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 Showing 1 - 82 of 82 Journals sorted alphabetically Advances in Applied Mathematics       (Followers: 15) Advances in Applied Mathematics and Mechanics       (Followers: 6) Advances in Applied Mechanics       (Followers: 15) AKCE International Journal of Graphs and Combinatorics American Journal of Applied Mathematics and Statistics       (Followers: 11) American Journal of Applied Sciences       (Followers: 22) American Journal of Modeling and Optimization       (Followers: 3) Annals of Actuarial Science       (Followers: 2) Applied Mathematical Modelling       (Followers: 22) Applied Mathematics and Computation       (Followers: 31) Applied Mathematics and Mechanics       (Followers: 4) Applied Mathematics and Nonlinear Sciences Applied Mathematics and Physics       (Followers: 2) Biometrical Letters British Actuarial Journal       (Followers: 2) Bulletin of Mathematical Sciences and Applications Communication in Biomathematical Sciences       (Followers: 2) Communications in Applied and Industrial Mathematics       (Followers: 1) Communications on Applied Mathematics and Computation       (Followers: 1) Differential Geometry and its Applications       (Followers: 4) Discrete and Continuous Models and Applied Computational Science Discrete Applied Mathematics       (Followers: 10) Doğuş Üniversitesi Dergisi e-Journal of Analysis and Applied Mathematics Engineering Mathematics Letters       (Followers: 1) European Actuarial Journal Foundations and Trends® in Optimization       (Followers: 3) Frontiers in Applied Mathematics and Statistics       (Followers: 1) Fundamental Journal of Mathematics and Applications International Journal of Advances in Applied Mathematics and Modeling       (Followers: 1) International Journal of Applied Mathematics and Statistics       (Followers: 3) International Journal of Computer Mathematics : Computer Systems Theory International Journal of Data Mining, Modelling and Management       (Followers: 10) International Journal of Engineering Mathematics       (Followers: 7) International Journal of Fuzzy Systems International Journal of Swarm Intelligence       (Followers: 2) International Journal of Theoretical and Mathematical Physics       (Followers: 13) International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems       (Followers: 3) Journal of Advanced Mathematics and Applications       (Followers: 1) Journal of Advances in Mathematics and Computer Science Journal of Applied & Computational Mathematics Journal of Applied Intelligent System Journal of Applied Mathematics & Bioinformatics       (Followers: 6) Journal of Applied Mathematics and Physics       (Followers: 9) Journal of Computational Geometry       (Followers: 3) Journal of Innovative Applied Mathematics and Computational Sciences       (Followers: 6) Journal of Mathematical Sciences and Applications       (Followers: 2) Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance       (Followers: 12) Journal of Mathematics and Statistics Studies Journal of Physical Mathematics       (Followers: 2) Journal of Symbolic Logic       (Followers: 2) Letters in Biomathematics       (Followers: 1) Mathematical and Computational Applications       (Followers: 3) Mathematical Models and Computer Simulations       (Followers: 3) Mathematics and Computers in Simulation       (Followers: 3) Modeling Earth Systems and Environment       (Followers: 1) Moscow University Computational Mathematics and Cybernetics Multiscale Modeling and Simulation       (Followers: 2) Pacific Journal of Mathematics for Industry Partial Differential Equations in Applied Mathematics       (Followers: 1) Ratio Mathematica Results in Applied Mathematics       (Followers: 1) Scandinavian Actuarial Journal       (Followers: 2) SIAM Journal on Applied Dynamical Systems       (Followers: 3) SIAM Journal on Applied Mathematics       (Followers: 11) SIAM Journal on Computing       (Followers: 11) SIAM Journal on Control and Optimization       (Followers: 18) SIAM Journal on Discrete Mathematics       (Followers: 8) SIAM Journal on Financial Mathematics       (Followers: 3) SIAM Journal on Imaging Sciences       (Followers: 7) SIAM Journal on Mathematical Analysis       (Followers: 4) SIAM Journal on Matrix Analysis and Applications       (Followers: 3) SIAM Journal on Numerical Analysis       (Followers: 7) SIAM Journal on Optimization       (Followers: 12) SIAM Journal on Scientific Computing       (Followers: 16) SIAM Review       (Followers: 9) SIAM/ASA Journal on Uncertainty Quantification       (Followers: 2) Swarm Intelligence       (Followers: 3) Theory of Probability and its Applications       (Followers: 2) Uniform Distribution Theory Universal Journal of Applied Mathematics       (Followers: 2) Universal Journal of Computational Mathematics       (Followers: 3)
Similar Journals
 SIAM Journal on Applied Dynamical SystemsJournal Prestige (SJR): 1.04 Citation Impact (citeScore): 2Number of Followers: 3      Hybrid journal (It can contain Open Access articles) ISSN (Online) 1536-0040 Published by Society for Industrial and Applied Mathematics  [17 journals]
• Efficient Computation of Linear Response of Chaotic Attractors with
One-Dimensional Unstable Manifolds

