Society for Industrial and Applied Mathematics: SIAM Journal on Applied Dynamical Systems: Table of Contents
Table of Contents for SIAM Journal on Applied Dynamical Systems. List of articles from both the latest and ahead of print issues.
https://epubs.siam.org/loi/sjaday?af=R
Society for Industrial and Applied Mathematics: SIAM Journal on Applied Dynamical Systems: Table of Contents
Society for Industrial and Applied Mathematics
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SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg
https://epubs.siam.org/loi/sjaday?af=R

Closed Geodesics on Weingarten Surfaces with [math]
https://epubs.siam.org/doi/abs/10.1137/23M1608616?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 17051719, September 2024. <br/> Abstract.In 2006, Alexander proved a result that implies for a Weingarten surface [math], if [math] is the number of times a closed geodesic winds around the axis of rotation and [math] is the number of times the geodesic oscillates about the equator, then [math] when [math] and [math] when [math]. In this paper, we present another proof of Alexander’s result for the Weingarten surfaces [math] that is simpler and more direct. Our approach uses sharp estimates of certain improper integrals to obtain the intervals for permissible ratios [math]. We numerically compute a number of closed geodesics for various combinations of [math] to illustrate the variety of patterns that are possible.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 17051719, September 2024. <br/> Abstract.In 2006, Alexander proved a result that implies for a Weingarten surface [math], if [math] is the number of times a closed geodesic winds around the axis of rotation and [math] is the number of times the geodesic oscillates about the equator, then [math] when [math] and [math] when [math]. In this paper, we present another proof of Alexander’s result for the Weingarten surfaces [math] that is simpler and more direct. Our approach uses sharp estimates of certain improper integrals to obtain the intervals for permissible ratios [math]. We numerically compute a number of closed geodesics for various combinations of [math] to illustrate the variety of patterns that are possible. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Closed Geodesics on Weingarten Surfaces with [math]
10.1137/23M1608616
SIAM Journal on Applied Dynamical Systems
20240703T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Frank E. Baginski
Valério Ramos Batista
Closed Geodesics on Weingarten Surfaces with [math]
23
3
1705
1719
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1608616
https://epubs.siam.org/doi/abs/10.1137/23M1608616?af=R
© 2024 Society for Industrial and Applied Mathematics

Complete and Partial Synchronization of TwoGroup and ThreeGroup Kuramoto Oscillators
https://epubs.siam.org/doi/abs/10.1137/23M1586227?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 17201765, September 2024. <br/> Abstract.This paper is to investigate synchronization theories of a twogroup Kuramoto model and a threegroup Kuramoto model. In the settings of these models, every oscillator directly interacts with each other in the same group. In each group, only one oscillator directly interacts with one oscillator in another group. We prove that if the coupling strength is large and the initial configuration of each group is confined to a sector with the arc length less than [math], then all oscillators achieve a complete frequency synchronization asymptotically. We emphasize that there is no need to impose any initial condition on the connection between different groups. If, in addition, the natural frequencies in one group are the same, then partial phase synchronization occurs. Moreover, if all natural frequencies are identical, we prove that all oscillators either achieve a complete phase synchronization asymptotically or tend to a bipolar phaselocking state. We also provide several numerical simulations to support the main results.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 17201765, September 2024. <br/> Abstract.This paper is to investigate synchronization theories of a twogroup Kuramoto model and a threegroup Kuramoto model. In the settings of these models, every oscillator directly interacts with each other in the same group. In each group, only one oscillator directly interacts with one oscillator in another group. We prove that if the coupling strength is large and the initial configuration of each group is confined to a sector with the arc length less than [math], then all oscillators achieve a complete frequency synchronization asymptotically. We emphasize that there is no need to impose any initial condition on the connection between different groups. If, in addition, the natural frequencies in one group are the same, then partial phase synchronization occurs. Moreover, if all natural frequencies are identical, we prove that all oscillators either achieve a complete phase synchronization asymptotically or tend to a bipolar phaselocking state. We also provide several numerical simulations to support the main results.<p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Complete and Partial Synchronization of TwoGroup and ThreeGroup Kuramoto Oscillators
10.1137/23M1586227
SIAM Journal on Applied Dynamical Systems
20240709T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
ShihHsin Chen
ChunHsiung Hsia
TingYang Hsiao
Complete and Partial Synchronization of TwoGroup and ThreeGroup Kuramoto Oscillators
23
3
1720
1765
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1586227
https://epubs.siam.org/doi/abs/10.1137/23M1586227?af=R
© 2024 Society for Industrial and Applied Mathematics

