SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2179-2207, September 2024. Abstract.An important problem in biological modeling is choosing the right model. Given experimental data, one is supposed to find the best mathematical representation to describe the real-world phenomena. However, there may not be a unique model representing that real-world phenomena. Two distinct models could yield the same exact dynamics. In this case, these models are called indistinguishable. In this work, we consider the indistinguishability problem for linear compartmental models, which are used in many areas, such as pharmacokinetics, physiology, cell biology, toxicology, and ecology. We exhibit sufficient conditions for indistinguishability for models with a certain graph structure: paths from input to output with “detours.” The benefit of applying our results is that indistinguishability can be proven using only the graph structure of the models, without the use of any symbolic computation. This can be very helpful for medium-to-large sized linear compartmental models. These are the first sufficient conditions for the indistinguishability of linear compartmental models based on graph structure alone, as previously only necessary conditions for indistinguishability of linear compartmental models existed based on graph structure alone. We prove our results by showing that the indistinguishable models are the same up to a renaming of parameters, which we call permutation indistinguishability.
{"title":"Graph-Based Sufficient Conditions for the Indistinguishability of Linear Compartmental Models","authors":"Cashous Bortner, Nicolette Meshkat","doi":"10.1137/23m1614663","DOIUrl":"https://doi.org/10.1137/23m1614663","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2179-2207, September 2024. <br/> Abstract.An important problem in biological modeling is choosing the right model. Given experimental data, one is supposed to find the best mathematical representation to describe the real-world phenomena. However, there may not be a unique model representing that real-world phenomena. Two distinct models could yield the same exact dynamics. In this case, these models are called indistinguishable. In this work, we consider the indistinguishability problem for linear compartmental models, which are used in many areas, such as pharmacokinetics, physiology, cell biology, toxicology, and ecology. We exhibit sufficient conditions for indistinguishability for models with a certain graph structure: paths from input to output with “detours.” The benefit of applying our results is that indistinguishability can be proven using only the graph structure of the models, without the use of any symbolic computation. This can be very helpful for medium-to-large sized linear compartmental models. These are the first sufficient conditions for the indistinguishability of linear compartmental models based on graph structure alone, as previously only necessary conditions for indistinguishability of linear compartmental models existed based on graph structure alone. We prove our results by showing that the indistinguishable models are the same up to a renaming of parameters, which we call permutation indistinguishability.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"44 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141881052","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 3, Page 2138-2178, September 2024. Abstract.This paper investigates the intricate connection between visual perception and the mathematical modeling of neural activity in the primary visual cortex (V1). The focus is on modeling the visual MacKay effect [D. M. MacKay, Nature, 180 (1957), pp. 849–850]. While bifurcation theory has been a prominent mathematical approach for addressing issues in neuroscience, especially in describing spontaneous pattern formations in V1 due to parameter changes, it faces challenges in scenarios with localized sensory inputs. This is evident, for instance, in MacKay’s psychophysical experiments, where the redundancy of visual stimuli information results in irregular shapes, making bifurcation theory and multiscale analysis less effective. To address this, we follow a mathematical viewpoint based on the input-output controllability of an Amari-type neural fields model. In this framework, we consider sensory input as a control function, a cortical representation via the retino-cortical map of the visual stimulus that captures its distinct features. This includes highly localized information in the center of MacKay’s funnel pattern “MacKay rays.” From a control theory point of view, the Amari-type equation’s exact controllability property is discussed for linear and nonlinear response functions. For the visual MacKay effect modeling, we adjust the parameter representing intra-neuron connectivity to ensure that cortical activity exponentially stabilizes to the stationary state in the absence of sensory input. Then, we perform quantitative and qualitative studies to demonstrate that they capture all the essential features of the induced after-image reported by MacKay.