Authors: Nisha Chandramoorthy, Qiqi Wang
Pages: 735 - 781
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 735-781, June 2022.
The sensitivity of time averages in a chaotic system to an infinitesimal parameter perturbation grows exponentially with the averaging time. However, long-term averages or ensemble statistics often vary differentiably with system parameters. Ruelle's response theory gives a rigorous formula for these parametric derivatives of statistics or linear response. But the direct evaluation of this formula is ill-conditioned, and hence linear response and downstream applications of sensitivity analysis, such as optimization and uncertainty quantification, have been a computational challenge in chaotic dynamical systems. This paper presents the space-split sensitivity (S3) algorithm to transform Ruelle's formula into a well-conditioned ergodic-averaging computation. We prove a decomposition of Ruelle's formula that is differentiable on the unstable manifold, which we assume to be one-dimensional. This decomposition of Ruelle's formula ensures that one of the resulting terms, the stable contribution, can be computed using a regularized tangent equation, similarly as in a nonchaotic system. The remaining term, known as the unstable contribution, is regularized and converted into an efficiently computable ergodic average. In this process, we develop new algorithms, which may be useful beyond linear response, to compute the unstable derivatives of the regularized tangent vector field and the unstable direction. We prove that the S3 algorithm, which combines these computational ingredients that enter the stable and unstable contributions, converges like a Monte Carlo approximation of Ruelle's formula. The algorithm presented here is hence a first step toward full-fledged applications of sensitivity analysis in chaotic systems, wherever such applications have been limited due to lack of availability of long-term sensitivities.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-04-07T07:00:00Z
DOI: 10.1137/21M1405599
Issue No: Vol. 21, No. 2 (2022)

• Quasi-Steady-State and Singular Perturbation Reduction for Reaction
Networks with Noninteracting Species

Authors: Elisenda Feliu, Christian Lax, Sebastian Walcher, Carsten Wiuf
Pages: 782 - 816
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 782-816, June 2022.
Quasi-steady state (QSS) reduction is a commonly used method to lower the dimension of a differential equation model of a chemical reaction network. From a mathematical perspective, QSS reduction is generally interpreted as a special type of singular perturbation reduction, but in many instances the correspondence is not worked out rigorously, and the QSS reduction may yield incorrect results. The present paper contains a thorough discussion of QSS reduction and its relation to singular perturbation reduction for the special, but important, case when the right-hand side of the differential equation is linear in the variables to be eliminated (but the differential equation model might otherwise be nonlinear). For this class we give necessary and sufficient conditions for a singular perturbation reduction (in the sense of Tikhonov and Fenichel) to exist, and to agree with QSS reduction. We then apply the general results to chemical reaction networks with noninteracting species, generalizing earlier results and methods for steady states to QSS scenarios. We provide easy-to-check graphical conditions to select parameter values for which the singular perturbation reduction applies, and additionally, we identify when the singular perturbation reduction agrees with the QSS reduction. Finally we consider a number of examples.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-04-11T07:00:00Z
DOI: 10.1137/20M1364503
Issue No: Vol. 21, No. 2 (2022)

• Persistence of Conley--Morse Graphs in Combinatorial Dynamical Systems

Authors: Tamal K. Dey, Marian Mrozek, Ryan Slechta
Pages: 817 - 839
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 817-839, June 2022.
Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The first captures changes in the Conley index, while the second captures changes in the Morse decomposition. However, such approaches have limitations. The former approach only describes how the Conley index changes across a selected isolated invariant set though the dynamics can be much more complicated than the behavior of a single isolated invariant set. Likewise, considering a Morse decomposition omits much information about the individual Morse sets. In this paper, we propose a method to summarize changes in combinatorial dynamical systems by capturing changes in the so-called Conley--Morse graphs. A Conley--Morse graph contains information about both the structure of a selected Morse decomposition and about the Conley index at each Morse set in the decomposition. Hence, our method summarizes the changing structure of a sequence of dynamical systems at a finer granularity than previous approaches.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-04-11T07:00:00Z
DOI: 10.1137/21M143162X
Issue No: Vol. 21, No. 2 (2022)

• Analysis of a Dynamical System Modeling Lasers and Applications for
Optical Neural Networks

