SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2533-2556, September 2024. Abstract. This paper is concerned with global dynamics, especially periodicity, for a piecewise smooth system incorporating a new control strategy. The switches take place at discrete times and depend on the status. By employing the approach of Poincaré maps, the existence, exact number, and asymptotical stability of periodic solutions are investigated thoroughly in some parameter regions. The periodic solutions are induced by the control strategy. As applications, convergence and periodicity are studied both for a fishery model and for an SIS model with discrete time on-off control. Numerical simulations are performed to verify our results.
{"title":"Global Dynamics of Piecewise Smooth Systems with Switches Depending on Both Discrete Times and Status","authors":"Lihong Huang, Jiafu Wang","doi":"10.1137/24m1634941","DOIUrl":"https://doi.org/10.1137/24m1634941","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2533-2556, September 2024. <br/> Abstract. This paper is concerned with global dynamics, especially periodicity, for a piecewise smooth system incorporating a new control strategy. The switches take place at discrete times and depend on the status. By employing the approach of Poincaré maps, the existence, exact number, and asymptotical stability of periodic solutions are investigated thoroughly in some parameter regions. The periodic solutions are induced by the control strategy. As applications, convergence and periodicity are studied both for a fishery model and for an SIS model with discrete time on-off control. Numerical simulations are performed to verify our results.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250385","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 2489-2532, September 2024. Abstract.We consider a family of perturbed Hamiltonian systems with Hamiltonian [math] in 1:1:2 resonance, where [math] is a polynomial which is axially symmetric with respect to the [math]-axis. Here, [math] is a homogeneous polynomial of degree [math], and we note that our analysis is carried out considering the polynomials [math] and [math]. We initially perform a Lie–Deprit normalization (truncation of the higher-order terms), and a singular reduction by the oscillator symmetry is done. Considering the averaging method for Hamiltonian systems, the existence and an approximation of two families of periodic solutions are proved together with their linear stability. A third family of periodic solutions is found by using the Lyapunov center theorem. In addition, the existence of KAM 3-tori is obtained by enclosing the stable periodic solutions. After that, since the Hamiltonian is axially symmetric, we carry out another reduction induced by this exact symmetry. Studying its Poisson vector field on the reduced space by the exact symmetry, we show the existence of two equilibrium points. We reconstruct these points as two families of periodic solutions of the complete Hamiltonian system together with their linear stability. Next, we make a second singular reduction using the axial symmetry. A geometrical study of the twice-reduced space is done to characterize the singularities. Precisely, we study the critical points (relative equilibria) on the twice-reduced space together with the stability, and parametric bifurcations are determined. The equilibria occurring in the twice-reduced space are reconstructed as 3-tori filled by quasi-periodic solutions of the full system. Our analysis permits us to determine the main representative parameters of the cubic ([math]) and quartic ([math]) terms to get our results. Important differences with the case of resonance 1:1:1 are detected.