{"title":"A Mathematical Model of the Visual MacKay Effect","authors":"Cyprien Tamekue, Dario Prandi, Yacine Chitour","doi":"10.1137/23m1616686","DOIUrl":"https://doi.org/10.1137/23m1616686","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2138-2178, September 2024. <br/> Abstract.This paper investigates the intricate connection between visual perception and the mathematical modeling of neural activity in the primary visual cortex (V1). The focus is on modeling the visual MacKay effect [D. M. MacKay, Nature, 180 (1957), pp. 849–850]. While bifurcation theory has been a prominent mathematical approach for addressing issues in neuroscience, especially in describing spontaneous pattern formations in V1 due to parameter changes, it faces challenges in scenarios with localized sensory inputs. This is evident, for instance, in MacKay’s psychophysical experiments, where the redundancy of visual stimuli information results in irregular shapes, making bifurcation theory and multiscale analysis less effective. To address this, we follow a mathematical viewpoint based on the input-output controllability of an Amari-type neural fields model. In this framework, we consider sensory input as a control function, a cortical representation via the retino-cortical map of the visual stimulus that captures its distinct features. This includes highly localized information in the center of MacKay’s funnel pattern “MacKay rays.” From a control theory point of view, the Amari-type equation’s exact controllability property is discussed for linear and nonlinear response functions. For the visual MacKay effect modeling, we adjust the parameter representing intra-neuron connectivity to ensure that cortical activity exponentially stabilizes to the stationary state in the absence of sensory input. Then, we perform quantitative and qualitative studies to demonstrate that they capture all the essential features of the induced after-image reported by MacKay.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"21 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141881051","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 3, Page 2018-2060, September 2024. Abstract. A hybrid asymptotic-numerical approach is developed to study the existence and linear stability of steady-state hotspot patterns for a three-component one-dimensional reaction-diffusion (RD) system that models urban crime with police intervention. Our analysis is focused on a new scaling regime in the RD system where there are two distinct competing mechanisms of hotspot annihilation and creation that, when coincident in a parameter space, lead to complex spatio-temporal dynamics of hotspot patterns. Hotspot annihilation events are shown numerically to be triggered by an asynchronous oscillatory instability of the hotspot amplitudes that arises from a secondary instability on the branch of periodic solutions that emerges from a Hopf bifurcation of the steady-state solution. In addition, hotspots can be nucleated from a quiescent background when the criminal diffusivity is below a saddle-node bifurcation threshold of hotspot equilibria, which we estimate from our asymptotic analysis. To investigate instabilities of hotspot steady states, the spectrum of the linearization around a two-boundary hotspot pattern is computed, and instability thresholds due to either zero-eigenvalue crossings or Hopf bifurcations are shown. The bifurcation software pde2path is used to follow the branch of periodic solutions and detect the onset of the secondary instability. Overall, these results provide a phase diagram in parameter space where distinct types of dynamical behaviors occur. In one region of this phase diagram, where the police diffusivity is small, a two-boundary hotspot steady state is unstable to an asynchronous oscillatory instability in the hotspot amplitudes. This instability typically triggers a nonlinear process leading to the annihilation of one of the hotspots. However, for parameter values where this instability is coincident with the nonexistence of a one-hotspot steady state, we show that hotspot patterns undergo complex “nucleation-annihilation” dynamics that are characterized by large-scale persistent oscillations of the hotspot amplitudes. In this way, our results identify parameter ranges in the three-component crime model where the effect of police intervention is to simply displace crime between adjacent hotspots and where new crime hotspots regularly emerge “spontaneously” from regions that were previously free of crime. More generally, it is suggested that when these annihilation and nucleation mechanisms are coincident for other multihotspot patterns, the problem of predicting the spatial-temporal distribution of crime is largely intractable.