Authors: Lauri Ylinen, Tuomo von Lerber, Franko Küppers, Matti Lassas
Pages: 840 - 878
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 840-878, June 2022.
An analytical study of dynamical properties of a semiconductor laser with optical injection of arbitrary polarization is presented. It is shown that if the injected field is sufficiently weak, then the laser has nine equilibrium points; however, only one of them is stable. Even if the injected field is linearly polarized, six of the equilibrium points have a state of polarization that is elliptical. Dependence of the equilibrium points on the injected field is described, and it is shown that as the intensity of the injected field increases, the number of equilibrium points decreases, with only a single equilibrium point remaining for strong enough injected fields. As an application, a complex-valued optical neural network with working principle based on injection locking is proposed.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-04-11T07:00:00Z
DOI: 10.1137/21M1405976
Issue No: Vol. 21, No. 2 (2022)

• Shadowing-Based Data Assimilation Method for Partially Observed Models

Authors: Bart M. de Leeuw, Svetlana Dubinkina
Pages: 879 - 902
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 879-902, June 2022.
In this article we develop further an algorithm for data assimilation based upon a shadowing refinement technique [de Leeuw et al., SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 2446--2477] to take partial observations into account. Our method is based on a regularized Gauss--Newton method. We prove local convergence to the solution manifold and provide a lower bound on the algorithmic time step. We use numerical experiments with the Lorenz 63 and Lorenz 96 models to illustrate convergence of the algorithm and show that the results compare favorably with a variational technique---weak-constraint four-dimensional variational method---and a shadowing technique--pseudo-orbit data assimilation. Numerical experiments show that a preconditioner chosen based on a cost function allows the algorithm to find an orbit of the dynamical system in the vicinity of the true solution.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-04-11T07:00:00Z
DOI: 10.1137/18M1223897
Issue No: Vol. 21, No. 2 (2022)

• Stationary and Time-Dependent Molecular Distributions in Slow-Fast
Feedback Circuits

Authors: Pavol Bokes
Pages: 903 - 931
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 903-931, June 2022.
Biochemical reaction systems, in particular those that govern the expression of genes in single cells, involve discrete numbers of molecules and are inherently stochastic. This study concerns reaction systems with two molecular species and an arbitrary number of reactions, the rates of which conform to a scaling of the classical type in the first species (but not the second). For limiting values of the scaling parameter, the first species is highly abundant, evolves slowly, and buffers the relatively fast fluctuation of the second, scarce, species. The scale separation facilitates the construction of asymptotic approximations to the molecular distributions: the large-time behavior is described by a WKB approximation, which is further resolved into a tractable mixture of metastable modes; the earlier-time dynamics are described by quasi-stationary and linear noise approximations. The paper presents the theory and the algebraic recipes that are required to implement it. The framework is applied on two reaction systems with positive feedback: a delayed feedback circuit and a gene autoregulation model with cooperative binding of the protein to the promoter. For the former, the approximation scheme results into a mixture of Gaussian/Poisson modes for the inactive/active protein distribution. For the latter, the analysis elucidates the effects of bursty gene expression, promoter responsivity, and protein sequestration on bimodality of protein distributions.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-04-13T07:00:00Z
DOI: 10.1137/21M1404338
Issue No: Vol. 21, No. 2 (2022)

• Stochastically Adaptive Control and Synchronization: From Globally
One-Sided Lipschitzian to Only Locally Lipschitzian Systems

Authors: Shijie Zhou, Ying-Cheng Lai, Wei Lin
Pages: 932 - 959
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 932-959, June 2022.
The mathematical framework of stochastically adaptive feedback control, which is generally applicable to significant problems in nonlinear dynamics such as stabilization and synchronization, has been previously established but only for systems whose vector fields satisfy the global Lipschitzian condition. Nonlinear dynamical systems arising from physical, chemical, or biological situations are typically described by vector fields that are only locally Lipschitzian. To generalize the mathematical theory of stochastically adaptive control to realistic systems is quite a challenging and formidable task. We meet this challenge by proving rigorously that stabilization and synchronization can be achieved with probability one for only locally Lipschitzian systems. The result holds not only for one-dimensional but also for any finite-dimensional white noises. Representative examples and an application to synchronization-based parameter identification are presented to illustrate the broad applicability of the developed mathematical criteria. Our successful relaxation of the mathematical condition from globally to locally Lipschitzian provides a rigorous guarantee of the stability of stochastically adaptive control in physical systems with significant implications to the design and realization of engineering control systems.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-04-13T07:00:00Z
DOI: 10.1137/21M1402042
Issue No: Vol. 21, No. 2 (2022)