{"title":"Reduction and Reconstruction of the Oscillator in 1:1:2 Resonance plus an Axially Symmetric Polynomial Perturbation","authors":"Yocelyn Pérez Rothen, Claudio Vidal","doi":"10.1137/23m1621885","DOIUrl":"https://doi.org/10.1137/23m1621885","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2489-2532, September 2024. <br/> Abstract.We consider a family of perturbed Hamiltonian systems with Hamiltonian [math] in 1:1:2 resonance, where [math] is a polynomial which is axially symmetric with respect to the [math]-axis. Here, [math] is a homogeneous polynomial of degree [math], and we note that our analysis is carried out considering the polynomials [math] and [math]. We initially perform a Lie–Deprit normalization (truncation of the higher-order terms), and a singular reduction by the oscillator symmetry is done. Considering the averaging method for Hamiltonian systems, the existence and an approximation of two families of periodic solutions are proved together with their linear stability. A third family of periodic solutions is found by using the Lyapunov center theorem. In addition, the existence of KAM 3-tori is obtained by enclosing the stable periodic solutions. After that, since the Hamiltonian is axially symmetric, we carry out another reduction induced by this exact symmetry. Studying its Poisson vector field on the reduced space by the exact symmetry, we show the existence of two equilibrium points. We reconstruct these points as two families of periodic solutions of the complete Hamiltonian system together with their linear stability. Next, we make a second singular reduction using the axial symmetry. A geometrical study of the twice-reduced space is done to characterize the singularities. Precisely, we study the critical points (relative equilibria) on the twice-reduced space together with the stability, and parametric bifurcations are determined. The equilibria occurring in the twice-reduced space are reconstructed as 3-tori filled by quasi-periodic solutions of the full system. Our analysis permits us to determine the main representative parameters of the cubic ([math]) and quartic ([math]) terms to get our results. Important differences with the case of resonance 1:1:1 are detected.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250386","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}
Hongyong Cui, Rodiak N. Figueroa-López, José A. Langa, Marcelo J. D. Nascimento
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2407-2443, September 2024. Abstract. In this paper we studied the forward dynamics of nonautonomous dynamical systems in terms of forward attractors. We first reviewed the well-known uniform attractor theory, and then by weakening the uniformity of attraction we introduced semiuniform forward attractors and minimal (nonuniform) forward attractors. With these semiuniform attractors, a characterization of the structure of uniform attractors was given: a uniform attractor is composed of two semiuniform attractors and bounded complete trajectories connecting them. As a consequence, the nature of the forward attraction of a dissipative nonautonomous dynamical system was then revealed: the vector field in the distant future of the system determines the (nonuniform) forward asymptotic behavior. A criterion for certain semiuniform attractors to have finite fractal dimension was given and the finite dimensionality of uniform attractors was discussed. Forward attracting time-dependent sets were studied also. A sufficient condition and a necessary condition for a time-dependent set to be forward attracting were given with illustrative counterexamples. Forward attractors of a Navier–Stokes equation with asymptotically vanishing viscosity (with an Euler equation as the limit equation) and with time-dependent forcing were studied as applications.
{"title":"Forward Attraction of Nonautonomous Dynamical Systems and Applications to Navier–Stokes Equations","authors":"Hongyong Cui, Rodiak N. Figueroa-López, José A. Langa, Marcelo J. D. Nascimento","doi":"10.1137/23m1626384","DOIUrl":"https://doi.org/10.1137/23m1626384","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2407-2443, September 2024. <br/> Abstract. In this paper we studied the forward dynamics of nonautonomous dynamical systems in terms of forward attractors. We first reviewed the well-known uniform attractor theory, and then by weakening the uniformity of attraction we introduced semiuniform forward attractors and minimal (nonuniform) forward attractors. With these semiuniform attractors, a characterization of the structure of uniform attractors was given: a uniform attractor is composed of two semiuniform attractors and bounded complete trajectories connecting them. As a consequence, the nature of the forward attraction of a dissipative nonautonomous dynamical system was then revealed: the vector field in the distant future of the system determines the (nonuniform) forward asymptotic behavior. A criterion for certain semiuniform attractors to have finite fractal dimension was given and the finite dimensionality of uniform attractors was discussed. Forward attracting time-dependent sets were studied also. A sufficient condition and a necessary condition for a time-dependent set to be forward attracting were given with illustrative counterexamples. Forward attractors of a Navier–Stokes equation with asymptotically vanishing viscosity (with an Euler equation as the limit equation) and with time-dependent forcing were studied as applications.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191998","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}
Mark Sinzger-D’Angelo, Jan Hasenauer, Heinz Koeppl
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2444-2488, September 2024. Abstract.Cellular processes are open systems, situated in a heterogeneous context, rather than operating in isolation. Chemical reaction networks (CRNs) whose reaction rates are modelled as external stochastic processes account for the heterogeneous environment when describing the embedded process. A marginal description of the embedded process is of interest for (i) marginal simulations that bypass the co-simulation of the environment, (ii) obtaining new process equations from which moment equations can be derived, (iii) the computation of information-theoretic quantities, and (iv) state estimation. It is known since Snyder’s and related works that marginalization over a stochastic intensity turns point processes into self-exciting ones. While the Snyder filter specifies the exact history-dependent propensities in the framework of CRNs in a Markov environment, it was recently suggested to use approximate filters for the marginal description. By regarding the chemical reactions as events, we establish a link between CRNs in a linear random environment and Hawkes processes, a class of self-exciting counting processes widely used in event analysis. The Hawkes approximation can be obtained via a moment closure scheme or as the optimal linear approximation under the quadratic criterion. We show the equivalence of both approaches. Furthermore, we use martingale techniques to provide results on the agreement of the Hawkes process and the exact marginal process in their second-order statistics, i.e., covariance, auto/cross-correlation. We introduce an approximate marginal simulation algorithm and illustrate it in case studies.
