstructing complex networks and predicting the dynamics are particularly challenging in real-world applications because the available information and data are incomplete. We develop a unified collaborative deep-learning framework consisting of three modules: network inference, state estimation, and dynamical learning. The complete network structure is first inferred and the states of the unobserved nodes are estimated, based on which the dynamical learning module is activated to determine the dynamical evolution rules. An alternating parameter updating strategy is deployed to improve the inference and prediction accuracy. Our framework outperforms baseline methods for synthetic and empirical networks hosting a variety of dynamical processes. A reciprocity emerges between network inference and dynamical prediction: better inference of network structure improves the accuracy of dynamical prediction, and vice versa. We demonstrate the superior performance of our framework on an influenza dataset consisting of 37 US States and a PM2.5 dataset covering 184 cities in China.

Constructing low-dimensional ordinary differential equations from chaotic time series of high- or infinite-dimensional systems using radial-function-based regression

In our previous study [N. Tsutsumi, K. Nakai, and Y. Saiki, Chaos 32, 091101 (2022)1054-150010.1063/5.0100166] we proposed a method of constructing a system of ordinary differential equations of chaotic behavior only from observable deterministic time series, which we will call the radial-function-based regression (RfR) method. The RfR method employs a regression using Gaussian radial basis functions together with polynomial terms to facilitate the robust modeling of chaotic behavior. In this paper, we apply the RfR method to several example time series of high- or infinite-dimensional deterministic systems, and we construct a system of relatively low-dimensional ordinary differential equations with a large number of terms. The examples include time series generated from a partial differential equation, a delay differential equation, a turbulence model, and intermittent dynamics. The case when the observation includes noise is also tested. We have effectively constructed a system of differential equations for each of these examples, which is assessed from the point of view of time series forecast, reconstruction of invariant sets, and invariant densities. We find that in some of the models, an appropriate trajectory is realized on the chaotic saddle and is identified by the stagger-and-step method.

Learning dynamics on invariant measures using PDE-constrained optimization

We extend the methodology in Yang et al. [SIAM J. Appl. Dyn. Syst. 22, 269-310 (2023)] to learn autonomous continuous-time dynamical systems from invariant measures. The highlight of our approach is to reformulate the inverse problem of learning ODEs or SDEs from data as a PDE-constrained optimization problem. This shift in perspective allows us to learn from slowly sampled inference trajectories and perform uncertainty quantification for the forecasted dynamics. Our approach also yields a forward model with better stability than direct trajectory simulation in certain situations. We present numerical results for the Van der Pol oscillator and the Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, to demonstrate the effectiveness of the proposed approach.

Bayesian modeling of pattern formation from one snapshot of pattern

Partial differential equations (PDEs) have been widely used to reproduce patterns in nature and to give insight into the mechanism underlying pattern formation. Although many PDE models have been proposed, they rely on the pre-request knowledge of physical laws and symmetries, and developing a model to reproduce a given desired pattern remains difficult. We propose a method, referred to as Bayesian modeling of PDEs (BM-PDEs), to estimate the best dynamical PDE for one snapshot of a objective pattern under the stationary state without ground truth. We apply BM-PDEs to nontrivial patterns, such as quasicrystals (QCs), a double gyroid, and Frank-Kasper structures. We also generate three-dimensional dodecagonal QCs from a PDE model. This is done by using the estimated parameters for the Frank-Kasper A15 structure, which closely approximates the local structures of QCs. Our method works for noisy patterns and the pattern synthesized without the ground-truth parameters, which are required for the application toward experimental data.

Constructing differential equations using only a scalar time-series about continuous time chaotic dynamics

We propose a simple method of constructing a system of differential equations of chaotic behavior based on the regression only from scalar observable time-series data. The estimated system enables us to reconstruct invariant sets and statistical properties as well as to infer short time-series. Our successful modeling relies on the introduction of a set of Gaussian radial basis functions to capture local structures. The proposed method is used to construct a system of ordinary differential equations whose orbit reconstructs a time-series of a variable of the well-known Lorenz system as a simple but typical example. A system for a macroscopic fluid variable is also constructed.