{"title":"The Nucleation-Annihilation Dynamics of Hotspot Patterns for a Reaction-Diffusion System of Urban Crime with Police Deployment","authors":"Chunyi Gai, Michael J. Ward","doi":"10.1137/23m1562330","DOIUrl":"https://doi.org/10.1137/23m1562330","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2018-2060, September 2024. <br/> Abstract. A hybrid asymptotic-numerical approach is developed to study the existence and linear stability of steady-state hotspot patterns for a three-component one-dimensional reaction-diffusion (RD) system that models urban crime with police intervention. Our analysis is focused on a new scaling regime in the RD system where there are two distinct competing mechanisms of hotspot annihilation and creation that, when coincident in a parameter space, lead to complex spatio-temporal dynamics of hotspot patterns. Hotspot annihilation events are shown numerically to be triggered by an asynchronous oscillatory instability of the hotspot amplitudes that arises from a secondary instability on the branch of periodic solutions that emerges from a Hopf bifurcation of the steady-state solution. In addition, hotspots can be nucleated from a quiescent background when the criminal diffusivity is below a saddle-node bifurcation threshold of hotspot equilibria, which we estimate from our asymptotic analysis. To investigate instabilities of hotspot steady states, the spectrum of the linearization around a two-boundary hotspot pattern is computed, and instability thresholds due to either zero-eigenvalue crossings or Hopf bifurcations are shown. The bifurcation software pde2path is used to follow the branch of periodic solutions and detect the onset of the secondary instability. Overall, these results provide a phase diagram in parameter space where distinct types of dynamical behaviors occur. In one region of this phase diagram, where the police diffusivity is small, a two-boundary hotspot steady state is unstable to an asynchronous oscillatory instability in the hotspot amplitudes. This instability typically triggers a nonlinear process leading to the annihilation of one of the hotspots. However, for parameter values where this instability is coincident with the nonexistence of a one-hotspot steady state, we show that hotspot patterns undergo complex “nucleation-annihilation” dynamics that are characterized by large-scale persistent oscillations of the hotspot amplitudes. In this way, our results identify parameter ranges in the three-component crime model where the effect of police intervention is to simply displace crime between adjacent hotspots and where new crime hotspots regularly emerge “spontaneously” from regions that were previously free of crime. More generally, it is suggested that when these annihilation and nucleation mechanisms are coincident for other multihotspot patterns, the problem of predicting the spatial-temporal distribution of crime is largely intractable.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"86 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865110","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 3, Page 2061-2098, September 2024. Abstract.In [SIAM J. Appl. Dyn. Syst., 12 (2013), pp. 2068–2092], Widiasih proposed and analyzed a deterministic one-dimensional Budyko–Sellers energy-balance model with a moving ice line. In this paper, we extend this model to the stochastic setting and analyze it within the framework of stochastic slow-fast systems. We derive the dynamics for the ice line in the limit of a small parameter as a solution to a stochastic differential equation. The stochastic approach enables the study of co-existing (metastable) climate states as well as the transition dynamics between them.
{"title":"Stochastic Energy-Balance Model With A Moving Ice Line","authors":"Ilya Pavlyukevich, Marian Ritsch","doi":"10.1137/23m1619873","DOIUrl":"https://doi.org/10.1137/23m1619873","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2061-2098, September 2024. <br/> Abstract.In [SIAM J. Appl. Dyn. Syst., 12 (2013), pp. 2068–2092], Widiasih proposed and analyzed a deterministic one-dimensional Budyko–Sellers energy-balance model with a moving ice line. In this paper, we extend this model to the stochastic setting and analyze it within the framework of stochastic slow-fast systems. We derive the dynamics for the ice line in the limit of a small parameter as a solution to a stochastic differential equation. The stochastic approach enables the study of co-existing (metastable) climate states as well as the transition dynamics between them.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"143 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865182","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}