Weighted Birkhoff Averages and the Parameterization Method
https://epubs.siam.org/doi/abs/10.1137/23M1579546?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 17661804, 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.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 17661804, 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. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Weighted Birkhoff Averages and the Parameterization Method
10.1137/23M1579546
SIAM Journal on Applied Dynamical Systems
20240710T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
David Blessing
J. D. Mireles James
Weighted Birkhoff Averages and the Parameterization Method
23
3
1766
1804
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1579546
https://epubs.siam.org/doi/abs/10.1137/23M1579546?af=R
© 2024 Society for Industrial and Applied Mathematics

Inertial Focusing Dynamics of Spherical Particles in Curved Microfluidic Ducts with a Trapezoidal Cross Section
https://epubs.siam.org/doi/abs/10.1137/23M1613220?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 18051835, 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 crosssection 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 crosssection 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.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 18051835, 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 crosssection 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 crosssection 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.<p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Inertial Focusing Dynamics of Spherical Particles in Curved Microfluidic Ducts with a Trapezoidal Cross Section
10.1137/23M1613220
SIAM Journal on Applied Dynamical Systems
20240712T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Brendan Harding
Yvonne M. Stokes
Rahil N. Valani
Inertial Focusing Dynamics of Spherical Particles in Curved Microfluidic Ducts with a Trapezoidal Cross Section
23
3
1805
1835
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1613220
https://epubs.siam.org/doi/abs/10.1137/23M1613220?af=R
© 2024 Society for Industrial and Applied Mathematics

Rate and Bifurcation Induced Transitions in Asymptotically SlowFast Systems
https://epubs.siam.org/doi/abs/10.1137/24M1632000?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 18361869, 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 slowfast 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 blowup 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 blowup, 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.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 18361869, 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 slowfast 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 blowup 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 blowup, 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. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Rate and Bifurcation Induced Transitions in Asymptotically SlowFast Systems
10.1137/24M1632000
SIAM Journal on Applied Dynamical Systems
20240715T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Samuel Jelbart
Rate and Bifurcation Induced Transitions in Asymptotically SlowFast Systems
23
3
1836
1869
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/24M1632000
https://epubs.siam.org/doi/abs/10.1137/24M1632000?af=R
© 2024 Society for Industrial and Applied Mathematics

Less Interaction with Forward Models in Langevin Dynamics: Enrichment and Homotopy
https://epubs.siam.org/doi/abs/10.1137/23M1546841?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 18701908, September 2024. <br/>Abstract.Ensemble methods have become ubiquitous for the solution of Bayesian inference problems. Stateoftheart Langevin samplers such as the ensemble Kalman sampler (EKS) or the affine invariant Langevin dynamics (ALDI) rely on successive evaluations of the forward model or its gradient. A main drawback of these methods hence is their vast number of required forward calls as well as their possible lack of convergence in the case of more involved posterior measures such as multimodal distributions. The goal of this paper is to address these challenges to some extent. First, several possible adaptive ensemble enrichment strategies that successively enlarge the number of particles in the underlying Langevin dynamics are discussed that in turn lead to a significant reduction of the total number of forward calls. Second, analytical consistency guarantees of the ensemble enrichment method are provided for linear forward models. Third, to address more involved target distributions, the method is extended by applying adapted Langevin dynamics based on a homotopy formalism for which convergence is proved. Finally, numerical investigations of several benchmark problems illustrate the possible gain of the proposed method, comparing it to stateoftheart Langevin samplers.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 18701908, September 2024. <br/>Abstract.Ensemble methods have become ubiquitous for the solution of Bayesian inference problems. Stateoftheart Langevin samplers such as the ensemble Kalman sampler (EKS) or the affine invariant Langevin dynamics (ALDI) rely on successive evaluations of the forward model or its gradient. A main drawback of these methods hence is their vast number of required forward calls as well as their possible lack of convergence in the case of more involved posterior measures such as multimodal distributions. The goal of this paper is to address these challenges to some extent. First, several possible adaptive ensemble enrichment strategies that successively enlarge the number of particles in the underlying Langevin dynamics are discussed that in turn lead to a significant reduction of the total number of forward calls. Second, analytical consistency guarantees of the ensemble enrichment method are provided for linear forward models. Third, to address more involved target distributions, the method is extended by applying adapted Langevin dynamics based on a homotopy formalism for which convergence is proved. Finally, numerical investigations of several benchmark problems illustrate the possible gain of the proposed method, comparing it to stateoftheart Langevin samplers. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Less Interaction with Forward Models in Langevin Dynamics: Enrichment and Homotopy
10.1137/23M1546841
SIAM Journal on Applied Dynamical Systems
20240715T07:00:00Z
© 2024 The Authors
Martin Eigel
Robert Gruhlke
David Sommer
Less Interaction with Forward Models in Langevin Dynamics: Enrichment and Homotopy
23
3
1870
1908
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1546841
https://epubs.siam.org/doi/abs/10.1137/23M1546841?af=R
© 2024 The Authors