Data-driven prediction in dynamical systems: recent developments

In recent years, we have witnessed a significant shift toward ever-more complex and ever-larger-scale systems in the majority of the grand societal challenges tackled in applied sciences. The need to comprehend and predict the dynamics of complex systems have spurred developments in large-scale simulations and a multitude of methods across several disciplines. The goals of understanding and prediction in complex dynamical systems, however, have been hindered by high dimensionality, complexity and chaotic behaviours. Recent advances in data-driven techniques and machine-learning approaches have revolutionized how we model and analyse complex systems. The integration of these techniques with dynamical systems theory opens up opportunities to tackle previously unattainable challenges in modelling and prediction of dynamical systems. While data-driven prediction methods have made great strides in recent years, it is still necessary to develop new techniques to improve their applicability to a wider range of complex systems in science and engineering. This focus issue shares recent developments in the field of complex dynamical systems with emphasis on data-driven, data-assisted and artificial intelligence-based discovery of dynamical systems. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Noise robust approach to reconstruction of van der Pol-like oscillators and its application to Granger causality

Van der Pol oscillators and their generalizations are known to be a fundamental model in the theory of oscillations and their applications. Many objects of a different nature can be described using van der Pol-like equations under some circumstances; therefore, methods of reconstruction of such equations from experimental data can be of significant importance for tasks of model verification, indirect parameter estimation, coupling analysis, system classification, etc. The previously reported techniques were not applicable to time series with large measurement noise, which is usual in biological, climatological, and many other experiments. Here, we present a new approach based on the use of numerical integration instead of the differentiation and implicit approximation of a nonlinear dissipation function. We show that this new technique can work for noise levels up to 30% by standard deviation from the signal for different types of autonomous van der Pol-like systems and for ensembles of such systems, providing a new approach to the realization of the Granger-causality idea.

Finding nonlinear system equations and complex network structures from data: A sparse optimization approach

In applications of nonlinear and complex dynamical systems, a common situation is that the system can be measured, but its structure and the detailed rules of dynamical evolution are unknown. The inverse problem is to determine the system equations and structure from time series. The principle of exploiting sparse optimization to find the equations of dynamical systems from data was first articulated in 2011 by the ASU group. The basic idea is to expand the system equations into a power series or a Fourier series of a finite number of terms and then to determine the vector of the expansion coefficients based solely on data through sparse optimization. This Tutorial presents a brief review of the recent progress in this area. Issues discussed include discovering the equations of stationary or nonstationary chaotic systems to enable the prediction of critical transition and system collapse, inferring the full topology of complex oscillator networks and social networks hosting evolutionary game dynamics, and identifying partial differential equations for spatiotemporal dynamical systems. Situations where sparse optimization works or fails are pointed out. The relation with the traditional delay-coordinate embedding method is discussed, and the recent development of a model-free, data-driven prediction framework based on machine learning is mentioned.

Nonlinear reconstruction of bioclimatic outdoor-environment dynamics for the Lower Silesia region (SW Poland)

Measured meteorological time series are frequently used to obtain information about climate dynamics. We use time series analysis and nonlinear system identification methods in order to assess outdoor-environment bioclimatic conditions starting from the analysis of long historical meteorological data records. We investigate and model the stochastic and deterministic properties of 117 years (1891-2007) of monthly measurements of air temperature, precipitation and sunshine duration by separating their slow and fast components of the dynamics. In particular, we reconstruct the trend behaviour at long terms by modelling its dynamics via a phase space dynamical systems approach. The long-term reconstruction method reveals that an underlying dynamical system would drive the trend behaviour of the meteorological variables and in turn of the calculated Universal Thermal Climatic Index (UTCI), as representative of bioclimatic conditions. At longer terms, the system would slowly be attracted to a limit cycle characterized by 50-60 years cycle fluctuations that is reminiscent of the Atlantic Multidecadal Oscillation (AMO). Because of lack of information about long historical wind speed data we performed a sensitivity analysis of the UTCI to three constant wind speed scenarios (i.e. 0.5, 1 and 5 m/s). This methodology may be transferred to model bioclimatic conditions of nearby regions lacking of measured data but experiencing similar climatic conditions.

