SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 668-695, March 2024. Abstract.We present a geometrical description of the symmetries and reduction of the full gravitational 2-body problem after complete averaging over fast angles. Our variables allow for a well-suited formulation in action-angle type coordinates associated with the averaged angles, which provide geometric insight into the problem. After introducing extra fictitious variables and through a symplectic transformation, we move to a singularity-free quaternionic triple-chart. This choice allows for a global chart to avoid the classical singularities associated with angles and renders all the invariants as homogeneous quadratic polynomials. Additionally, it permits one to quickly write the Hamiltonian of the system in terms of the invariants and the Poisson structure at each stage of the reduction process. In contrast with existing literature, the geometrical approach of this research completely describes all the dynamical aspects of the full reduced space since it involves the relative position of the rotational and orbital angular momenta and their orientation, which has yet to be considered in previous studies. Our program includes a preliminary parametric analysis of relative equilibria and a complete description of the fibers in the reconstruction of the reduced system.
{"title":"[math]-Reduction, Relative Equilibria, and Bifurcations for the Full Averaged Model of Two Interacting Rigid Bodies","authors":"F. Crespo, D. E. Espejo, J. C. van der Meer","doi":"10.1137/23m158125x","DOIUrl":"https://doi.org/10.1137/23m158125x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 668-695, March 2024. <br/> Abstract.We present a geometrical description of the symmetries and reduction of the full gravitational 2-body problem after complete averaging over fast angles. Our variables allow for a well-suited formulation in action-angle type coordinates associated with the averaged angles, which provide geometric insight into the problem. After introducing extra fictitious variables and through a symplectic transformation, we move to a singularity-free quaternionic triple-chart. This choice allows for a global chart to avoid the classical singularities associated with angles and renders all the invariants as homogeneous quadratic polynomials. Additionally, it permits one to quickly write the Hamiltonian of the system in terms of the invariants and the Poisson structure at each stage of the reduction process. In contrast with existing literature, the geometrical approach of this research completely describes all the dynamical aspects of the full reduced space since it involves the relative position of the rotational and orbital angular momenta and their orientation, which has yet to be considered in previous studies. Our program includes a preliminary parametric analysis of relative equilibria and a complete description of the fibers in the reconstruction of the reduced system.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"70 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139925455","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}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 641-667, March 2024. Abstract.To model a single-species cognitive movement, we formulate a reaction-diffusion equation with nonlocal spatial memory and investigate its dynamics. We explore the influence of the perceptual scale on the stability and Turing bifurcation. When the random diffusion is dominant, the perceptual scale does not affect the stability, but when the memory-based diffusion is dominant, there exist Turing bifurcations induced by the perceptual scale. Then the joint effect of the perceptual scale and the memory delay on the stability and spatio-temporal dynamics is investigated to show rich spatio-temporal dynamics via Turing–Hopf bifurcation and double Hopf bifurcation. Finally, we apply our analysis to an application and illustrate our theoretical results with numerical simulations.
{"title":"Spatio-temporal Dynamics in a Reaction-Diffusion Equation with Nonlocal Spatial Memory","authors":"Shuyang Xue, Yongli Song, Hao Wang","doi":"10.1137/22m1543860","DOIUrl":"https://doi.org/10.1137/22m1543860","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 641-667, March 2024. <br/>Abstract.To model a single-species cognitive movement, we formulate a reaction-diffusion equation with nonlocal spatial memory and investigate its dynamics. We explore the influence of the perceptual scale on the stability and Turing bifurcation. When the random diffusion is dominant, the perceptual scale does not affect the stability, but when the memory-based diffusion is dominant, there exist Turing bifurcations induced by the perceptual scale. Then the joint effect of the perceptual scale and the memory delay on the stability and spatio-temporal dynamics is investigated to show rich spatio-temporal dynamics via Turing–Hopf bifurcation and double Hopf bifurcation. Finally, we apply our analysis to an application and illustrate our theoretical results with numerical simulations.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"12 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139925694","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}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 696-720, March 2024. Abstract.In this paper, we study a two stream species Lotka–Volterra competition patch model with the patches aligned along a line. The two species are supposed to be identical except for the diffusion rates. For each species, the diffusion rates between patches are the same, while the drift rates vary. Our results show that the convexity of the drift rates has a significant impact on the competition outcomes: if the drift rates are convex, then the species with the larger diffusion rate wins the competition; if the drift rates are concave, then the species with the smaller diffusion rate wins the competition.