{"title":"Hawkes Process Modelling for Chemical Reaction Networks in a Random Environment","authors":"Mark Sinzger-D’Angelo, Jan Hasenauer, Heinz Koeppl","doi":"10.1137/23m1588573","DOIUrl":"https://doi.org/10.1137/23m1588573","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2444-2488, September 2024. <br/> Abstract.Cellular processes are open systems, situated in a heterogeneous context, rather than operating in isolation. Chemical reaction networks (CRNs) whose reaction rates are modelled as external stochastic processes account for the heterogeneous environment when describing the embedded process. A marginal description of the embedded process is of interest for (i) marginal simulations that bypass the co-simulation of the environment, (ii) obtaining new process equations from which moment equations can be derived, (iii) the computation of information-theoretic quantities, and (iv) state estimation. It is known since Snyder’s and related works that marginalization over a stochastic intensity turns point processes into self-exciting ones. While the Snyder filter specifies the exact history-dependent propensities in the framework of CRNs in a Markov environment, it was recently suggested to use approximate filters for the marginal description. By regarding the chemical reactions as events, we establish a link between CRNs in a linear random environment and Hawkes processes, a class of self-exciting counting processes widely used in event analysis. The Hawkes approximation can be obtained via a moment closure scheme or as the optimal linear approximation under the quadratic criterion. We show the equivalence of both approaches. Furthermore, we use martingale techniques to provide results on the agreement of the Hawkes process and the exact marginal process in their second-order statistics, i.e., covariance, auto/cross-correlation. We introduce an approximate marginal simulation algorithm and illustrate it in case studies.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191999","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}
Emilio Cruciani, Emanuela L. Giacomelli, Jinyeop Lee
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2364-2406, September 2024. Abstract.Complex networked systems in fields such as physics, biology, and social sciences often involve interactions that extend beyond simple pairwise ones. Hypergraphs serve as powerful modeling tools for describing and analyzing the intricate behaviors of systems with multibody interactions. Herein, we investigate discrete-time dynamics with three-body interactions, described by an underlying 3-uniform hypergraph, where vertices update their states through a nonlinearly weighted average depending on their neighboring pairs’ states. These dynamics capture reinforcing group effects, such as peer pressure, and exhibit higher-order dynamical effects resulting from a complex interplay between initial states, hypergraph topology, and nonlinearity of the update. Differently from linear averaging dynamics on graphs with two-body interactions, this model does not converge to the average of the initial states but rather induces a shift. By assuming random initial states and by making some regularity and density assumptions on the hypergraph, we prove that the dynamics converge to a multiplicatively shifted average of the initial states, with high probability. We further characterize the shift as a function of two parameters describing the initial state and interaction strength, as well as the convergence time as a function of the hypergraph structure.