Hopf Bifurcations of Two Population Neural Fields on the Sphere with Diffusion and Distributed Delays
https://epubs.siam.org/doi/abs/10.1137/23M1554011?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 19091945, September 2024. <br/> Abstract.A common model for studying pattern formation in large groups of neurons is the neural field. We investigate a neural field with excitatory and inhibitory neurons, like [H. R. Wilson and J. D. Cowan (1972), Biophys. J., 12, pp. 1–24], with transmission delays and gap junctions. We build on the work of [S. Visser, R. Nicks, O. Faugeras, and S. Coombes (2017), Phys. D, 349, pp. 27–45] by investigating pattern formation in these models on the sphere. Specifically, we investigate how gap junctions, modelled by a diffusion term, influence the behavior of the neural field. We look in detail at the periodic orbits that are generated by Hopf bifurcations in the presence of spherical symmetry. To this end, we derive general formulas to compute the normal form coefficients for these bifurcations up to third order and predict the stability of the resulting branches. A novel numerical method to solve delay equations with diffusion on the sphere is formulated and applied in simulations.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 19091945, September 2024. <br/> Abstract.A common model for studying pattern formation in large groups of neurons is the neural field. We investigate a neural field with excitatory and inhibitory neurons, like [H. R. Wilson and J. D. Cowan (1972), Biophys. J., 12, pp. 1–24], with transmission delays and gap junctions. We build on the work of [S. Visser, R. Nicks, O. Faugeras, and S. Coombes (2017), Phys. D, 349, pp. 27–45] by investigating pattern formation in these models on the sphere. Specifically, we investigate how gap junctions, modelled by a diffusion term, influence the behavior of the neural field. We look in detail at the periodic orbits that are generated by Hopf bifurcations in the presence of spherical symmetry. To this end, we derive general formulas to compute the normal form coefficients for these bifurcations up to third order and predict the stability of the resulting branches. A novel numerical method to solve delay equations with diffusion on the sphere is formulated and applied in simulations. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Hopf Bifurcations of Two Population Neural Fields on the Sphere with Diffusion and Distributed Delays
10.1137/23M1554011
SIAM Journal on Applied Dynamical Systems
20240715T07:00:00Z
© 2024 Len Spek
Len Spek
Stephan A. van Gils
Yuri A. Kuznetsov
Mónika Polner
Hopf Bifurcations of Two Population Neural Fields on the Sphere with Diffusion and Distributed Delays
23
3
1909
1945
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1554011
https://epubs.siam.org/doi/abs/10.1137/23M1554011?af=R
© 2024 Len Spek

Transverse Lyapunov Exponent and Chimeras in Globally Coupled Maps
https://epubs.siam.org/doi/abs/10.1137/23M1603339?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 19461965, September 2024. <br/> Abstract.We study the stability properties and the longterm 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.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 19461965, September 2024. <br/> Abstract.We study the stability properties and the longterm 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. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Transverse Lyapunov Exponent and Chimeras in Globally Coupled Maps
10.1137/23M1603339
SIAM Journal on Applied Dynamical Systems
20240719T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Théophile Caby
Pierre Guiraud
Transverse Lyapunov Exponent and Chimeras in Globally Coupled Maps
23
3
1946
1965
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1603339
https://epubs.siam.org/doi/abs/10.1137/23M1603339?af=R
© 2024 Society for Industrial and Applied Mathematics

Rigorous Computation of Solutions of Semilinear PDEs on Unbounded Domains via Spectral Methods
https://epubs.siam.org/doi/abs/10.1137/23M1607507?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 19662017, 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 computerassisted 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 finitedimensional 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.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 19662017, 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 computerassisted 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 finitedimensional 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. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Rigorous Computation of Solutions of Semilinear PDEs on Unbounded Domains via Spectral Methods
10.1137/23M1607507
SIAM Journal on Applied Dynamical Systems
20240722T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Matthieu Cadiot
JeanPhilippe Lessard
JeanChristophe Nave
Rigorous Computation of Solutions of Semilinear PDEs on Unbounded Domains via Spectral Methods
23
3
1966
2017
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1607507
https://epubs.siam.org/doi/abs/10.1137/23M1607507?af=R
© 2024 Society for Industrial and Applied Mathematics