Reconstruction of parameters and unobserved variables of a semiconductor laser with optical feedback from intensity time series

We propose a method for the reconstruction of time-delayed feedback systems having unobserved variables from scalar time series. The method is based on the modified initial condition approach, which allows one to significantly reduce the number of starting guesses for an unobserved variable with a time delay. The proposed method is applied to the reconstruction of the Lang-Kobayashi equations, which describe the dynamics of a single-mode semiconductor laser with external optical feedback. We consider the case where only the time series of laser intensity is observable and the other two variables of the model are hidden. The dependence of the quality of the system reconstruction on the accuracy of assignment of starting guesses for unobserved variables and unknown laser parameters is studied. The method could be used for testing the security of information transmission in laser-based chaotic communication systems.

A practical method for estimating coupling functions in complex dynamical systems

A foremost challenge in modern network science is the inverse problem of reconstruction (inference) of coupling equations and network topology from the measurements of the network dynamics. Of particular interest are the methods that can operate on real (empirical) data without interfering with the system. One such earlier attempt (Tokuda et al. 2007 Phys. Rev. Lett. 99, 064101. (doi:10.1103/PhysRevLett.99.064101)) was a method suited for general limit-cycle oscillators, yielding both oscillators' natural frequencies and coupling functions between them (phase equations) from empirically measured time series. The present paper reviews the above method in a way comprehensive to domain-scientists other than physics. It also presents applications of the method to (i) detection of the network connectivity, (ii) inference of the phase sensitivity function, (iii) approximation of the interaction among phase-coherent chaotic oscillators, and (iv) experimental data from a forced Van der Pol electric circuit. This reaffirms the range of applicability of the method for reconstructing coupling functions and makes it accessible to a much wider scientific community. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Bayesian inference for parameter estimation in lactoferrin-mediated iron transport across blood-brain barrier

BACKGROUND

In neurodegenerative diseases such as Alzheimer's and Parkinson's, excessive irons as well as lactoferrin (Lf), but not transferrin (Tf), have been found in and around the affected regions of the brain. These evidences suggest that lactoferrin plays a critical role during neurodegenerative diseases, although Lf-mediated iron transport across blood-brain barrier (BBB) is negligible compared to that of transferrin in normal condition. However, the kinetics of lactoferrins and lactoferrin-mediated iron transport are still unknown.

METHOD

To determine the kinetic rate constants of lactoferrin-mediated iron transport through BBB, a mass-action based ordinary differential equation model has been presented. A Bayesian framework is developed to estimate the kinetic rate parameters from posterior probability density functions. The iron transport across BBB is studied by considering both Lf- and Tf-mediated pathways for both normal and pathologic conditions.

RESULTS

Using the point estimates of kinetic parameters, our model can effectively reproduce the experimental data of iron transport through BBB endothelial cells. The robustness of the model and parameter estimation process are further verified by perturbation of kinetic parameters. Our results show that surge in high-affinity receptor density increases lactoferrin as well as iron in the brain.

CONCLUSIONS

Due to the lack of a feedback loop such as iron regulatory proteins (IRPs) for lactoferrin, iron can transport to the brain continuously, which might increase brain iron to pathological levels and can contribute to neurodegeneration.

GENERAL SIGNIFICANCE

This study provides an improved understanding of presence of lactoferrin and iron in the brain during neurodegenerative diseases.

Unwinding the model manifold: Choosing similarity measures to remove local minima in sloppy dynamical systems

In this paper, we consider the problem of parameter sensitivity in models of complex dynamical systems through the lens of information geometry. We calculate the sensitivity of model behavior to variations in parameters. In most cases, models are sloppy, that is, exhibit an exponential hierarchy of parameter sensitivities. We propose a parameter classification scheme based on how the sensitivities scale at long observation times. We show that for oscillatory models, either with a limit cycle or a strange attractor, sensitivities can become arbitrarily large, which implies a high effective dimensionality on the model manifold. Sloppy models with a single fixed point have model manifolds with low effective dimensionality, previously described as a "hyper-ribbon." In contrast, models with high effective dimensionality translate into multimodal fitting problems. We define a measure of curvature on the model manifold which we call the winding frequency that estimates the density of local minima in the model's parameter space. We then show how alternative choices of fitting metrics can "unwind" the model manifold and give low winding frequencies. This prescription translates the model manifold from one of high effective dimensionality into the hyper-ribbon structures observed elsewhere. This translation opens the door for applications of sloppy model analysis and model reduction methods developed for models with low effective dimensionality.