{"title":"On the Convergence of Nonlinear Averaging Dynamics with Three-Body Interactions on Hypergraphs","authors":"Emilio Cruciani, Emanuela L. Giacomelli, Jinyeop Lee","doi":"10.1137/23m1568338","DOIUrl":"https://doi.org/10.1137/23m1568338","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2364-2406, September 2024. <br/> Abstract.Complex networked systems in fields such as physics, biology, and social sciences often involve interactions that extend beyond simple pairwise ones. Hypergraphs serve as powerful modeling tools for describing and analyzing the intricate behaviors of systems with multibody interactions. Herein, we investigate discrete-time dynamics with three-body interactions, described by an underlying 3-uniform hypergraph, where vertices update their states through a nonlinearly weighted average depending on their neighboring pairs’ states. These dynamics capture reinforcing group effects, such as peer pressure, and exhibit higher-order dynamical effects resulting from a complex interplay between initial states, hypergraph topology, and nonlinearity of the update. Differently from linear averaging dynamics on graphs with two-body interactions, this model does not converge to the average of the initial states but rather induces a shift. By assuming random initial states and by making some regularity and density assumptions on the hypergraph, we prove that the dynamics converge to a multiplicatively shifted average of the initial states, with high probability. We further characterize the shift as a function of two parameters describing the initial state and interaction strength, as well as the convergence time as a function of the hypergraph structure.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192000","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 2323-2363, September 2024. Abstract.We are interested in the dynamics of interfaces, or zeros, of shock waves in general scalar viscous conservation laws with a locally Lipschitz continuous flux function, such as the modular Burgers equation. We prove that all interfaces coalesce within finite time, leaving behind either a single interface or no interface at all. Our proof relies on mass and energy estimates, regularization of the flux function, and an application of the Sturm theorems on the number of zeros of solutions of parabolic problems. Our analysis yields an explicit upper bound on the time of extinction in terms of the initial condition and the flux function. Moreover, in the case of a smooth flux function, we characterize the generic bifurcations arising at a coalescence event with and without the presence of odd symmetry. We identify associated scaling laws describing the local interface dynamics near collision. Finally, we present an extension of these results to the case of antishock waves converging to asymptotic limits of opposite signs. Our analysis is corroborated by numerical simulations of the modular Burgers equation.
{"title":"On the Extinction of Multiple Shocks in Scalar Viscous Conservation Laws","authors":"Jeanne Lin, Dmitry E. Pelinovsky, Björn de Rijk","doi":"10.1137/24m1640628","DOIUrl":"https://doi.org/10.1137/24m1640628","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2323-2363, September 2024. <br/> Abstract.We are interested in the dynamics of interfaces, or zeros, of shock waves in general scalar viscous conservation laws with a locally Lipschitz continuous flux function, such as the modular Burgers equation. We prove that all interfaces coalesce within finite time, leaving behind either a single interface or no interface at all. Our proof relies on mass and energy estimates, regularization of the flux function, and an application of the Sturm theorems on the number of zeros of solutions of parabolic problems. Our analysis yields an explicit upper bound on the time of extinction in terms of the initial condition and the flux function. Moreover, in the case of a smooth flux function, we characterize the generic bifurcations arising at a coalescence event with and without the presence of odd symmetry. We identify associated scaling laws describing the local interface dynamics near collision. Finally, we present an extension of these results to the case of antishock waves converging to asymptotic limits of opposite signs. Our analysis is corroborated by numerical simulations of the modular Burgers equation.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191836","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 2293-2322, September 2024. Abstract.We investigate the collective dynamics of a network of heterogeneous Izhikevich neurons with global constant-delay coupling using a mean-field approximation, valid in the thermodynamic limit. The introduction of a biologically motivated synaptic current expression and a spike frequency adaptation mechanism give rise to significantly different bifurcation structures. Our study emphasizes the impact of heterogeneity in the quenched current, adaptation intensity, and synaptic delay on the emergence of collective oscillations. The effects of heterogeneity and adaptation vary across different scenarios but essentially result from the balance of excitatory drives, including input currents that cause neurons to spike, adaptation currents that terminate spiking, and synaptic currents that predominantly favor spiking in excitatory networks but hinder it in inhibitory cases. Our perturbation and bifurcation analysis reveal interesting transitions in the behavior in both limits of extremely weak heterogeneity and coupling strength. Finally, our analysis indicates that synaptic delays exhibit little impact on the generation of collective oscillations in weakly coupled heterogeneous networks. This effect becomes more pronounced with increasing heterogeneity. Moreover, a larger delay does not necessarily enhance the likelihood of oscillations, especially in weakly adapting neural networks. Beyond that, delays primarily function as an excitatory drive, promoting the emergence of oscillations and even inducing new macroscopic dynamics. Specifically, torus bifurcations may occur in a single population of neurons without an external drive, serving as a crucial mechanism for the emergence of population bursting with two nested frequencies.