The NucleationAnnihilation Dynamics of Hotspot Patterns for a ReactionDiffusion System of Urban Crime with Police Deployment
https://epubs.siam.org/doi/abs/10.1137/23M1562330?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 20182060, September 2024. <br/> Abstract. A hybrid asymptoticnumerical approach is developed to study the existence and linear stability of steadystate hotspot patterns for a threecomponent onedimensional reactiondiffusion (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 spatiotemporal 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 steadystate solution. In addition, hotspots can be nucleated from a quiescent background when the criminal diffusivity is below a saddlenode 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 twoboundary hotspot pattern is computed, and instability thresholds due to either zeroeigenvalue 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 twoboundary 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 onehotspot steady state, we show that hotspot patterns undergo complex “nucleationannihilation” dynamics that are characterized by largescale persistent oscillations of the hotspot amplitudes. In this way, our results identify parameter ranges in the threecomponent 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 spatialtemporal distribution of crime is largely intractable.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 20182060, September 2024. <br/> Abstract. A hybrid asymptoticnumerical approach is developed to study the existence and linear stability of steadystate hotspot patterns for a threecomponent onedimensional reactiondiffusion (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 spatiotemporal 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 steadystate solution. In addition, hotspots can be nucleated from a quiescent background when the criminal diffusivity is below a saddlenode 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 twoboundary hotspot pattern is computed, and instability thresholds due to either zeroeigenvalue 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 twoboundary 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 onehotspot steady state, we show that hotspot patterns undergo complex “nucleationannihilation” dynamics that are characterized by largescale persistent oscillations of the hotspot amplitudes. In this way, our results identify parameter ranges in the threecomponent 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 spatialtemporal distribution of crime is largely intractable. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
The NucleationAnnihilation Dynamics of Hotspot Patterns for a ReactionDiffusion System of Urban Crime with Police Deployment
10.1137/23M1562330
SIAM Journal on Applied Dynamical Systems
20240729T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Chunyi Gai
Michael J. Ward
The NucleationAnnihilation Dynamics of Hotspot Patterns for a ReactionDiffusion System of Urban Crime with Police Deployment
23
3
2018
2060
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1562330
https://epubs.siam.org/doi/abs/10.1137/23M1562330?af=R
© 2024 Society for Industrial and Applied Mathematics

Stochastic EnergyBalance Model With A Moving Ice Line
https://epubs.siam.org/doi/abs/10.1137/23M1619873?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 20612098, September 2024. <br/> Abstract.In [SIAM J. Appl. Dyn. Syst., 12 (2013), pp. 2068–2092], Widiasih proposed and analyzed a deterministic onedimensional Budyko–Sellers energybalance 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 slowfast 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 coexisting (metastable) climate states as well as the transition dynamics between them.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 20612098, September 2024. <br/> Abstract.In [SIAM J. Appl. Dyn. Syst., 12 (2013), pp. 2068–2092], Widiasih proposed and analyzed a deterministic onedimensional Budyko–Sellers energybalance 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 slowfast 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 coexisting (metastable) climate states as well as the transition dynamics between them. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Stochastic EnergyBalance Model With A Moving Ice Line
10.1137/23M1619873
SIAM Journal on Applied Dynamical Systems
20240729T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Ilya Pavlyukevich
Marian Ritsch
Stochastic EnergyBalance Model With A Moving Ice Line
23
3
2061
2098
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1619873
https://epubs.siam.org/doi/abs/10.1137/23M1619873?af=R
© 2024 Society for Industrial and Applied Mathematics

A Geometric Singular Perturbation Analysis of Shock Selection Rules in Composite Regularized ReactionNonlinear Diffusion Models
https://epubs.siam.org/doi/abs/10.1137/23M1591803?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 20992137, September 2024. <br/> Abstract.Reactionnonlinear 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 steepfronted traveling wave solutions in RND PDE models with forwardbackwardforward 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 steepfronted traveling waves which limit to their shockfronted 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 shockfronted 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 piecewisesmooth 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 shockfronted solutions, namely, nonmonotone shockfronted waves as well as shockfronted 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” steepfronted 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 multiplescale nature of the composite regularized RND PDE model continues to play an important role in the numerical analysis of the spatial eigenvalue problem.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 20992137, September 2024. <br/> Abstract.Reactionnonlinear 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 steepfronted traveling wave solutions in RND PDE models with forwardbackwardforward 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 steepfronted traveling waves which limit to their shockfronted 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 shockfronted 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 piecewisesmooth 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 shockfronted solutions, namely, nonmonotone shockfronted waves as well as shockfronted 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” steepfronted 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 multiplescale nature of the composite regularized RND PDE model continues to play an important role in the numerical analysis of the spatial eigenvalue problem.<p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
A Geometric Singular Perturbation Analysis of Shock Selection Rules in Composite Regularized ReactionNonlinear Diffusion Models
10.1137/23M1591803
SIAM Journal on Applied Dynamical Systems
20240729T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Bronwyn H. BradshawHajek
Ian Lizarraga
Robert Marangell
Martin Wechselberger
A Geometric Singular Perturbation Analysis of Shock Selection Rules in Composite Regularized ReactionNonlinear Diffusion Models
23
3
2099
2137
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1591803
https://epubs.siam.org/doi/abs/10.1137/23M1591803?af=R
© 2024 Society for Industrial and Applied Mathematics