• Intensity---A Metric Approach to Quantifying Attractor Robustness in ODEs

Authors: Katherine J. Meyer, Richard P. McGehee
Pages: 960 - 981
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 960-981, June 2022.
Although mathematical models do not fully match reality, robustness of dynamical objects to perturbation helps bridge from theoretical to real-world dynamical systems. Classical theories of structural stability and isolated invariant sets treat robustness of qualitative dynamics to sufficiently small errors. But they do not indicate just how large a perturbation can become before the qualitative behavior of our system changes fundamentally. Here we introduce a quantity, intensity of attraction, that measures the robustness of attractors in metric terms. Working in the setting of ordinary differential equations on $\mathbb{R}^n$, we consider robustness to vector field perturbations that are time dependent or independent. We define intensity in a control-theoretic framework, based on the magnitude of control needed to steer trajectories out of a domain of attraction. Our main result is that intensity also quantifies the robustness of an attractor to time-independent vector field perturbations; we prove this by connecting the reachable sets of control theory to isolating blocks of Conley theory. In addition to treating classical questions of robustness in a new metric framework, intensity of attraction offers a novel tool for resilience quantification in ecological applications. Unlike many measurements of resilience, intensity detects the strength of transient dynamics in a domain of attraction.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-04-25T07:00:00Z
DOI: 10.1137/20M138689X
Issue No: Vol. 21, No. 2 (2022)

• Process-Oriented Geometric Singular Perturbation Theory and Calcium
Dynamics

Authors: Samuel Jelbart, Nathan Pages, Vivien Kirk, James Sneyd, Martin Wechselberger
Pages: 982 - 1029
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 982-1029, June 2022.
Phenomena in chemistry, biology, and neuroscience are often modeled using ordinary differential equations (ODEs) in which the right-hand side is comprised of terms which correspond to individual “processes” or “fluxes.” Frequently, these ODEs are characterized by multiple time-scale phenomena due to order of magnitude differences between contributing processes and the presence of switching, i.e., dominance or subdominance of particular terms as a function of state variables. We outline a heuristic procedure for the identification of small parameters in ODE models of this kind, with a particular emphasis on the identification of small parameters relating to switching behaviors. This procedure is outlined informally in generality, and applied in detail to a model for intracellular calcium dynamics characterized by switching and multiple (more than two) time-scale dynamics. A total of five small parameters are identified, and related to a single perturbation parameter by a polynomial scaling law based on order of magnitude comparisons. The resulting singular perturbation problem has a time-scale separation which depends on the region of state space. We prove the existence and uniqueness of stable relaxation oscillations with three distinct time scales using a coordinate-independent formulation of geometric singular perturbation theory in combination with the blowup method. We also provide an estimate for the period of the oscillations, and consider a number of possibilities for their onset under parameter variation.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-04-25T07:00:00Z
DOI: 10.1137/21M1412402
Issue No: Vol. 21, No. 2 (2022)

• Discriminant Dynamic Mode Decomposition for Labeled Spatiotemporal Data
Collections

Authors: Naoya Takeishi, Keisuke Fujii, Koh Takeuchi, Yoshinobu Kawahara
Pages: 1030 - 1058
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1030-1058, June 2022.
Extracting coherent patterns is one of the standard approaches toward understanding spatiotemporal data. Dynamic mode decomposition (DMD) is a powerful tool for extracting coherent patterns, but the original DMD and most of its variants do not consider label information, which is often available as side information of spatiotemporal data. In this work, we propose a new method for extracting distinctive coherent patterns from labeled spatiotemporal data collections such that they contribute to major differences in a labeled set of dynamics. We achieve such pattern extraction by incorporating discriminant analysis into DMD. To this end, we define a kernel function on subspaces spanned by sets of dynamic modes and develop an objective to take both reconstruction goodness as DMD and class-separation goodness as discriminant analysis into account. We illustrate our method using a synthetic dataset and several real-world datasets. The proposed method can be a useful tool for exploratory data analysis for understanding spatiotemporal data.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-05-02T07:00:00Z
DOI: 10.1137/21M1399907
Issue No: Vol. 21, No. 2 (2022)

• Canards Underlie Both Electrical and Ca$^{2+}$-Induced Early
Afterdepolarizations in a Model for Cardiac Myocytes