Bronwyn H. Bradshaw-Hajek, Ian Lizarraga, Robert Marangell, Martin Wechselberger
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2099-2137, September 2024. Abstract.Reaction-nonlinear diffusion partial differential equations (RND PDEs) have recently been developed as a powerful and flexible modeling tool in order to investigate the emergence of steep fronts in biological and ecological contexts. In this work, we demonstrate the utility and scope of regularization as a technique to investigate the existence and uniqueness of steep-fronted traveling wave solutions in RND PDE models with forward-backward-forward diffusion. In a recent work (see [Y. Li et al., Phys. D, 423 (2021), 132916]), geometric singular perturbation theory (GSPT) was introduced as a framework to analyze these regularized RND PDEs. Using the GSPT toolbox, different regularizations were shown to give rise to distinct families of monotone steep-fronted traveling waves which limit to their shock-fronted singular counterparts, obeying either the equal area or extremal area (i.e., algebraic decay) rules that are well known in the shockwave literature. In this work, we extend those earlier results by showing that composite regularizations can be used to construct families of monotone shock-fronted traveling waves sweeping out distinct generalized area rules, which smoothly interpolate between these two extremal rules for shock selection. Our analysis blends Melnikov methods—including a new variant of the method which can be applied to autonomous piecewise-smooth systems—with GSPT techniques applied to the traveling wave problem of the regularized RND model over distinct spatiotemporal scales. We further demonstrate using numerical continuation that our composite model supports more exotic shock-fronted solutions, namely, nonmonotone shock-fronted waves as well as shock-fronted waves containing slow tails in the aggregation (backward diffusion) regime. We complement these existence results with a numerical spectral stability analysis of some of these new “interpolated” steep-fronted waves. Using techniques from geometric spectral stability theory, our numerical results suggest that the monotone families remain spectrally stable in the “interpolation” regime, which extends recent stability results by some of the authors in [I. Lizarraga and R. Marangell, Phys. D, 460 (2024), 134069], [I. Lizarraga and R. Marangell, J. Nonlinear Sci., 33 (2023), 82]. The multiple-scale nature of the composite regularized RND PDE model continues to play an important role in the numerical analysis of the spatial eigenvalue problem.
SIAM 应用动力系统期刊》，第 23 卷第 3 期，第 2099-2137 页，2024 年 9 月。 摘要.反应非线性扩散偏微分方程（RND PDEs）最近被开发成一种强大而灵活的建模工具，用于研究生物和生态环境中陡峭前沿的出现。在这项工作中，我们展示了正则化作为一种技术在研究具有前向-后向-前向扩散的 RND PDE 模型中陡锋行波解的存在性和唯一性方面的实用性和范围。在最近的一项工作中（见 [Y. Li et al., Phys. D, 423 (2021), 132916]），引入了几何奇异扰动理论（GSPT）作为分析这些正则化 RND PDE 的框架。利用 GSPT 工具箱，不同的正则化被证明会产生不同的单调陡前行波系列，这些行波会极限到它们的冲击前奇异对应波，服从冲击波文献中众所周知的等面积或极值面积（即代数衰减）规则。在这项研究中，我们扩展了之前的研究成果，证明复合正则化可以用来构建单调冲击前行波族，扫出不同的广义面积规则，在这两种极端规则之间平滑插值，以进行冲击选择。我们的分析融合了梅尔尼科夫方法（包括该方法的一个新变体，它可应用于自主片状光滑系统）和应用于不同时空尺度上正则化 RND 模型行波问题的 GSPT 技术。我们通过数值延续进一步证明，我们的复合模型支持更奇特的冲击波前解，即非单调冲击波前解以及在聚集（后向扩散）机制中包含慢尾的冲击波前解。我们对这些新的 "插值 "陡前波进行了数值谱稳定性分析，以补充这些存在性结果。利用几何谱稳定性理论的技术，我们的数值结果表明，单调族在 "插值 "机制中保持谱稳定性，这扩展了[I. Lizarraga and R. Marangell, Phys. D, 460 (2024), 134069]、[I. Lizarraga and R. Marangell, J. Nonlinear Sci., 33 (2023), 82]中一些作者的最新稳定性结果。复合正则化 RND PDE 模型的多尺度性质在空间特征值问题的数值分析中继续发挥着重要作用。