{"title":"Evolution of Dispersal in Advective Patchy Environments with Varying Drift Rates","authors":"Shanshan Chen, Jie Liu, Yixiang Wu","doi":"10.1137/22m1542027","DOIUrl":"https://doi.org/10.1137/22m1542027","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 696-720, March 2024. <br/> Abstract.In this paper, we study a two stream species Lotka–Volterra competition patch model with the patches aligned along a line. The two species are supposed to be identical except for the diffusion rates. For each species, the diffusion rates between patches are the same, while the drift rates vary. Our results show that the convexity of the drift rates has a significant impact on the competition outcomes: if the drift rates are convex, then the species with the larger diffusion rate wins the competition; if the drift rates are concave, then the species with the smaller diffusion rate wins the competition.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"164 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139925457","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}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 616-640, March 2024. Abstract. We previously showed that three-dimensional quadratic diffeomorphisms have anti-integrable (AI) limits that correspond to a quadratic correspondence, a pair of one-dimensional maps. At the AI limit the dynamics is conjugate to a full shift on two symbols. Here we consider a more general AI limit, allowing two parameters of the map to go to infinity. We prove the existence of AI states for each symbol sequence for three cases of the quadratic correspondence: parabolas, ellipses, and hyperbolas. A contraction argument gives parameter domains such that this is a bijection, but the correspondence also is observed to apply more generally. We show that orbits of the original map can be obtained by numerical continuation for a volume-contracting case. These results show that periodic AI states evolve into the observed periodic attractors of the diffeomorphism. We also continue a periodic AI state with a symbol sequence chosen so that it continues to an orbit resembling a chaotic attractor that is a 3D version of the classical 2D Hénon attractor.
SIAM 应用动力系统期刊》第 23 卷第 1 期第 616-640 页，2024 年 3 月。 摘要。我们之前证明了三维二次差分变分具有反可积分（AI）极限，它对应于二次对应关系，即一对一维映射。在 AI 极限，动力学共轭于两个符号上的全移。在这里，我们考虑了更一般的 AI 极限，允许映射的两个参数达到无穷大。我们证明了在抛物线、椭圆和双曲线这三种二次对应的情况下，每个符号序列都存在人工智能状态。收缩论证给出了参数域，因此这是一个双射，但也观察到对应关系适用于更广泛的情况。我们证明，在体积收缩的情况下，可以通过数值延续得到原始映射的轨道。这些结果表明，周期性人工智能状态会演化成所观察到的衍射周期性吸引子。我们还用符号序列延续了一个周期性人工智能状态，使其延续到一个类似于混沌吸引子的轨道，而混沌吸引子是经典二维赫农吸引子的三维版本。
{"title":"Connecting Anti-integrability to Attractors for Three-Dimensional Quadratic Diffeomorphisms","authors":"Amanda E. Hampton, James D. Meiss","doi":"10.1137/23m1571897","DOIUrl":"https://doi.org/10.1137/23m1571897","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 616-640, March 2024. <br/> Abstract. We previously showed that three-dimensional quadratic diffeomorphisms have anti-integrable (AI) limits that correspond to a quadratic correspondence, a pair of one-dimensional maps. At the AI limit the dynamics is conjugate to a full shift on two symbols. Here we consider a more general AI limit, allowing two parameters of the map to go to infinity. We prove the existence of AI states for each symbol sequence for three cases of the quadratic correspondence: parabolas, ellipses, and hyperbolas. A contraction argument gives parameter domains such that this is a bijection, but the correspondence also is observed to apply more generally. We show that orbits of the original map can be obtained by numerical continuation for a volume-contracting case. These results show that periodic AI states evolve into the observed periodic attractors of the diffeomorphism. We also continue a periodic AI state with a symbol sequence chosen so that it continues to an orbit resembling a chaotic attractor that is a 3D version of the classical 2D Hénon attractor.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"3 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770434","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}
Alan Veliz-Cuba, Vanessa Newsome-Slade, Elena S. Dimitrova
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 592-615, March 2024. Abstract.Due to cost concerns, it is optimal to gain insight into the connectivity of biological and other networks using as few experiments as possible. Data selection for unique network connectivity identification has been an open problem since the introduction of algebraic methods for reverse engineering for almost two decades. In this manuscript we determine what data sets uniquely identify the unsigned wiring diagram corresponding to a system that is discrete in time and space. Furthermore, we answer the question of uniqueness for signed wiring diagrams for Boolean networks. Computationally, unsigned and signed wiring diagrams have been studied separately, and in this manuscript we also show that there exists an ideal capable of encoding both unsigned and signed information. This provides a unified approach to studying reverse engineering that also gives significant computational benefits.
