Developing a model to capture all aspects of a complex dynamical system is an immense task, and each model will have deficiencies in some areas, such as global climate models having difficulty in capturing tropical intraseasonal variability such as the Madden–Julian oscillation. Besides complex models, it is possible to create simplified, low-dimensional models to capture specific phenomena while ignoring many aspects of the full system. Here, we propose a strategy to allow complex models to communicate with simplified models throughout a simulation. The communication allows one to leverage the strengths of each model, without needing to change their dynamics, to mitigate model error. Furthermore, to ensure ease of implementation in complex systems, the strategy is based on common data assimilation techniques that are normally used to combine models and real-world data. This strategy is investigated here in a test case that is nonlinear, non-Gaussian, and high-dimensional (approximately degrees of freedom), and the multiple models have different state spaces. In particular, it is an idealized tropical climate model in three spatial dimensions. The multimodel communication strategy is seen to mitigate model error and reproduce statistical features akin to those of the truth model when the communication is sufficiently frequent. In these tests, the low-dimensional model contributes only two degrees of freedom, which suggests that, in some systems, large amounts of model error can possibly be reduced by focusing on a small set of model components.
{"title":"Mitigating Model Error via a Multimodel Method and Application to Tropical Intraseasonal Oscillations","authors":"Jason L. Torchinsky, Samuel Stechmann","doi":"10.1137/22m152551x","DOIUrl":"https://doi.org/10.1137/22m152551x","url":null,"abstract":"Developing a model to capture all aspects of a complex dynamical system is an immense task, and each model will have deficiencies in some areas, such as global climate models having difficulty in capturing tropical intraseasonal variability such as the Madden–Julian oscillation. Besides complex models, it is possible to create simplified, low-dimensional models to capture specific phenomena while ignoring many aspects of the full system. Here, we propose a strategy to allow complex models to communicate with simplified models throughout a simulation. The communication allows one to leverage the strengths of each model, without needing to change their dynamics, to mitigate model error. Furthermore, to ensure ease of implementation in complex systems, the strategy is based on common data assimilation techniques that are normally used to combine models and real-world data. This strategy is investigated here in a test case that is nonlinear, non-Gaussian, and high-dimensional (approximately degrees of freedom), and the multiple models have different state spaces. In particular, it is an idealized tropical climate model in three spatial dimensions. The multimodel communication strategy is seen to mitigate model error and reproduce statistical features akin to those of the truth model when the communication is sufficiently frequent. In these tests, the low-dimensional model contributes only two degrees of freedom, which suggests that, in some systems, large amounts of model error can possibly be reduced by focusing on a small set of model components.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cris R. Hasan, Ruaidhrí Mac Cárthaigh, Sebastian Wieczorek
{"title":"Rate-Induced Tipping in Heterogeneous Reaction-Diffusion Systems: An Invariant Manifold Framework and Geographically Shifting Ecosystems","authors":"Cris R. Hasan, Ruaidhrí Mac Cárthaigh, Sebastian Wieczorek","doi":"10.1137/22m1536625","DOIUrl":"https://doi.org/10.1137/22m1536625","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136068099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Joshua W. Moore, Trevor C. Dale, Thomas E. Woolley
Recent advances in high-resolution experimental methods have highlighted the significance of cell signal pathway crosstalk and localized signaling activity in the development and disease of numerous biological systems. The investigation of multiple signal pathways often introduces different methods of cell-cell communication, i.e., contact-based or diffusive signaling, which generates both a spatial and a temporal dependence on cell behaviors. Motivated by cellular mechanisms that control cell-fate decisions in developing bilayer tissues, we use dynamical systems coupled with multilayer graphs to analyze the role of signaling polarity and pathway crosstalk in fine-grain pattern formation of protein activity. Specifically, we study how multilayer graph edge structures and weights influence the layerwise (laminar) patterning of cells in bilayer structures, which are commonly found in glandular tissues. We present sufficient conditions for the existence, uniqueness, and instability of homogeneous cell states in the large-scale spatially discrete dynamical system. Using methods of pattern templating by graph partitioning to generate quotient systems, in combination with concepts from monotone dynamical systems, we exploit the extensive dimensionality reduction to provide existence conditions for the polarity required to induce fine-grain laminar patterns with multiple spatially dependent intracellular components. We then explore the spectral links between the quotient and large-scale dynamical systems to extend the laminar patterning criteria from existence to convergence for sufficiently large amounts of cellular polarity in the large-scale dynamical system, independent of spatial dimension and number of cells in the tissue.