{"title":"A Geometric Singular Perturbation Analysis of Shock Selection Rules in Composite Regularized Reaction-Nonlinear Diffusion Models","authors":"Bronwyn H. Bradshaw-Hajek, Ian Lizarraga, Robert Marangell, Martin Wechselberger","doi":"10.1137/23m1591803","DOIUrl":"https://doi.org/10.1137/23m1591803","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2099-2137, September 2024. <br/> Abstract.Reaction-nonlinear diffusion partial differential equations (RND PDEs) have recently been developed as a powerful and flexible modeling tool in order to investigate the emergence of steep fronts in biological and ecological contexts. In this work, we demonstrate the utility and scope of regularization as a technique to investigate the existence and uniqueness of steep-fronted traveling wave solutions in RND PDE models with forward-backward-forward diffusion. In a recent work (see [Y. Li et al., Phys. D, 423 (2021), 132916]), geometric singular perturbation theory (GSPT) was introduced as a framework to analyze these regularized RND PDEs. Using the GSPT toolbox, different regularizations were shown to give rise to distinct families of monotone steep-fronted traveling waves which limit to their shock-fronted singular counterparts, obeying either the equal area or extremal area (i.e., algebraic decay) rules that are well known in the shockwave literature. In this work, we extend those earlier results by showing that composite regularizations can be used to construct families of monotone shock-fronted traveling waves sweeping out distinct generalized area rules, which smoothly interpolate between these two extremal rules for shock selection. Our analysis blends Melnikov methods—including a new variant of the method which can be applied to autonomous piecewise-smooth systems—with GSPT techniques applied to the traveling wave problem of the regularized RND model over distinct spatiotemporal scales. We further demonstrate using numerical continuation that our composite model supports more exotic shock-fronted solutions, namely, nonmonotone shock-fronted waves as well as shock-fronted waves containing slow tails in the aggregation (backward diffusion) regime. We complement these existence results with a numerical spectral stability analysis of some of these new “interpolated” steep-fronted waves. Using techniques from geometric spectral stability theory, our numerical results suggest that the monotone families remain spectrally stable in the “interpolation” regime, which extends recent stability results by some of the authors in [I. Lizarraga and R. Marangell, Phys. D, 460 (2024), 134069], [I. Lizarraga and R. Marangell, J. Nonlinear Sci., 33 (2023), 82]. The multiple-scale nature of the composite regularized RND PDE model continues to play an important role in the numerical analysis of the spatial eigenvalue problem.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865181","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 3, Page 1966-2017, September 2024. Abstract.In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semilinear PDEs in a Hilbert space [math] ([math]) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in [math] as well as bounded linear operators from [math] to [math]. In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton–Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations/parameters, such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in [math] as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave.
{"title":"Rigorous Computation of Solutions of Semilinear PDEs on Unbounded Domains via Spectral Methods","authors":"Matthieu Cadiot, Jean-Philippe Lessard, Jean-Christophe Nave","doi":"10.1137/23m1607507","DOIUrl":"https://doi.org/10.1137/23m1607507","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1966-2017, September 2024. <br/> Abstract.In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semilinear PDEs in a Hilbert space [math] ([math]) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in [math] as well as bounded linear operators from [math] to [math]. In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton–Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations/parameters, such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in [math] as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"93 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743114","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 3, Page 1946-1965, September 2024. Abstract.We study the stability properties and the long-term dynamics of chimeras in systems of globally coupled maps. In particular, we establish a formula for the transverse Lyapunov exponent of the states of the system containing synchronized units. We use this formula to present numerical evidence of attracting chimeras having chaotic dynamics as well as periodic behaviors. We also show that, at least for polynomial local maps, attracting periodic cycles tend to belong to cluster spaces, and, more generally, limit sets of chimera orbits have zero Lebesgue measure for strong coupling regimes.