{"title":"A Unified Approach to Reverse Engineering and Data Selection for Unique Network Identification","authors":"Alan Veliz-Cuba, Vanessa Newsome-Slade, Elena S. Dimitrova","doi":"10.1137/22m1540570","DOIUrl":"https://doi.org/10.1137/22m1540570","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 592-615, March 2024. <br/> Abstract.Due to cost concerns, it is optimal to gain insight into the connectivity of biological and other networks using as few experiments as possible. Data selection for unique network connectivity identification has been an open problem since the introduction of algebraic methods for reverse engineering for almost two decades. In this manuscript we determine what data sets uniquely identify the unsigned wiring diagram corresponding to a system that is discrete in time and space. Furthermore, we answer the question of uniqueness for signed wiring diagrams for Boolean networks. Computationally, unsigned and signed wiring diagrams have been studied separately, and in this manuscript we also show that there exists an ideal capable of encoding both unsigned and signed information. This provides a unified approach to studying reverse engineering that also gives significant computational benefits.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"6 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770627","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}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 553-591, March 2024. Abstract. In this paper, we will perform the parameter-dependent center manifold reduction near the generic and transcritical codimension two Bogdanov–Takens bifurcation in classical delay differential equations. Using an approximation to the homoclinic solutions derived with a generalized Lindstedt–Poincaré method, we develop a method to initialize the continuation of the homoclinic bifurcation curves emanating from these points. The normal form transformation is derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas, which have been implemented in the freely available bifurcation software package DDE-BifTool. The effectiveness is demonstrated on various models
{"title":"Bifurcation Analysis of Bogdanov–Takens Bifurcations in Delay Differential Equations","authors":"M. M. Bosschaert, Yu. A. Kuznetsov","doi":"10.1137/22m1527532","DOIUrl":"https://doi.org/10.1137/22m1527532","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 553-591, March 2024. <br/> Abstract. In this paper, we will perform the parameter-dependent center manifold reduction near the generic and transcritical codimension two Bogdanov–Takens bifurcation in classical delay differential equations. Using an approximation to the homoclinic solutions derived with a generalized Lindstedt–Poincaré method, we develop a method to initialize the continuation of the homoclinic bifurcation curves emanating from these points. The normal form transformation is derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas, which have been implemented in the freely available bifurcation software package DDE-BifTool. The effectiveness is demonstrated on various models","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"33 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646702","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}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 470-504, March 2024. Abstract. We propose a general strategy for reduced order modeling of systems that display highly nonlinear oscillations. By considering a continuous family of forced periodic orbits defined in relation to a stable fixed point and subsequently leveraging phase-amplitude-based reduction strategies, we arrive at a low order model capable of accurately capturing nonlinear oscillations resulting from arbitrary external inputs. In the limit that oscillations are small, the system dynamics relax to those obtained from local linearization, i.e., that can be fully described using linear eigenmodes. For larger amplitude oscillations, the behavior can be understood in terms of the dynamics of a small number of nonlinear modes. We illustrate the proposed strategy in a variety of examples yielding results that are substantially better than those obtained using standard linearization-based techniques.