PARAMETER AND UNCERTAINTY ESTIMATION FOR DYNAMICAL SYSTEMS USING SURROGATE STOCHASTIC PROCESSES

Inference on unknown quantities in dynamical systems via observational data is essential for providing meaningful insight, furnishing accurate predictions, enabling robust control, and establishing appropriate designs for future experiments. Merging mathematical theory with empirical measurements in a statistically coherent way is critical and challenges abound, e.g., ill-posedness of the parameter estimation problem, proper regularization and incorporation of prior knowledge, and computational limitations. To address these issues, we propose a new method for learning parameterized dynamical systems from data. We first customize and fit a surrogate stochastic process directly to observational data, front-loading with statistical learning to respect prior knowledge (e.g., smoothness), cope with challenging data features like heteroskedasticity, heavy tails, and censoring. Then, samples of the stochastic process are used as "surrogate data" and point estimates are computed via ordinary point estimation methods in a modular fashion. Attractive features of this two-step approach include modularity and trivial parallelizability. We demonstrate its advantages on a predator-prey simulation study and on a real-world application involving within-host influenza virus infection data paired with a viral kinetic model, with comparisons to a more conventional Markov chain Monte Carlo (MCMC) based Bayesian approach.

Identification of microbiota dynamics using robust parameter estimation methods

The compositions of in-host microbial communities (microbiota) play a significant role in host health, and a better understanding of the microbiota's role in a host's transition from health to disease or vice versa could lead to novel medical treatments. One of the first steps toward this understanding is modeling interaction dynamics of the microbiota, which can be exceedingly challenging given the complexity of the dynamics and difficulties in collecting sufficient data. Methods such as principal differential analysis, dynamic flux estimation, and others have been developed to overcome these challenges. Despite their advantages, these methods are still vastly underutilized in fields such as mathematical biology, and one potential reason for this is their sophisticated implementation. While this paper focuses on applying principal differential analysis to microbiota data, we also provide comprehensive details regarding the derivation and numerics of this method and include a functional implementation for readers' benefit. For further validation of these methods, we demonstrate the feasibility of principal differential analysis using simulation studies and then apply the method to intestinal and vaginal microbiota data. In working with these data, we capture experimentally confirmed dynamics while also revealing potential new insights into the system dynamics.

Hybrid modeling and prediction of dynamical systems

Scientific analysis often relies on the ability to make accurate predictions of a system's dynamics. Mechanistic models, parameterized by a number of unknown parameters, are often used for this purpose. Accurate estimation of the model state and parameters prior to prediction is necessary, but may be complicated by issues such as noisy data and uncertainty in parameters and initial conditions. At the other end of the spectrum exist nonparametric methods, which rely solely on data to build their predictions. While these nonparametric methods do not require a model of the system, their performance is strongly influenced by the amount and noisiness of the data. In this article, we consider a hybrid approach to modeling and prediction which merges recent advancements in nonparametric analysis with standard parametric methods. The general idea is to replace a subset of a mechanistic model's equations with their corresponding nonparametric representations, resulting in a hybrid modeling and prediction scheme. Overall, we find that this hybrid approach allows for more robust parameter estimation and improved short-term prediction in situations where there is a large uncertainty in model parameters. We demonstrate these advantages in the classical Lorenz-63 chaotic system and in networks of Hindmarsh-Rose neurons before application to experimentally collected structured population data.