{"title":"Modeling Polarity-Driven Laminar Patterns in Bilayer Tissues with Mixed Signaling Mechanisms","authors":"Joshua W. Moore, Trevor C. Dale, Thomas E. Woolley","doi":"10.1137/22m1522565","DOIUrl":"https://doi.org/10.1137/22m1522565","url":null,"abstract":"Recent advances in high-resolution experimental methods have highlighted the significance of cell signal pathway crosstalk and localized signaling activity in the development and disease of numerous biological systems. The investigation of multiple signal pathways often introduces different methods of cell-cell communication, i.e., contact-based or diffusive signaling, which generates both a spatial and a temporal dependence on cell behaviors. Motivated by cellular mechanisms that control cell-fate decisions in developing bilayer tissues, we use dynamical systems coupled with multilayer graphs to analyze the role of signaling polarity and pathway crosstalk in fine-grain pattern formation of protein activity. Specifically, we study how multilayer graph edge structures and weights influence the layerwise (laminar) patterning of cells in bilayer structures, which are commonly found in glandular tissues. We present sufficient conditions for the existence, uniqueness, and instability of homogeneous cell states in the large-scale spatially discrete dynamical system. Using methods of pattern templating by graph partitioning to generate quotient systems, in combination with concepts from monotone dynamical systems, we exploit the extensive dimensionality reduction to provide existence conditions for the polarity required to induce fine-grain laminar patterns with multiple spatially dependent intracellular components. We then explore the spectral links between the quotient and large-scale dynamical systems to extend the laminar patterning criteria from existence to convergence for sufficiently large amounts of cellular polarity in the large-scale dynamical system, independent of spatial dimension and number of cells in the tissue.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136115189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aaron Kirtland, Jonah Botvinick-Greenhouse, Marianne DeBrito, Megan Osborne, Casey Johnson, Robert S. Martin, Samuel J. Araki, Daniel Q. Eckhardt
To address noise inherent in electronic data acquisition systems and real-world sources, Araki et al. [Phys. D, 417 (2021), 132819] demonstrated a grid-based nonlinear technique to remove noise from a chaotic signal, leveraging a clean high-fidelity signal from the same dynamical system and ensemble averaging in multidimensional phase space. This method achieved denoising of a time series data with 100% added noise but suffered in regions of low data density. To improve this grid-based method, here an unstructured mesh based on triangulations and Voronoi diagrams is used to accomplish the same task. The unstructured mesh more uniformly distributes data samples over mesh cells to improve the accuracy of the reconstructed signal. By empirically balancing bias and variance errors in selecting the number of unstructured cells as a function of the number of available samples, the method achieves asymptotic statistical convergence with known test data and reduces synthetic noise on experimental signals from Hall effect thrusters with greater success than the original grid-based strategy.
{"title":"An Unstructured Mesh Approach to Nonlinear Noise Reduction for Coupled Systems","authors":"Aaron Kirtland, Jonah Botvinick-Greenhouse, Marianne DeBrito, Megan Osborne, Casey Johnson, Robert S. Martin, Samuel J. Araki, Daniel Q. Eckhardt","doi":"10.1137/22m152092x","DOIUrl":"https://doi.org/10.1137/22m152092x","url":null,"abstract":"To address noise inherent in electronic data acquisition systems and real-world sources, Araki et al. [Phys. D, 417 (2021), 132819] demonstrated a grid-based nonlinear technique to remove noise from a chaotic signal, leveraging a clean high-fidelity signal from the same dynamical system and ensemble averaging in multidimensional phase space. This method achieved denoising of a time series data with 100% added noise but suffered in regions of low data density. To improve this grid-based method, here an unstructured mesh based on triangulations and Voronoi diagrams is used to accomplish the same task. The unstructured mesh more uniformly distributes data samples over mesh cells to improve the accuracy of the reconstructed signal. By empirically balancing bias and variance errors in selecting the number of unstructured cells as a function of the number of available samples, the method achieves asymptotic statistical convergence with known test data and reduces synthetic noise on experimental signals from Hall effect thrusters with greater success than the original grid-based strategy.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135919878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yen Ting Lin, Yifeng Tian, Danny Perez, Daniel Livescu
We propose to adopt statistical regression as the projection operator to enable data-driven learning of the operators in the Mori–Zwanzig formalism. We present a principled method to extract the Markov and memory operators for any regression models. We show that the choice of linear regression results in a recently proposed data-driven learning algorithm based on Mori’s projection operator, which is a higher-order approximate Koopman learning method. We show that more expressive nonlinear regression models naturally fill in the gap between the highly idealized and computationally efficient Mori’s projection operator and the most optimal yet computationally infeasible Zwanzig’s projection operator. We performed numerical experiments and extracted the operators for an array of regression-based projections, including linear, polynomial, spline, and neural network–based regressions, showing a progressive improvement as the complexity of the regression model increased. Our proposition provides a general framework to extract memory-dependent corrections and can be readily applied to an array of data-driven learning methods for stationary dynamical systems in the literature.