{"title":"Transverse Lyapunov Exponent and Chimeras in Globally Coupled Maps","authors":"Théophile Caby, Pierre Guiraud","doi":"10.1137/23m1603339","DOIUrl":"https://doi.org/10.1137/23m1603339","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1946-1965, September 2024. <br/> Abstract.We study the stability properties and the long-term dynamics of chimeras in systems of globally coupled maps. In particular, we establish a formula for the transverse Lyapunov exponent of the states of the system containing synchronized units. We use this formula to present numerical evidence of attracting chimeras having chaotic dynamics as well as periodic behaviors. We also show that, at least for polynomial local maps, attracting periodic cycles tend to belong to cluster spaces, and, more generally, limit sets of chimera orbits have zero Lebesgue measure for strong coupling regimes.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"25 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743223","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 3, Page 1836-1869, September 2024. Abstract.This work provides a geometric approach to the study of bifurcation and rate induced transitions in a class of nonautonomous systems referred to herein as asymptotically slow-fast systems, which may be viewed as “intermediate” between the (smaller, resp., larger) classes of asymptotically autonomous and nonautonomous systems. After showing that the relevant systems can be viewed as singular perturbations of a limiting system with a discontinuity in time, we develop an analytical framework for their analysis based on geometric blow-up techniques. We then provide sufficient conditions for the occurrence of bifurcation and rate induced transitions in low dimensions, as well as sufficient conditions for “tracking” in arbitrary (finite) dimensions, i.e., the persistence of an attracting and normally hyperbolic manifold through the transitionary regime. The proofs rely on geometric blow-up, a variant of the Melnikov method which applies on noncompact domains, and general invariant manifold theory. The formalism is applicable in arbitrary (finite) dimensions, and for systems with forward and backward attractors characterized by nontrivial (i.e., nonconstant) dependence on time. The results are demonstrated for low dimensional applications.
{"title":"Rate and Bifurcation Induced Transitions in Asymptotically Slow-Fast Systems","authors":"Samuel Jelbart","doi":"10.1137/24m1632000","DOIUrl":"https://doi.org/10.1137/24m1632000","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1836-1869, September 2024. <br/> Abstract.This work provides a geometric approach to the study of bifurcation and rate induced transitions in a class of nonautonomous systems referred to herein as asymptotically slow-fast systems, which may be viewed as “intermediate” between the (smaller, resp., larger) classes of asymptotically autonomous and nonautonomous systems. After showing that the relevant systems can be viewed as singular perturbations of a limiting system with a discontinuity in time, we develop an analytical framework for their analysis based on geometric blow-up techniques. We then provide sufficient conditions for the occurrence of bifurcation and rate induced transitions in low dimensions, as well as sufficient conditions for “tracking” in arbitrary (finite) dimensions, i.e., the persistence of an attracting and normally hyperbolic manifold through the transitionary regime. The proofs rely on geometric blow-up, a variant of the Melnikov method which applies on noncompact domains, and general invariant manifold theory. The formalism is applicable in arbitrary (finite) dimensions, and for systems with forward and backward attractors characterized by nontrivial (i.e., nonconstant) dependence on time. The results are demonstrated for low dimensional applications.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"26 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721243","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}
Brendan Harding, Yvonne M. Stokes, Rahil N. Valani
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1805-1835, September 2024. Abstract.Inertial focusing in curved microfluidic ducts exploits the interaction of the drag force from the Dean flow with the inertial lift force to separate particles or cells laterally across the cross-section width according to their size. Experimental work has identified that using a trapezoidal cross section, as opposed to a rectangular one, can enhance the sized based separation of particles/cells over a wide range of flow rates. Using our model, derived by carefully examining the way the Dean drag and inertial lift forces interact at low flow rates, we calculate the leading order approximation of these forces for a range of trapezoidal ducts, both vertically symmetric and nonsymmetric, with an increasing amount of skew towards the outside wall. We then conduct a systematic study to examine the bifurcations in the particle equilbira that occur with respect to a shape parameter characterizing the trapezoidal cross section. We reveal how the dynamics associated with particle migration are modified by the degree of skew in the cross-section shape, and show the existence of cusp bifurcations (with the bend radius as a second parameter). Additionally, our investigation suggests an optimal amount of skew for the trapezoidal cross section for the purposes of maximizing particle separation over a wide range of bend radii.