Inverse problem studies of biochemical systems with structure identification of S-systems by embedding training functions in a genetic algorithm

An efficient inverse problem approach for parameter estimation, state and structure identification from dynamic data by embedding training functions in a genetic algorithm methodology (ETFGA) is proposed for nonlinear dynamical biosystems using S-system canonical models. Use of multiple shooting and decomposition approach as training functions has been shown for handling of noisy datasets and computational efficiency in studying the inverse problem. The advantages of the methodology are brought out systematically by studying it for three biochemical model systems of interest. By studying a small-scale gene regulatory system described by a S-system model, the first example demonstrates the use of ETFGA for the multifold aims of the inverse problem. The estimation of a large number of parameters with simultaneous state and network identification is shown by training a generalized S-system canonical model with noisy datasets. The results of this study bring out the superior performance of ETFGA on comparison with other metaheuristic approaches. The second example studies the regulation of cAMP oscillations in Dictyostelium cells now assuming limited availability of noisy data. Here, flexibility of the approach to incorporate partial system information in the identification process is shown and its effect on accuracy and predictive ability of the estimated model are studied. The third example studies the phenomenological toy model of the regulation of circadian oscillations in Drosophila that follows rate laws different from S-system power-law. For the limited noisy data, using a priori information about properties of the system, we could estimate an alternate S-system model that showed robust oscillatory behavior with predictive abilities.

Deterministic inference for stochastic systems using multiple shooting and a linear noise approximation for the transition probabilities

Estimating model parameters from experimental data is a crucial technique for working with computational models in systems biology. Since stochastic models are increasingly important, parameter estimation methods for stochastic modelling are also of increasing interest. This study presents an extension to the ‘multiple shooting for stochastic systems (MSS)’ method for parameter estimation. The transition probabilities of the likelihood function are approximated with normal distributions. Means and variances are calculated with a linear noise approximation on the interval between succeeding measurements. The fact that the system is only approximated on intervals which are short in comparison with the total observation horizon allows to deal with effects of the intrinsic stochasticity. The study presents scenarios in which the extension is essential for successfully estimating the parameters and scenarios in which the extension is of modest benefit. Furthermore, it compares the estimation results with reversible jump techniques showing that the approximation does not lead to a loss of accuracy. Since the method is not based on stochastic simulations or approximative sampling of distributions, its computational speed is comparable with conventional least‐squares parameter estimation methods.

Forecasting the future: is it possible for adiabatically time-varying nonlinear dynamical systems?

Nonlinear dynamical systems in reality are often under environmental influences that are time-dependent. To assess whether such a system can perform as desired or as designed and is sustainable requires forecasting its future states and attractors based solely on time series. We propose a viable solution to this challenging problem by resorting to the compressive-sensing paradigm. In particular, we demonstrate that, for a dynamical system whose equations are unknown, a series expansion in both dynamical and time variables allows the forecasting problem to be formulated and solved in the framework of compressive sensing using only a few measurements. We expect our method to be useful in addressing issues of significant current concern such as the sustainability of various natural and man-made systems.

Single-shooting homotopy method for parameter identification in dynamical systems

An algorithm for identifying parameters in dynamical systems is developed in this work using homotopy transformations and the single-shooting method. The equations governing the dynamics of the mathematical model are augmented with observer-like homotopy terms that smooth the objective function. As a result, premature convergence to a local minimum is avoided and the obtained parameter estimates are globally optimal. Numerical examples are presented to demonstrate the application of the proposed approach to chaotic systems.

Global modeling of complex data series using the term-ranking approach and its application to voice synthesis

A term-ranking approach is proposed to globally model the underlying dynamics of a chaotic series. The basic idea of this approach is to rank candidate bases before they are used to construct the global model. The ranked bases are involved in the global model one by one in a sequence from high to low until the best model is found. Simulations show that the model obtained by the term-ranking approach has a much longer prediction time, but fewer coefficients, than the widely used standard model. The proposed approach is also successfully applied to coding and synthesis of chaoslike voice data, showing promise for its use with truly noisy experimental data.

Optimized synchronization of chaotic and hyperchaotic systems

A method of synchronization is presented which, unlike existing methods, can, for generic dynamical systems, force all conditional Lyapunov exponents to go to -∞ . It also has improved noise immunity compared to existing methods, and unlike most of them it can synchronize hyperchaotic systems with almost any single coupling variable from the drive system. Results are presented for the Rossler hyperchaos system and the Lorenz system.