{"title":"Reduced Order Characterization of Nonlinear Oscillations Using an Adaptive Phase-Amplitude Coordinate Framework","authors":"Dan Wilson, Kai Sun","doi":"10.1137/23m1551699","DOIUrl":"https://doi.org/10.1137/23m1551699","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 470-504, March 2024. <br/> Abstract. We propose a general strategy for reduced order modeling of systems that display highly nonlinear oscillations. By considering a continuous family of forced periodic orbits defined in relation to a stable fixed point and subsequently leveraging phase-amplitude-based reduction strategies, we arrive at a low order model capable of accurately capturing nonlinear oscillations resulting from arbitrary external inputs. In the limit that oscillations are small, the system dynamics relax to those obtained from local linearization, i.e., that can be fully described using linear eigenmodes. For larger amplitude oscillations, the behavior can be understood in terms of the dynamics of a small number of nonlinear modes. We illustrate the proposed strategy in a variety of examples yielding results that are substantially better than those obtained using standard linearization-based techniques.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"84 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139583644","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}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 505-552, March 2024. Abstract. Traditional sensitivity analysis methods fail for chaotic systems due to the unstable characteristics of the linearized equations. To overcome these issues two methods have been developed in the literature, one being the shadowing approach, which results in a minimization problem, and the other being numerical viscosity, where a damping term is added to the linearized equations to suppress the instability. The shadowing approach is computationally expensive but produces accurate sensitivities, while numerical viscosity can produce less accurate sensitivities but with significantly reduced computational cost. However, it is not fully clear how the solutions generated by these two approaches compare to each other. In this work we aim to bridge this gap by introducing a control term, found with optimal control theory techniques, to prevent the exponential growth of solution of the linearized equations. We will refer to this method as optimal control shadowing. We investigate the computational aspects and performance of this new method on the Lorenz and Kuramoto–Sivashinsky systems and compare its performance with simple numerical viscosity schemes. We show that the tangent solution generated by the proposed approach is similar to that generated by shadowing methods, suggesting that optimal control attempts to stabilize the unstable shadowing direction. Further, for the spatially extended system, we examine the energy budget of the tangent equation and show that the control term found via the solution of the optimal control problem acts only at length scales where production of tangent energy dominates dissipation, which is not necessarily the case for the numerical viscosity methods.
{"title":"On the Application of Optimal Control Techniques to the Shadowing Approach for Time Averaged Sensitivity Analysis of Chaotic Systems","authors":"Rhys E. Gilbert, Davide Lasagna","doi":"10.1137/23m1550219","DOIUrl":"https://doi.org/10.1137/23m1550219","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 505-552, March 2024. <br/> Abstract. Traditional sensitivity analysis methods fail for chaotic systems due to the unstable characteristics of the linearized equations. To overcome these issues two methods have been developed in the literature, one being the shadowing approach, which results in a minimization problem, and the other being numerical viscosity, where a damping term is added to the linearized equations to suppress the instability. The shadowing approach is computationally expensive but produces accurate sensitivities, while numerical viscosity can produce less accurate sensitivities but with significantly reduced computational cost. However, it is not fully clear how the solutions generated by these two approaches compare to each other. In this work we aim to bridge this gap by introducing a control term, found with optimal control theory techniques, to prevent the exponential growth of solution of the linearized equations. We will refer to this method as optimal control shadowing. We investigate the computational aspects and performance of this new method on the Lorenz and Kuramoto–Sivashinsky systems and compare its performance with simple numerical viscosity schemes. We show that the tangent solution generated by the proposed approach is similar to that generated by shadowing methods, suggesting that optimal control attempts to stabilize the unstable shadowing direction. Further, for the spatially extended system, we examine the energy budget of the tangent equation and show that the control term found via the solution of the optimal control problem acts only at length scales where production of tangent energy dominates dissipation, which is not necessarily the case for the numerical viscosity methods.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"19 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139583339","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}
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 410-439, March 2024. Abstract.This paper provides for the first time correct third-order homoclinic predictors in [math]-dimensional ODEs near a generic Bogdanov–Takens bifurcation point, which can be used to start the numerical continuation of the appearing homoclinic orbits. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt–Poincaré method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is derived rigorously. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt–Poincaré) to approximate the homoclinic solution near Bogdanov–Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open-source MATLAB/GNU Octave continuation package MatCont.
