Reduction of dimension for nonlinear dynamical systems

Harrington, Heather A.; Gorder, Robert A. Van

doi:10.1007/s11071-016-3272-5

Cited by 13 publications

(7 citation statements)

References 82 publications

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“…In [ 43 ], Harrington and van Gorder present a method to algebraically convert systems of coupled differential equations from one form to another. As an example from that paper, consider the Lorenz system of three ODEs:

Through algebraic substitutions, this can be converted to a single third-order nonlinear differential equation in only the variable x and its derivatives.…”

Section: Appendix A1 Algebraic Methodsmentioning

confidence: 99%

Data-Driven Corrections of Partial Lotka–Volterra Models

Morrison

2020

Entropy

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In many applications of interacting systems, we are only interested in the dynamic behavior of a subset of all possible active species. For example, this is true in combustion models (many transient chemical species are not of interest in a given reaction) and in epidemiological models (only certain subpopulations are consequential). Thus, it is common to use greatly reduced or partial models in which only the interactions among the species of interest are known. In this work, we explore the use of an embedded, sparse, and data-driven discrepancy operator to augment these partial interaction models. Preliminary results show that the model error caused by severe reductions—e.g., elimination of hundreds of terms—can be captured with sparse operators, built with only a small fraction of that number. The operator is embedded within the differential equations of the model, which allows the action of the operator to be interpretable. Moreover, it is constrained by available physical information and calibrated over many scenarios. These qualities of the discrepancy model—interpretability, physical consistency, and robustness to different scenarios—are intended to support reliable predictions under extrapolative conditions.

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Through algebraic substitutions, this can be converted to a single third-order nonlinear differential equation in only the variable x and its derivatives.…”

Section: Appendix A1 Algebraic Methodsmentioning

confidence: 99%

Data-Driven Corrections of Partial Lotka–Volterra Models

Morrison

2020

Entropy

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show abstract

“…The Rössler system consists of a set of ordinary differential equations defined as follows:

\begin{matrix} lefttrue \\ \dot{x} = - y - z \\ \dot{y} = x + ay \\ \dot{z} = b + z (x - c) \end{matrix}

where a , b , and c are parameters that specify the chaotic system. We investigate the system parameters with parameters value a = 0.2, b = 0.2, and c = 5.7, which is known to produce a deterministic chaotic time series (Strogatz, ; Harrington & Van Gorder, ). The time range is fixed from 0 to 50, and a total number of N = 100,000 paired observations ( x t , y t , z t ) was generated from an initial condition of (−2, −10, 0.2) with low noise level ( s = 0.1).…”

Section: Application To a Dynamic Examplementioning

confidence: 99%

Refining Predictor Spectral Representation Using Wavelet Theory for Improved Natural System Modeling

Jiang

Sharma

2020

Water Resources Research

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Predicting future surpluses or shortages of water is a long‐standing problem having considerable ramifications to water management across the world. Any prediction model for a natural system such as one that estimates water surpluses or shortages requires a two‐step approach. These are the following: first, identify and select meaningful predictor variables from a large number of potential predictors and second, formulate an accurate, efficient, and robust predictive model between selected predictors and the response. Recognizing that the timescales at which a response may operate is usually different from that of the predictors being identified, we introduce here a wavelet‐based unique variance transformation to each of the multiple predictor variables in the system which ensures an improved regression relationship to the modeled response. All existing methods assume no change in predictors even if they characterize variability at markedly different timescales, a deficiency that is addressed using the variance‐transformed predictor which can explain maximal information in an associated response. Using this unique variance transformation, additional predictor variables can be selected by assessing their ability to characterize residual information in the response that accounts for the effect of preidentified predictors. We demonstrate the utility of the wavelet‐based method using synthetically generated data sets from known linear and nonlinear systems with parametric and nonparametric predictive models. Applications to a dynamic system and a real‐world example to downscale a drought indicator over the Sydney region confirm its utility in an applied setting.

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“…Algebraic method. In [9], Harrington and van Gorder present a method to algebraically convert systems of coupled differential equations from one form to another. As an example from that paper, consider the Lorenz system of three ODEs:…”

Section: Model Conversionmentioning

confidence: 99%

Embedded discrepancy operators in reduced models of interacting species

Morrison¹

2019

Preprint

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In many applications of interacting systems, we are only interested in the dynamic behavior of a subset of all possible active species. For example, this is true in combustion models (many transient chemical species are not of interest in a given reaction) and in epidemiological models (only certain critical populations are truly consequential). Thus it is common to use greatly reduced models, in which only the interactions among the species of interest are retained. However, reduction introduces a model error, or discrepancy, which typically is not well characterized. In this work, we explore the use of an embedded and statistically calibrated discrepancy operator to represent model error. The operator is embedded within the differential equations of the model, which allows the action of the operator to be interpretable. Moreover, it is constrained by available physical information, and calibrated over many scenarios. These qualities of the discrepancy model-interpretability, physicalconsistency, and robustness to different scenarios-are intended to support reliable predictions under extrapolative conditions.

show abstract

Reduction of dimension for nonlinear dynamical systems

Cited by 13 publications

References 82 publications

Data-Driven Corrections of Partial Lotka–Volterra Models

Data-Driven Corrections of Partial Lotka–Volterra Models

Refining Predictor Spectral Representation Using Wavelet Theory for Improved Natural System Modeling

Embedded discrepancy operators in reduced models of interacting species

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