{"title":"Population Dynamics in Networks of Izhikevich Neurons with Global Delayed Coupling","authors":"Liang Chen, Sue Ann Campbell","doi":"10.1137/24m1631146","DOIUrl":"https://doi.org/10.1137/24m1631146","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2293-2322, September 2024. <br/> Abstract.We investigate the collective dynamics of a network of heterogeneous Izhikevich neurons with global constant-delay coupling using a mean-field approximation, valid in the thermodynamic limit. The introduction of a biologically motivated synaptic current expression and a spike frequency adaptation mechanism give rise to significantly different bifurcation structures. Our study emphasizes the impact of heterogeneity in the quenched current, adaptation intensity, and synaptic delay on the emergence of collective oscillations. The effects of heterogeneity and adaptation vary across different scenarios but essentially result from the balance of excitatory drives, including input currents that cause neurons to spike, adaptation currents that terminate spiking, and synaptic currents that predominantly favor spiking in excitatory networks but hinder it in inhibitory cases. Our perturbation and bifurcation analysis reveal interesting transitions in the behavior in both limits of extremely weak heterogeneity and coupling strength. Finally, our analysis indicates that synaptic delays exhibit little impact on the generation of collective oscillations in weakly coupled heterogeneous networks. This effect becomes more pronounced with increasing heterogeneity. Moreover, a larger delay does not necessarily enhance the likelihood of oscillations, especially in weakly adapting neural networks. Beyond that, delays primarily function as an excitatory drive, promoting the emergence of oscillations and even inducing new macroscopic dynamics. Specifically, torus bifurcations may occur in a single population of neurons without an external drive, serving as a crucial mechanism for the emergence of population bursting with two nested frequencies.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191837","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}
William Duncan, Fernando Antoneli, Janet Best, Martin Golubitsky, Jiaxin Jin, H. Frederik Nijhout, Mike Reed, Ian Stewart
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2262-2292, September 2024. Abstract.Homeostasis is a regulatory mechanism that keeps a specific variable close to a set value as other variables fluctuate. The notion of homeostasis can be rigorously formulated when the model of interest is represented as an input-output network, with distinguished input and output nodes, and the dynamics of the network determines the corresponding input-output function of the system. In this context, homeostasis can be defined as an “infinitesimal” notion, namely, the derivative of the input-output function is zero at an isolated point. Combining this approach with graph-theoretic ideas from combinatorial matrix theory provides a systematic framework for calculating homeostasis points in models and classifying the different homeostasis types in input-output networks. In this paper we extend this theory by introducing the notion of a homeostasis pattern, defined as a set of nodes, in addition to the output node, that are simultaneously infinitesimally homeostatic. We prove that each homeostasis type leads to a distinct homeostasis pattern. Moreover, we describe all homeostasis patterns supported by a given input-output network in terms of a combinatorial structure associated to the input-output network. We call this structure the homeostasis pattern network.
{"title":"Homeostasis Patterns","authors":"William Duncan, Fernando Antoneli, Janet Best, Martin Golubitsky, Jiaxin Jin, H. Frederik Nijhout, Mike Reed, Ian Stewart","doi":"10.1137/23m158807x","DOIUrl":"https://doi.org/10.1137/23m158807x","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2262-2292, September 2024. <br/> Abstract.Homeostasis is a regulatory mechanism that keeps a specific variable close to a set value as other variables fluctuate. The notion of homeostasis can be rigorously formulated when the model of interest is represented as an input-output network, with distinguished input and output nodes, and the dynamics of the network determines the corresponding input-output function of the system. In this context, homeostasis can be defined as an “infinitesimal” notion, namely, the derivative of the input-output function is zero at an isolated point. Combining this approach with graph-theoretic ideas from combinatorial matrix theory provides a systematic framework for calculating homeostasis points in models and classifying the different homeostasis types in input-output networks. In this paper we extend this theory by introducing the notion of a homeostasis pattern, defined as a set of nodes, in addition to the output node, that are simultaneously infinitesimally homeostatic. We prove that each homeostasis type leads to a distinct homeostasis pattern. Moreover, we describe all homeostasis patterns supported by a given input-output network in terms of a combinatorial structure associated to the input-output network. We call this structure the homeostasis pattern network.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191839","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}
P. Glendinning, S. J. Hogan, M. E. Homer, M. R. Jeffrey, R. Szalai
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2242-2261, September 2024. Abstract.We classify Filippov’s type 3 singular points of planar bimodal piecewise smooth systems. These singular points consist of fold or cusp tangencies of the vector fields to both sides of a switching surface. For isolated analytic type 3 singular points there are 25 topological classes, up to time reversal. For isolated general type 3 singular points there are 40 topological classes, up to time reversal.