{"title":"Interplay between Normal Forms and Center Manifold Reduction for Homoclinic Predictors near Bogdanov–Takens Bifurcation","authors":"Maikel M. Bosschaert, Yuri A. Kuznetsov","doi":"10.1137/22m151354x","DOIUrl":"https://doi.org/10.1137/22m151354x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 410-439, March 2024. <br/> Abstract.This paper provides for the first time correct third-order homoclinic predictors in [math]-dimensional ODEs near a generic Bogdanov–Takens bifurcation point, which can be used to start the numerical continuation of the appearing homoclinic orbits. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt–Poincaré method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is derived rigorously. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt–Poincaré) to approximate the homoclinic solution near Bogdanov–Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open-source MATLAB/GNU Octave continuation package MatCont.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"3 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554944","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}
Megan R. Ebers, Katherine M. Steele, J. Nathan Kutz
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 440-469, March 2024. Abstract.Physics-based and first-principles models pervade the engineering and physical sciences, allowing for the ability to model the dynamics of complex systems with a prescribed accuracy. The approximations used in deriving governing equations often result in discrepancies between the model and sensor-based measurements of the system, revealing the approximate nature of the equations and/or the signal-to-noise ratio of the sensor itself. In modern dynamical systems, such discrepancies between model and measurement can lead to poor quantification, often undermining the ability to produce accurate and precise control algorithms. We introduce a discrepancy modeling framework to identify the missing physics and resolve the model-measurement mismatch with two distinct approaches: (i) by learning a model for the evolution of systematic state-space residual, and (ii) by discovering a model for the deterministic dynamical error. Regardless of approach, a common suite of data-driven model discovery methods can be used. Specifically, we use four fundamentally different methods to demonstrate the mathematical implementations of discrepancy modeling: (i) the sparse identification of nonlinear dynamics, (ii) dynamic mode decomposition, (iii) Gaussian process regression, and (iv) neural networks. The choice of method depends on one’s intent (e.g., mechanistic interpretability) for discrepancy modeling, sensor measurement characteristics (e.g., quantity, quality, resolution), and constraints imposed by practical applications (e.g., state- or dynamical-space operability). We demonstrate the utility and suitability for discrepancy modeling using the suite of data-driven modeling methods on four dynamical systems under varying signal-to-noise ratios. Finally, we emphasize structural shortcomings of each discrepancy modeling approach depending on error type. In summary, if the true dynamics are unknown (i.e., an imperfect model), one should learn a discrepancy model of the missing physics in the dynamical space. Yet, if the true dynamics are known yet model-measurement mismatch still exists, one should learn a discrepancy model in the state space.
{"title":"Discrepancy Modeling Framework: Learning Missing Physics, Modeling Systematic Residuals, and Disambiguating between Deterministic and Random Effects","authors":"Megan R. Ebers, Katherine M. Steele, J. Nathan Kutz","doi":"10.1137/22m148375x","DOIUrl":"https://doi.org/10.1137/22m148375x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 440-469, March 2024. <br/> Abstract.Physics-based and first-principles models pervade the engineering and physical sciences, allowing for the ability to model the dynamics of complex systems with a prescribed accuracy. The approximations used in deriving governing equations often result in discrepancies between the model and sensor-based measurements of the system, revealing the approximate nature of the equations and/or the signal-to-noise ratio of the sensor itself. In modern dynamical systems, such discrepancies between model and measurement can lead to poor quantification, often undermining the ability to produce accurate and precise control algorithms. We introduce a discrepancy modeling framework to identify the missing physics and resolve the model-measurement mismatch with two distinct approaches: (i) by learning a model for the evolution of systematic state-space residual, and (ii) by discovering a model for the deterministic dynamical error. Regardless of approach, a common suite of data-driven model discovery methods can be used. Specifically, we use four fundamentally different methods to demonstrate the mathematical implementations of discrepancy modeling: (i) the sparse identification of nonlinear dynamics, (ii) dynamic mode decomposition, (iii) Gaussian process regression, and (iv) neural networks. The choice of method depends on one’s intent (e.g., mechanistic interpretability) for discrepancy modeling, sensor measurement characteristics (e.g., quantity, quality, resolution), and constraints imposed by practical applications (e.g., state- or dynamical-space operability). We demonstrate the utility and suitability for discrepancy modeling using the suite of data-driven modeling methods on four dynamical systems under varying signal-to-noise ratios. Finally, we emphasize structural shortcomings of each discrepancy modeling approach depending on error type. In summary, if the true dynamics are unknown (i.e., an imperfect model), one should learn a discrepancy model of the missing physics in the dynamical space. Yet, if the true dynamics are known yet model-measurement mismatch still exists, one should learn a discrepancy model in the state space.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"2 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139555271","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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