Structure identification based on steady-state control: experimental results and applications

We report experimental results on structure identification of nonlinear systems by a steady-state control method. The idea underlying the method is to drive the nonlinear system to steady state by applying a suitable feedback control input. It turns out experimentally that this control-based structure identification method can be used for some applications, such as estimation of initial conditions and state variables of nonlinear systems and structure identification of some special elements. Two attractors of the Chua oscillator are presented to illustrate the reliability of the suggested techniques under the hypotheses of measurable state variables and physical access to the system for implementing the proportional feedback.

Spatiotemporal system identification on nonperiodic domains using Chebyshev spectral operators and system reduction algorithms

A system identification methodology based on Chebyshev spectral operators and an orthogonal system reduction algorithm is proposed, leading to a new approach for data-driven modeling of nonlinear spatiotemporal systems on nonperiodic domains. A continuous model structure is devised allowing for terms of arbitrary derivative order and nonlinearity degree. Chebyshev spectral operators are introduced to realm of inverse problems to discretize that continuous structure and arrive with spectral accuracy at a discrete form. Finally, least squares combined with an orthogonal system reduction algorithm are employed to solve for the parameters and eliminate the redundancies to achieve a parsimonious model. A numerical case study of identifying the Allen-Cahn metastable equation demonstrates the superior accuracy of the proposed Chebyshev spectral identification over its finite difference counterpart.

Monte Carlo method for adaptively estimating the unknown parameters and the dynamic state of chaotic systems

We propose a Monte Carlo methodology for the joint estimation of unobserved dynamic variables and unknown static parameters in chaotic systems. The technique is sequential, i.e., it updates the variable and parameter estimates recursively as new observations become available, and, hence, suitable for online implementation. We demonstrate the validity of the method by way of two examples. In the first one, we tackle the estimation of all the dynamic variables and one unknown parameter of a five-dimensional nonlinear model using a time series of scalar observations experimentally collected from a chaotic CO2 laser. In the second example, we address the estimation of the two dynamic variables and the phase parameter of a numerical model commonly employed to represent the dynamics of optoelectronic feedback loops designed for chaotic communications over fiber-optic links.

Spatiotemporal system reconstruction using Fourier spectral operators and structure selection techniques

A technique based on trigonometric spectral methods and structure selection is proposed for the reconstruction, from observed time series, of spatiotemporal systems governed by nonlinear partial differential equations of polynomial type with terms of arbitrary derivative order and nonlinearity degree. The system identification using Fourier spectral differentiation operators in conjunction with a structure selection procedure leads to a parsimonious model of the original system by detecting and eliminating the redundant parameters using orthogonal decomposition of the state data. Implementation of the technique is exemplified for a highly stiff reaction-diffusion system governed by the Kuramoto-Sivashinsky equation. Numerical experiments demonstrate the superior performance of the proposed technique in terms of accuracy as well as robustness, even with smaller sets of sampling data.

Detecting anomalous phase synchronization from time series

Modeling approaches are presented for detecting an anomalous route to phase synchronization from time series of two interacting nonlinear oscillators. The anomalous transition is characterized by an enlargement of the mean frequency difference between the oscillators with an initial increase in the coupling strength. Although such a structure is common in a large class of coupled nonisochronous oscillators, prediction of the anomalous transition is nontrivial for experimental systems, whose dynamical properties are unknown. Two approaches are examined; one is a phase equational modeling of coupled limit cycle oscillators and the other is a nonlinear predictive modeling of coupled chaotic oscillators. Application to prototypical models such as two interacting predator-prey systems in both limit cycle and chaotic regimes demonstrates the capability of detecting the anomalous structure from only a few sets of time series. Experimental data from two coupled Chua circuits shows its applicability to real experimental system.

Monte Carlo method for multiparameter estimation in coupled chaotic systems

We address the problem of estimating multiple parameters of a chaotic dynamical model from the observation of a scalar time series. We assume that the series is produced by a chaotic system with the same functional form as the model, so that synchronization between the two systems can be achieved by an adequate coupling. In this scenario, we propose an efficient Monte Carlo optimization algorithm that iteratively updates the model parameters in order to minimize the synchronization error. As an example, we apply it to jointly estimate the three static parameters of a chaotic Lorenz system.