A Mathematical Model of the Visual MacKay Effect
https://epubs.siam.org/doi/abs/10.1137/23M1616686?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 21382178, 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 inputoutput controllability of an Amaritype neural fields model. In this framework, we consider sensory input as a control function, a cortical representation via the retinocortical 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 Amaritype equation’s exact controllability property is discussed for linear and nonlinear response functions. For the visual MacKay effect modeling, we adjust the parameter representing intraneuron 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 afterimage reported by MacKay.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 21382178, 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 inputoutput controllability of an Amaritype neural fields model. In this framework, we consider sensory input as a control function, a cortical representation via the retinocortical 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 Amaritype equation’s exact controllability property is discussed for linear and nonlinear response functions. For the visual MacKay effect modeling, we adjust the parameter representing intraneuron 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 afterimage reported by MacKay. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
A Mathematical Model of the Visual MacKay Effect
10.1137/23M1616686
SIAM Journal on Applied Dynamical Systems
20240801T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Cyprien Tamekue
Dario Prandi
Yacine Chitour
A Mathematical Model of the Visual MacKay Effect
23
3
2138
2178
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1616686
https://epubs.siam.org/doi/abs/10.1137/23M1616686?af=R
© 2024 Society for Industrial and Applied Mathematics

GraphBased Sufficient Conditions for the Indistinguishability of Linear Compartmental Models
https://epubs.siam.org/doi/abs/10.1137/23M1614663?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 21792207, 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 realworld phenomena. However, there may not be a unique model representing that realworld 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 mediumtolarge 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.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 21792207, 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 realworld phenomena. However, there may not be a unique model representing that realworld 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 mediumtolarge 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. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
GraphBased Sufficient Conditions for the Indistinguishability of Linear Compartmental Models
10.1137/23M1614663
SIAM Journal on Applied Dynamical Systems
20240802T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Cashous Bortner
Nicolette Meshkat
GraphBased Sufficient Conditions for the Indistinguishability of Linear Compartmental Models
23
3
2179
2207
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1614663
https://epubs.siam.org/doi/abs/10.1137/23M1614663?af=R
© 2024 Society for Industrial and Applied Mathematics

Stochastic Mirror Descent for Convex Optimization with Consensus Constraints
https://epubs.siam.org/doi/abs/10.1137/22M1515197?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 22082241, 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.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 22082241, 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. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Stochastic Mirror Descent for Convex Optimization with Consensus Constraints
10.1137/22M1515197
SIAM Journal on Applied Dynamical Systems
20240805T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
A. Borovykh
N. Kantas
P. Parpas
G. A. Pavliotis
Stochastic Mirror Descent for Convex Optimization with Consensus Constraints
23
3
2208
2241
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/22M1515197
https://epubs.siam.org/doi/abs/10.1137/22M1515197?af=R
© 2024 Society for Industrial and Applied Mathematics

Classification of Filippov Type 3 Singular Points in Planar Bimodal Piecewise Smooth Systems
https://epubs.siam.org/doi/abs/10.1137/23M1622842?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 22422261, 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.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 22422261, 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. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Classification of Filippov Type 3 Singular Points in Planar Bimodal Piecewise Smooth Systems
10.1137/23M1622842
SIAM Journal on Applied Dynamical Systems
20240820T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
P. Glendinning
S. J. Hogan
M. E. Homer
M. R. Jeffrey
R. Szalai
Classification of Filippov Type 3 Singular Points in Planar Bimodal Piecewise Smooth Systems
23
3
2242
2261
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1622842
https://epubs.siam.org/doi/abs/10.1137/23M1622842?af=R
© 2024 Society for Industrial and Applied Mathematics