{"title":"Inertial Focusing Dynamics of Spherical Particles in Curved Microfluidic Ducts with a Trapezoidal Cross Section","authors":"Brendan Harding, Yvonne M. Stokes, Rahil N. Valani","doi":"10.1137/23m1613220","DOIUrl":"https://doi.org/10.1137/23m1613220","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1805-1835, September 2024. <br/> Abstract.Inertial focusing in curved microfluidic ducts exploits the interaction of the drag force from the Dean flow with the inertial lift force to separate particles or cells laterally across the cross-section width according to their size. Experimental work has identified that using a trapezoidal cross section, as opposed to a rectangular one, can enhance the sized based separation of particles/cells over a wide range of flow rates. Using our model, derived by carefully examining the way the Dean drag and inertial lift forces interact at low flow rates, we calculate the leading order approximation of these forces for a range of trapezoidal ducts, both vertically symmetric and nonsymmetric, with an increasing amount of skew towards the outside wall. We then conduct a systematic study to examine the bifurcations in the particle equilbira that occur with respect to a shape parameter characterizing the trapezoidal cross section. We reveal how the dynamics associated with particle migration are modified by the degree of skew in the cross-section shape, and show the existence of cusp bifurcations (with the bend radius as a second parameter). Additionally, our investigation suggests an optimal amount of skew for the trapezoidal cross section for the purposes of maximizing particle separation over a wide range of bend radii.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"46 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141609967","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 3, Page 1766-1804, September 2024. Abstract. This work provides a systematic recipe for computing accurate high order Fourier expansions of quasiperiodic invariant circles (and systems of such circles) in area preserving maps. The recipe requires only a finite data set sampled from the quasiperiodic circle. Our approach, being based on the parameterization method of [A. Haro and R. de la Llave, SIAM J. Appl. Dyn. Syst., 6 (2007), pp. 142–207; A. Haro and R. de la Llave, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261–1300; A. Haro and R. de la Llave, J. Differential Equations, 228 (2006), pp. 530–579], uses a Newton scheme to iteratively solve a conjugacy equation describing the invariant circle (or systems of circles). A critical step in properly formulating the conjugacy equation is to determine the rotation number of the quasiperiodic subsystem. For this we exploit the weighted Birkhoff averaging method of [S. Das et al., Nonlinearity, 30 (2017), pp. 4111–4140; S. Das et al., The Foundations of Chaos Revisited, Springer, Cham, 2016, pp. 103–118; S. Das and J. A. Yorke, Nonlinearity, 31 (2018), pp. 491–501]. This approach facilities accurate computation of the rotation number given nothing but the already mentioned orbit data. The weighted Birkhoff averages also facilitate the computation of other integral observables like Fourier coefficients of the parameterization of the invariant circle. Since, the parameterization method is based on a Newton scheme, we only need to approximate a small number of Fourier coefficients with low accuracy (say, a few correct digits) to find a good enough initial approximation so that Newton converges. Moreover, the Fourier coefficients may be computed independently, so we can sample the higher modes to guess the decay rate of the Fourier coefficients. This allows us to choose, a priori, an appropriate number of modes in the truncation. We illustrate the utility of the approach for explicit example systems including the area preserving Hénon map and the standard map (polynomial and trigonometric nonlinearity respectively). We present example computations for invariant circles and for systems of invariant circles with as many as 120 components. We also employ a numerical continuation scheme (where the rotation number is the continuation parameter) to compute large numbers of quasiperiodic circles in these systems. During the continuation we monitor the Sobolev norm of the parameterization, as explained in [R. Calleja and R. de la Llave, Nonlinearity, 23 (2010), pp. 2029–2058], to automatically detect the breakdown of the family.