Building dynamical models from data and prior knowledge: the case of the first period-doubling bifurcation

This paper reviews some aspects of nonlinear model building from data with (gray box) and without (black box) prior knowledge. The model class is very important because it determines two aspects of the final model, namely (i) the type of nonlinearity that can be accurately approximated and (ii) the type of prior knowledge that can be taken into account. Such features are usually in conflict when it comes to choosing the model class. The problem of model structure selection is also reviewed. It is argued that such a problem is philosophically different depending on the model class and it is suggested that the choice of model class should be performed based on the type of a priori available. A procedure is proposed to build polynomial models from data on a Poincaré section and prior knowledge about the first period-doubling bifurcation, for which the normal form is also polynomial. The final models approximate dynamical data in a least-squares sense and, by design, present the first period-doubling bifurcation at a specified value of parameters. The procedure is illustrated by means of simulated examples.

Inferring phase equations from multivariate time series

An approach is presented for extracting phase equations from multivariate time series data recorded from a network of weakly coupled limit cycle oscillators. Our aim is to estimate important properties of the phase equations including natural frequencies and interaction functions between the oscillators. Our approach requires the measurement of an experimental observable of the oscillators; in contrast with previous methods it does not require measurements in isolated single or two-oscillator setups. This noninvasive technique can be advantageous in biological systems, where extraction of few oscillators may be a difficult task. The method is most efficient when data are taken from the nonsynchronized regime. Applicability to experimental systems is demonstrated by using a network of electrochemical oscillators; the obtained phase model is utilized to predict the synchronization diagram of the system.

On parameter estimation of chaotic systems via symbolic time-series analysis

Symbolic time-series analysis is used for estimating the parameters of chaotic systems. It is assumed that a "target model" (i.e., a discrete- or continuous-time description of the data-generating mechanism) is available, but with unknown parameters. A time series, i.e., a noisy, finite sequence of a measured (output) variable, is given. The proposed method first prescribes to symbolize the time series, i.e., to transform it into a sequence of symbols, from which the statistics of symbols are readily derived. Then, a symbolic model (in the form of a Markov chain) is derived from the data. It allows one to predict, in a probabilistic fashion, the time evolution of the symbol sequence. The unknown parameters are derived by matching either the statistics of symbols, or the symbolic prediction derived from data, with those generated by the (parametrized) target model. Three examples of application (the Henon map, a population model, and the Duffing system) prove that satisfactory results can be obtained even with short time series.

Modeling global vector fields of chaotic systems from noisy time series with the aid of structure-selection techniques

We address the problem of reconstructing a set of nonlinear differential equations from chaotic time series. A method that combines the implicit Adams integration and the structure-selection technique of an error reduction ratio is proposed for system identification and corresponding parameter estimation of the model. The structure-selection technique identifies the significant terms from a pool of candidates of functional basis and determines the optimal model through orthogonal characteristics on data. The technique with the Adams integration algorithm makes the reconstruction available to data sampled with large time intervals. Numerical experiment on Lorenz and Rossler systems shows that the proposed strategy is effective in global vector field reconstruction from noisy time series.

Adaptive approximation method for joint parameter estimation and identical synchronization of chaotic systems

We introduce a numerical approximation method for estimating an unknown parameter of a (primary) chaotic system which is partially observed through a scalar time series. Specifically, we show that the recursive minimization of a suitably designed cost function that involves the dynamic state of a fully observed (secondary) system and the observed time series can lead to the identical synchronization of the two systems and the accurate estimation of the unknown parameter. The salient feature of the proposed technique is that the only external input to the secondary system is the unknown parameter which needs to be adjusted. We present numerical examples for the Lorenz system which show how our algorithm can be considerably faster than some previously proposed methods.