Homeostasis Patterns
https://epubs.siam.org/doi/abs/10.1137/23M158807X?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 22622292, 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 inputoutput network, with distinguished input and output nodes, and the dynamics of the network determines the corresponding inputoutput function of the system. In this context, homeostasis can be defined as an “infinitesimal” notion, namely, the derivative of the inputoutput function is zero at an isolated point. Combining this approach with graphtheoretic ideas from combinatorial matrix theory provides a systematic framework for calculating homeostasis points in models and classifying the different homeostasis types in inputoutput 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 inputoutput network in terms of a combinatorial structure associated to the inputoutput network. We call this structure the homeostasis pattern network.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 22622292, 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 inputoutput network, with distinguished input and output nodes, and the dynamics of the network determines the corresponding inputoutput function of the system. In this context, homeostasis can be defined as an “infinitesimal” notion, namely, the derivative of the inputoutput function is zero at an isolated point. Combining this approach with graphtheoretic ideas from combinatorial matrix theory provides a systematic framework for calculating homeostasis points in models and classifying the different homeostasis types in inputoutput 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 inputoutput network in terms of a combinatorial structure associated to the inputoutput network. We call this structure the homeostasis pattern network. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Homeostasis Patterns
10.1137/23M158807X
SIAM Journal on Applied Dynamical Systems
20240827T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
William Duncan
Fernando Antoneli
Janet Best
Martin Golubitsky
Jiaxin Jin
H. Frederik Nijhout
Mike Reed
Ian Stewart
Homeostasis Patterns
23
3
2262
2292
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M158807X
https://epubs.siam.org/doi/abs/10.1137/23M158807X?af=R
© 2024 Society for Industrial and Applied Mathematics

Population Dynamics in Networks of Izhikevich Neurons with Global Delayed Coupling
https://epubs.siam.org/doi/abs/10.1137/24M1631146?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 22932322, September 2024. <br/> Abstract.We investigate the collective dynamics of a network of heterogeneous Izhikevich neurons with global constantdelay coupling using a meanfield 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.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 22932322, September 2024. <br/> Abstract.We investigate the collective dynamics of a network of heterogeneous Izhikevich neurons with global constantdelay coupling using a meanfield 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. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Population Dynamics in Networks of Izhikevich Neurons with Global Delayed Coupling
10.1137/24M1631146
SIAM Journal on Applied Dynamical Systems
20240827T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Liang Chen
Sue Ann Campbell
Population Dynamics in Networks of Izhikevich Neurons with Global Delayed Coupling
23
3
2293
2322
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/24M1631146
https://epubs.siam.org/doi/abs/10.1137/24M1631146?af=R
© 2024 Society for Industrial and Applied Mathematics

On the Extinction of Multiple Shocks in Scalar Viscous Conservation Laws
https://epubs.siam.org/doi/abs/10.1137/24M1640628?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 23232363, 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.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 23232363, 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. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
On the Extinction of Multiple Shocks in Scalar Viscous Conservation Laws
10.1137/24M1640628
SIAM Journal on Applied Dynamical Systems
20240830T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Jeanne Lin
Dmitry E. Pelinovsky
Björn de Rijk
On the Extinction of Multiple Shocks in Scalar Viscous Conservation Laws
23
3
2323
2363
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/24M1640628
https://epubs.siam.org/doi/abs/10.1137/24M1640628?af=R
© 2024 Society for Industrial and Applied Mathematics

On the Convergence of Nonlinear Averaging Dynamics with ThreeBody Interactions on Hypergraphs
https://epubs.siam.org/doi/abs/10.1137/23M1568338?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 23642406, 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 discretetime dynamics with threebody interactions, described by an underlying 3uniform 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 higherorder 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 twobody 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.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 23642406, 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 discretetime dynamics with threebody interactions, described by an underlying 3uniform 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 higherorder 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 twobody 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. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
On the Convergence of Nonlinear Averaging Dynamics with ThreeBody Interactions on Hypergraphs
10.1137/23M1568338
SIAM Journal on Applied Dynamical Systems
20240904T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Emilio Cruciani
Emanuela L. Giacomelli
Jinyeop Lee
On the Convergence of Nonlinear Averaging Dynamics with ThreeBody Interactions on Hypergraphs
23
3
2364
2406
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1568338
https://epubs.siam.org/doi/abs/10.1137/23M1568338?af=R
© 2024 Society for Industrial and Applied Mathematics

Forward Attraction of Nonautonomous Dynamical Systems and Applications to Navier–Stokes Equations
https://epubs.siam.org/doi/abs/10.1137/23M1626384?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 24072443, 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 wellknown 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 timedependent sets were studied also. A sufficient condition and a necessary condition for a timedependent 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 timedependent forcing were studied as applications.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 24072443, 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 wellknown 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 timedependent sets were studied also. A sufficient condition and a necessary condition for a timedependent 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 timedependent forcing were studied as applications. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Forward Attraction of Nonautonomous Dynamical Systems and Applications to Navier–Stokes Equations
10.1137/23M1626384
SIAM Journal on Applied Dynamical Systems
20240912T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Hongyong Cui
Rodiak N. FigueroaLópez
José A. Langa
Marcelo J. D. Nascimento
Forward Attraction of Nonautonomous Dynamical Systems and Applications to Navier–Stokes Equations
23
3
2407
2443
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1626384
https://epubs.siam.org/doi/abs/10.1137/23M1626384?af=R
© 2024 Society for Industrial and Applied Mathematics