SIAM 应用动力系统期刊》，第 23 卷第 3 期，第 1766-1804 页，2024 年 9 月。 摘要。本研究提供了一种系统的方法，用于计算面积保留映射中准周期不变圆（以及此类圆的系统）的精确高阶傅里叶展开。该方法只需要从准周期圆中采样的有限数据集。我们的方法基于[A. Haro and R. de la LL]的参数化方法。Haro and R. de la Llave, SIAM J. Appl.Syst., 6 (2007), pp.Dyn.Syst.B, 6 (2006), pp. 1261-1300; A. Haro and R. de la Llave, J. Differential Equations, 228 (2006), pp.正确表述共轭方程的关键步骤是确定准周期子系统的旋转数。为此，我们利用了加权伯克霍夫平均法 [S. Das et al.S. Das 等人，《非线性》，30 (2017)，第 4111-4140 页；S. Das 等人，《混沌基础重温》，Springer, Cham, 2016, 第 103-118 页；S. Das 和 J. A. Yorke，《非线性》，31 (2018)，第 491-501 页]。除了上述轨道数据，这种方法还能准确计算旋转数。加权伯克霍夫平均数还有助于计算其他积分观测值，如不变圆参数化的傅里叶系数。由于参数化方法以牛顿方案为基础，我们只需对少量傅里叶系数进行低精度近似（例如几位正确数字），就能找到足够好的初始近似值，从而使牛顿收敛。此外，傅立叶系数可以独立计算，因此我们可以对高阶模式进行采样，从而猜测傅立叶系数的衰减率。这样，我们就可以先验地选择适当数量的截断模式。我们将举例说明这种方法对包括面积保持的赫农图谱和标准图谱（分别为多项式和三角非线性）在内的显式系统的实用性。我们举例说明了不变圆和多达 120 个分量的不变圆系统的计算。我们还采用数值延续方案（其中旋转数是延续参数）来计算这些系统中的大量准周期圆。在延续过程中，我们按照 [R. Calleja and R. de la Llave, Nonlinearity, 23 (2010), pp.
{"title":"Weighted Birkhoff Averages and the Parameterization Method","authors":"David Blessing, J. D. Mireles James","doi":"10.1137/23m1579546","DOIUrl":"https://doi.org/10.1137/23m1579546","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1766-1804, September 2024. <br/> Abstract. This work provides a systematic recipe for computing accurate high order Fourier expansions of quasiperiodic invariant circles (and systems of such circles) in area preserving maps. The recipe requires only a finite data set sampled from the quasiperiodic circle. Our approach, being based on the parameterization method of [A. Haro and R. de la Llave, SIAM J. Appl. Dyn. Syst., 6 (2007), pp. 142–207; A. Haro and R. de la Llave, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261–1300; A. Haro and R. de la Llave, J. Differential Equations, 228 (2006), pp. 530–579], uses a Newton scheme to iteratively solve a conjugacy equation describing the invariant circle (or systems of circles). A critical step in properly formulating the conjugacy equation is to determine the rotation number of the quasiperiodic subsystem. For this we exploit the weighted Birkhoff averaging method of [S. Das et al., Nonlinearity, 30 (2017), pp. 4111–4140; S. Das et al., The Foundations of Chaos Revisited, Springer, Cham, 2016, pp. 103–118; S. Das and J. A. Yorke, Nonlinearity, 31 (2018), pp. 491–501]. This approach facilities accurate computation of the rotation number given nothing but the already mentioned orbit data. The weighted Birkhoff averages also facilitate the computation of other integral observables like Fourier coefficients of the parameterization of the invariant circle. Since, the parameterization method is based on a Newton scheme, we only need to approximate a small number of Fourier coefficients with low accuracy (say, a few correct digits) to find a good enough initial approximation so that Newton converges. Moreover, the Fourier coefficients may be computed independently, so we can sample the higher modes to guess the decay rate of the Fourier coefficients. This allows us to choose, a priori, an appropriate number of modes in the truncation. We illustrate the utility of the approach for explicit example systems including the area preserving Hénon map and the standard map (polynomial and trigonometric nonlinearity respectively). We present example computations for invariant circles and for systems of invariant circles with as many as 120 components. We also employ a numerical continuation scheme (where the rotation number is the continuation parameter) to compute large numbers of quasiperiodic circles in these systems. During the continuation we monitor the Sobolev norm of the parameterization, as explained in [R. Calleja and R. de la Llave, Nonlinearity, 23 (2010), pp. 2029–2058], to automatically detect the breakdown of the family.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"80 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141585295","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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