Manipulation of surface reaction dynamics by global pressure and local temperature control: a model study

Specific catalyst design and external manipulation of surface reactions by controlling accessible physical or chemical parameters may be of great benefit for improving catalytic efficiencies and energetics, product yield, and selectivities in the field of heterogeneous catalysis. Studying a realistic spatiotemporal one-dimensional model for CO oxidation on Pt(110) we demonstrate the value and necessity of mathematical modeling and advanced numerical methods for directed external multiparameter control of surface reaction dynamics. At the model stage we show by means of optimal control techniques that species coverages can be adjusted to desired values, aperiodic oscillatory behavior for distinct coupled reaction sites can be synchronized, and overall reaction rates can be optimized by varying the surface temperature in space and time and the CO and O2 gas phase partial pressure with time. The control aims are formulated as objective functionals to be minimized which contain a suitable mathematical formulation for the deviation from the desired system behavior. The control functions pCO(t) (CO partial pressure), pO2(t) (O2 partial pressure), and T(x,t) (surface temperature distribution) are numerically computed by a specially tailored optimal control method based on a direct multiple shooting approach which is suitable to cope with the highly nonlinear unstable mode character of the CO oxidation model.

Choice of dynamical variables for global reconstruction of model equations from time series

The success of modeling from an experimental time series is determined to a significant extent by the choice of dynamical variables. We propose a method for preliminary investigation of a time series whose purpose is to find out whether a global dynamical model with smooth functions can be constructed for the chosen variables. The method consists in the estimation of single valuedness and continuity of relations between dynamical variables and variables to enter left-hand sides of model equations. The method is explained with numerical examples. Its efficiency is demonstrated by modeling a real nonlinear electric circuit.

Parameter estimation in spatially extended systems: the Karhunen-Lóeve and Galerkin multiple shooting approach

Parameter estimation for spatiotemporal dynamics for coupled map lattices and continuous time domain systems is shown using a combination of multiple shooting, Karhunen-Loéve decomposition and Galerkin's projection methodologies. The resulting advantages in estimating parameters have been studied and discussed for chaotic and turbulent dynamics using small amounts of data from subsystems, availability of only scalar and noisy time series data, effects of space-time parameter variations, and in the presence of multiple time scales.

Role of transient processes for reconstruction of model equations from time series

We perform a global reconstruction of differential and difference equations, which model an object in a wide domain of a phase space, from a time series. The efficiency of using time realizations of transient processes for this purpose is demonstrated. Time series of transients are shown to have some advantages for the realization of a procedure of model structure optimization.

Nonlinear dynamics and chaos theory: concepts and applications relevant to pharmacodynamics

The theory of nonlinear dynamical systems (chaos theory), which deals with deterministic systems that exhibit a complicated, apparently random-looking behavior, has formed an interdisciplinary area of research and has affected almost every field of science in the last 20 years. Life sciences are one of the most applicable areas for the ideas of chaos because of the complexity of biological systems. It is widely appreciated that chaotic behavior dominates physiological systems. This is suggested by experimental studies and has also been encouraged by very successful modeling. Pharmacodynamics are very tightly associated with complex physiological processes, and the implications of this relation demand that the new approach of nonlinear dynamics should be adopted in greater extent in pharmacodynamic studies. This is necessary not only for the sake of more detailed study, but mainly because nonlinear dynamics suggest a whole new rationale, fundamentally different from the classic approach. In this work the basic principles of dynamical systems are presented and applications of nonlinear dynamics in topics relevant to drug research and especially to pharmacodynamics are reviewed. Special attention is focused on three major fields of physiological systems with great importance in pharmacotherapy, namely cardiovascular, central nervous, and endocrine systems, where tools and concepts from nonlinear dynamics have been applied.

Constructing nonautonomous differential equations from experimental time series

An approach to constructing model differential equations of harmonically driven systems is proposed. It is a modification of the standard global reconstruction technique: an algebraic polynomial which coefficients depend on time is used for approximation. Efficiency and details of the approach are demonstrated by various numerical and natural examples.

Synchronization and control of spatiotemporal chaos using time-series data from local regions

In this paper we show that the analysis of the dynamics in localized regions, i.e., sub-systems can be used to characterize the chaotic dynamics and the synchronization ability of the spatiotemporal systems. Using noisy scalar time-series data for driving along with simultaneous self-adaptation of the control parameter representative control goals like suppressing spatiotemporal chaos and synchronization of spatiotemporally chaotic dynamics have been discussed. (c) 1998 American Institute of Physics.