Hawkes Process Modelling for Chemical Reaction Networks in a Random Environment
https://epubs.siam.org/doi/abs/10.1137/23M1588573?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 24442488, 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 cosimulation of the environment, (ii) obtaining new process equations from which moment equations can be derived, (iii) the computation of informationtheoretic quantities, and (iv) state estimation. It is known since Snyder’s and related works that marginalization over a stochastic intensity turns point processes into selfexciting ones. While the Snyder filter specifies the exact historydependent 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 selfexciting 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 secondorder statistics, i.e., covariance, auto/crosscorrelation. We introduce an approximate marginal simulation algorithm and illustrate it in case studies.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 24442488, 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 cosimulation of the environment, (ii) obtaining new process equations from which moment equations can be derived, (iii) the computation of informationtheoretic quantities, and (iv) state estimation. It is known since Snyder’s and related works that marginalization over a stochastic intensity turns point processes into selfexciting ones. While the Snyder filter specifies the exact historydependent 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 selfexciting 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 secondorder statistics, i.e., covariance, auto/crosscorrelation. We introduce an approximate marginal simulation algorithm and illustrate it in case studies. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Hawkes Process Modelling for Chemical Reaction Networks in a Random Environment
10.1137/23M1588573
SIAM Journal on Applied Dynamical Systems
20240912T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Mark SinzgerD’Angelo
Jan Hasenauer
Heinz Koeppl
Hawkes Process Modelling for Chemical Reaction Networks in a Random Environment
23
3
2444
2488
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1588573
https://epubs.siam.org/doi/abs/10.1137/23M1588573?af=R
© 2024 Society for Industrial and Applied Mathematics

Reduction and Reconstruction of the Oscillator in 1:1:2 Resonance plus an Axially Symmetric Polynomial Perturbation
https://epubs.siam.org/doi/abs/10.1137/23M1621885?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 24892532, 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 higherorder 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 3tori 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 twicereduced space is done to characterize the singularities. Precisely, we study the critical points (relative equilibria) on the twicereduced space together with the stability, and parametric bifurcations are determined. The equilibria occurring in the twicereduced space are reconstructed as 3tori filled by quasiperiodic 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.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 24892532, 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 higherorder 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 3tori 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 twicereduced space is done to characterize the singularities. Precisely, we study the critical points (relative equilibria) on the twicereduced space together with the stability, and parametric bifurcations are determined. The equilibria occurring in the twicereduced space are reconstructed as 3tori filled by quasiperiodic 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. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Reduction and Reconstruction of the Oscillator in 1:1:2 Resonance plus an Axially Symmetric Polynomial Perturbation
10.1137/23M1621885
SIAM Journal on Applied Dynamical Systems
20240913T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Yocelyn Pérez Rothen
Claudio Vidal
Reduction and Reconstruction of the Oscillator in 1:1:2 Resonance plus an Axially Symmetric Polynomial Perturbation
23
3
2489
2532
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1621885
https://epubs.siam.org/doi/abs/10.1137/23M1621885?af=R
© 2024 Society for Industrial and Applied Mathematics

Global Dynamics of Piecewise Smooth Systems with Switches Depending on Both Discrete Times and Status
https://epubs.siam.org/doi/abs/10.1137/24M1634941?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 25332556, 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 onoff control. Numerical simulations are performed to verify our results.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 25332556, 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 onoff control. Numerical simulations are performed to verify our results. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Global Dynamics of Piecewise Smooth Systems with Switches Depending on Both Discrete Times and Status
10.1137/24M1634941
SIAM Journal on Applied Dynamical Systems
20240917T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Lihong Huang
Jiafu Wang
Global Dynamics of Piecewise Smooth Systems with Switches Depending on Both Discrete Times and Status
23
3
2533
2556
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/24M1634941
https://epubs.siam.org/doi/abs/10.1137/24M1634941?af=R
© 2024 Society for Industrial and Applied Mathematics