{"title":"Classification of Filippov Type 3 Singular Points in Planar Bimodal Piecewise Smooth Systems","authors":"P. Glendinning, S. J. Hogan, M. E. Homer, M. R. Jeffrey, R. Szalai","doi":"10.1137/23m1622842","DOIUrl":"https://doi.org/10.1137/23m1622842","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2242-2261, September 2024. <br/> Abstract.We classify Filippov’s type 3 singular points of planar bimodal piecewise smooth systems. These singular points consist of fold or cusp tangencies of the vector fields to both sides of a switching surface. For isolated analytic type 3 singular points there are 25 topological classes, up to time reversal. For isolated general type 3 singular points there are 40 topological classes, up to time reversal.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191838","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}
A. Borovykh, N. Kantas, P. Parpas, G. A. Pavliotis
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2208-2241, September 2024. Abstract.The mirror descent algorithm is known to be effective in situations where it is beneficial to adapt the mirror map to the underlying geometry of the optimization model. However, the effect of mirror maps on the geometry of distributed optimization problems has not been previously addressed. In this paper we study an exact distributed mirror descent algorithm in continuous time under additive noise. We establish a linear convergence rate of the proposed dynamics for the setting of convex optimization. Our analysis draws motivation from the augmented Lagrangian and its relation to gradient tracking. To further explore the benefits of mirror maps in a distributed setting we present a preconditioned variant of our algorithm with an additional mirror map over the Lagrangian dual variables. This allows our method to adapt to both the geometry of the primal variables and the geometry of the consensus constraint. We also propose a Gauss–Seidel type discretization scheme for the proposed method and establish its linear convergence rate. For certain classes of problems we identify mirror maps that mitigate the effect of the graph’s spectral properties on the convergence rate of the algorithm. Using numerical experiments, we demonstrate the efficiency of the methodology on convex models, both with and without constraints. Our findings show that the proposed method outperforms other methods, especially in scenarios where the model’s geometry is not captured by the standard Euclidean norm.
{"title":"Stochastic Mirror Descent for Convex Optimization with Consensus Constraints","authors":"A. Borovykh, N. Kantas, P. Parpas, G. A. Pavliotis","doi":"10.1137/22m1515197","DOIUrl":"https://doi.org/10.1137/22m1515197","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2208-2241, September 2024. <br/> Abstract.The mirror descent algorithm is known to be effective in situations where it is beneficial to adapt the mirror map to the underlying geometry of the optimization model. However, the effect of mirror maps on the geometry of distributed optimization problems has not been previously addressed. In this paper we study an exact distributed mirror descent algorithm in continuous time under additive noise. We establish a linear convergence rate of the proposed dynamics for the setting of convex optimization. Our analysis draws motivation from the augmented Lagrangian and its relation to gradient tracking. To further explore the benefits of mirror maps in a distributed setting we present a preconditioned variant of our algorithm with an additional mirror map over the Lagrangian dual variables. This allows our method to adapt to both the geometry of the primal variables and the geometry of the consensus constraint. We also propose a Gauss–Seidel type discretization scheme for the proposed method and establish its linear convergence rate. For certain classes of problems we identify mirror maps that mitigate the effect of the graph’s spectral properties on the convergence rate of the algorithm. Using numerical experiments, we demonstrate the efficiency of the methodology on convex models, both with and without constraints. Our findings show that the proposed method outperforms other methods, especially in scenarios where the model’s geometry is not captured by the standard Euclidean norm.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943474","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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