{"title":"Regression-Based Projection for Learning Mori–Zwanzig Operators","authors":"Yen Ting Lin, Yifeng Tian, Danny Perez, Daniel Livescu","doi":"10.1137/22m1506146","DOIUrl":"https://doi.org/10.1137/22m1506146","url":null,"abstract":"We propose to adopt statistical regression as the projection operator to enable data-driven learning of the operators in the Mori–Zwanzig formalism. We present a principled method to extract the Markov and memory operators for any regression models. We show that the choice of linear regression results in a recently proposed data-driven learning algorithm based on Mori’s projection operator, which is a higher-order approximate Koopman learning method. We show that more expressive nonlinear regression models naturally fill in the gap between the highly idealized and computationally efficient Mori’s projection operator and the most optimal yet computationally infeasible Zwanzig’s projection operator. We performed numerical experiments and extracted the operators for an array of regression-based projections, including linear, polynomial, spline, and neural network–based regressions, showing a progressive improvement as the complexity of the regression model increased. Our proposition provides a general framework to extract memory-dependent corrections and can be readily applied to an array of data-driven learning methods for stationary dynamical systems in the literature.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fast Adjoint Algorithm for Linear Responses of Hyperbolic Chaos","authors":"Angxiu Ni","doi":"10.1137/22m1522383","DOIUrl":"https://doi.org/10.1137/22m1522383","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135970091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Equilibria Analysis of a Networked Bivirus Epidemic Model Using Poincaré–Hopf and Manifold Theory","authors":"Brian D. O. Anderson, Mengbin Ye","doi":"10.1137/22m1529981","DOIUrl":"https://doi.org/10.1137/22m1529981","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135969108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace of and its dual space . The three topological spaces are called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron–Frobenius operators of the one-sided and two-sided shifts of finite type (symbolic dynamical systems) are determined. A one-sided subshift of finite type which is conjugate to the multiplication with the golden ratio on modulo 1 is also considered. A new construction of the Gelfand triplet for the generalized spectrum of symbolic dynamical systems is proposed by means of an algebraic procedure. The asymptotic formula of the iteration of Perron–Frobenius operators is also given. The iteration converges to the mixing state whose rate of convergence is determined by the generalized spectrum.
{"title":"Generalized Eigenvalues of the Perron–Frobenius Operators of Symbolic Dynamical Systems","authors":"Hayato Chiba, Masahiro Ikeda, Isao Ishikawa","doi":"10.1137/22m1476204","DOIUrl":"https://doi.org/10.1137/22m1476204","url":null,"abstract":"The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace of and its dual space . The three topological spaces are called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron–Frobenius operators of the one-sided and two-sided shifts of finite type (symbolic dynamical systems) are determined. A one-sided subshift of finite type which is conjugate to the multiplication with the golden ratio on modulo 1 is also considered. A new construction of the Gelfand triplet for the generalized spectrum of symbolic dynamical systems is proposed by means of an algebraic procedure. The asymptotic formula of the iteration of Perron–Frobenius operators is also given. The iteration converges to the mixing state whose rate of convergence is determined by the generalized spectrum.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135969467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mauricio Ascencio, Esther Barrabés, Josep M. Cors, Claudio Vidal
We study the dynamics of an infinitesimal mass under the gravitational attraction of primaries arranged in a planar ring configuration plus the influence of the central mass with a Manev potential , , where is a parameter related to the oblaticity or radiation source (according to the sign of the parameter ). Specifically, we investigate the relative equilibria of the infinitesimal mass and their linear stability as functions of the parameter and the mass parameter , the ratio of mass of the central body to the mass of one of the remaining bodies. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass.
{"title":"Stability of Equilibrium Points in the Spatially Restricted ({boldsymbol{N+1}}) -Body Problem with Manev Potential","authors":"Mauricio Ascencio, Esther Barrabés, Josep M. Cors, Claudio Vidal","doi":"10.1137/23m1551912","DOIUrl":"https://doi.org/10.1137/23m1551912","url":null,"abstract":"We study the dynamics of an infinitesimal mass under the gravitational attraction of primaries arranged in a planar ring configuration plus the influence of the central mass with a Manev potential , , where is a parameter related to the oblaticity or radiation source (according to the sign of the parameter ). Specifically, we investigate the relative equilibria of the infinitesimal mass and their linear stability as functions of the parameter and the mass parameter , the ratio of mass of the central body to the mass of one of the remaining bodies. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136209243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Optimal Control of the Controlled Lotka–Volterra Equations with Applications. The Permanent Case","authors":"Bernard Bonnard, Jérémy Rouot","doi":"10.1137/22m151978x","DOIUrl":"https://doi.org/10.1137/22m151978x","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136058087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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