Adapting InfoMap to Absorbing Random Walks Using AbsorptionScaled Graphs
https://epubs.siam.org/doi/abs/10.1137/21M1466803?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 25572592, September 2024. <br/> Abstract.InfoMap is a popular approach to detect densely connected “communities” of nodes in networks. To detect such communities, InfoMap uses random walks and ideas from information theory. Motivated by the dynamics of disease spread on networks, whose nodes can have heterogeneous diseaseremoval rates, we adapt InfoMap to absorbing random walks. To do this, we use absorptionscaled graphs (in which edge weights are scaled according to absorption rates) and Markov time sweeping. One of our adaptations of InfoMap converges to the standard version of InfoMap in the limit in which the nodeabsorption rates approach 0. We demonstrate that the community structure that one obtains using our adaptations of InfoMap can differ markedly from the community structure that one detects using methods that do not account for nodeabsorption rates. We also illustrate that the community structure that is induced by heterogeneous absorption rates can have important implications for susceptible–infected–recovered (SIR) dynamics on ringlattice networks. For example, in some situations, the outbreak duration is maximized when a moderate number of nodes have large nodeabsorption rates.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 25572592, September 2024. <br/> Abstract.InfoMap is a popular approach to detect densely connected “communities” of nodes in networks. To detect such communities, InfoMap uses random walks and ideas from information theory. Motivated by the dynamics of disease spread on networks, whose nodes can have heterogeneous diseaseremoval rates, we adapt InfoMap to absorbing random walks. To do this, we use absorptionscaled graphs (in which edge weights are scaled according to absorption rates) and Markov time sweeping. One of our adaptations of InfoMap converges to the standard version of InfoMap in the limit in which the nodeabsorption rates approach 0. We demonstrate that the community structure that one obtains using our adaptations of InfoMap can differ markedly from the community structure that one detects using methods that do not account for nodeabsorption rates. We also illustrate that the community structure that is induced by heterogeneous absorption rates can have important implications for susceptible–infected–recovered (SIR) dynamics on ringlattice networks. For example, in some situations, the outbreak duration is maximized when a moderate number of nodes have large nodeabsorption rates. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
Adapting InfoMap to Absorbing Random Walks Using AbsorptionScaled Graphs
10.1137/21M1466803
SIAM Journal on Applied Dynamical Systems
20240919T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Esteban Vargas Bernal
Mason A. Porter
Joseph H. Tien
Adapting InfoMap to Absorbing Random Walks Using AbsorptionScaled Graphs
23
3
2557
2592
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/21M1466803
https://epubs.siam.org/doi/abs/10.1137/21M1466803?af=R
© 2024 Society for Industrial and Applied Mathematics

On the Number of Limit Cycles for Piecewise Polynomial Holomorphic Systems
https://epubs.siam.org/doi/abs/10.1137/23M1620922?af=R
SIAM Journal on Applied Dynamical Systems, <a href="https://epubs.siam.org/toc/sjaday/23/3">Volume 23, Issue 3</a>, Page 25932622, September 2024. <br/> Abstract.In this paper, we are concerned with determining lower bounds of the number of limit cycles for piecewise polynomial holomorphic systems with a straight line of discontinuity. We approach this problem with different points of view. Initially, we study the number of zeros of the first and secondorder averaging functions. We also use the Lyapunov quantities to produce limit cycles appearing from a monodromic equilibrium point via a degenerated Andronov–Hopf type bifurcation, adding at the very end the sliding effects. Finally, we use the Poincaré–Miranda theorem for obtaining an explicit piecewise linear holomorphic system with 3 limit cycles, a result that improves the known examples in the literature that had a single limit cycle.
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 25932622, September 2024. <br/> Abstract.In this paper, we are concerned with determining lower bounds of the number of limit cycles for piecewise polynomial holomorphic systems with a straight line of discontinuity. We approach this problem with different points of view. Initially, we study the number of zeros of the first and secondorder averaging functions. We also use the Lyapunov quantities to produce limit cycles appearing from a monodromic equilibrium point via a degenerated Andronov–Hopf type bifurcation, adding at the very end the sliding effects. Finally, we use the Poincaré–Miranda theorem for obtaining an explicit piecewise linear holomorphic system with 3 limit cycles, a result that improves the known examples in the literature that had a single limit cycle. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaday/cover.jpg" alttext="cover image"/></p>
On the Number of Limit Cycles for Piecewise Polynomial Holomorphic Systems
10.1137/23M1620922
SIAM Journal on Applied Dynamical Systems
20240924T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Armengol Gasull
Gabriel Rondón
Paulo Ricardo da Silva
On the Number of Limit Cycles for Piecewise Polynomial Holomorphic Systems
23
3
2593
2622
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1620922
https://epubs.siam.org/doi/abs/10.1137/23M1620922?af=R
© 2024 Society for Industrial and Applied Mathematics