Authors: Joshua Kimrey, Theodore Vo, Richard Bertram
Pages: 1059 - 1091
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1059-1091, June 2022.
Early afterdepolarizations (EADs) are voltage oscillations that can occur during the plateau phase of a cardiac action potential. EADs at the cellular level have been linked to potentially deadly tissue-level arrhythmias, and the mechanisms for their arisal are not fully understood. There is ongoing debate as to which is the predominant biophysical mechanism of EAD production: imbalanced interactions between voltage-gated transmembrane currents or overactive Ca$^{2+}$-dependent transmembrane currents brought about by pathological intracellular Ca$^{2+}$ release dynamics. In this article, we address this issue using a foundational 10-dimensional biophysical ventricular action potential model which contains both electrical and intracellular Ca$^{2+}$ components. Surprisingly, we find that the model can produce EADs through both biophysical mechanisms, which hints at a more fundamental dynamical mechanism for EAD production. Fast-slow analysis reveals EADs, in both cases, to be canard-induced mixed-mode oscillations. While the voltage-driven EADs arise from a fast-slow problem with two slow variables, the Ca$^{2+}$-driven EADs arise from the addition of a third slow variable. Hence, we adapt existing computational methods in order to compute 2D slow manifolds and 1D canard orbits in the reduced 7D model from which voltage-driven EADs arise. Further, we extend these computational methods in order to compute, for the first time, 2D sets of maximal canards which partition the 3D slow manifolds of the 8D problem from which Ca$^{2+}$-driven EADs arise. The canard viewpoint provides a unifying alternative to the voltage- or Ca$^{2+}$-driven viewpoints while also providing explanatory and predictive insights that cannot be obtained through the use of the traditional fast-slow approach.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-05-03T07:00:00Z
DOI: 10.1137/22M147757X
Issue No: Vol. 21, No. 2 (2022)

• A Graph-Theoretic Condition for Delay Stability of Reaction Systems

Authors: Polly Y. Yu, Gheorghe Craciun, Maya Mincheva, Casian Pantea
Pages: 1092 - 1118
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1092-1118, June 2022.
Delay mass-action systems provide a model of chemical kinetics when past states influence the current dynamics. In this work, we provide a graph-theoretic condition for delay stability, i.e., linear stability independent of both rate constants and delay parameters. In particular, the result applies when the system has no delay, implying asymptotic stability for the ODE system. The graph-theoretic condition is about cycles in the directed species-reaction graph of the network, which encodes how different species in the system interact.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-05-10T07:00:00Z
DOI: 10.1137/21M1420307
Issue No: Vol. 21, No. 2 (2022)

• Patterns and Quasipatterns from the Superposition of Two Hexagonal
Lattices

Authors: Gerard Iooss, Alastair M. Rucklidge
Pages: 1119 - 1165
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1119-1165, June 2022.
When two-dimensional pattern-forming problems are posed on a periodic domain, classical techniques (Lyapunov--Schmidt, equivariant bifurcation theory) give considerable information about what periodic patterns are formed in the transition where the featureless state loses stability. When the problem is posed on the whole plane, these periodic patterns are still present. Recent work on the Swift--Hohenberg equation (an archetypal pattern-forming partial differential equation) has proved the existence of quasipatterns, which are not spatially periodic and yet still have long-range order. Quasipatterns may have eight-fold, ten-fold, twelve-fold, and higher rotational symmetry, which preclude periodicity. There are also quasipatterns with six-fold rotational symmetry made up from the superposition of two equal amplitude hexagonal patterns rotated by almost any angle $\alpha$ with respect to each other. Here, we revisit the Swift--Hohenberg equation (with quadratic as well as cubic nonlinearities) and prove existence of several new quasipatterns. The most surprising are hexa-rolls: periodic and quasiperiodic patterns made from the superposition of hexagons and rolls (stripes) oriented in almost any direction with respect to each other and with any relative translation; these bifurcate directly from the featureless solution. In addition, we find quasipatterns made from the superposition of hexagons with unequal amplitude (provided the coefficient of the quadratic nonlinearity is small). We consider the periodic case as well, and extend the class of known solutions, including the superposition of hexagons and rolls. While we have focused on the Swift--Hohenberg equation, our work contributes to the general question of what periodic or quasiperiodic patterns should be found generically in pattern-forming problems on the plane.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-05-12T07:00:00Z
DOI: 10.1137/20M1372780
Issue No: Vol. 21, No. 2 (2022)

• Existence and Emergent Dynamics of Quadratically Separable States to the
Lohe Tensor Model

Authors: Seung-Yeal Ha, Dohyun Kim, Hansol Park
Pages: 1166 - 1208
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1166-1208, June 2022.
A tensor is a multidimensional array of complex numbers, and the Lohe tensor model is an aggregation model on the space of tensors with the same rank and size. It incorporates previously well-studied aggregation models on the space of low-rank tensors such as the Kuramoto model, Lohe sphere, and matrix models as special cases. Due to its structural complexities in cubic interactions for the Lohe tensor model, explicit construction of solutions with specific structures looks daunting. Recently, we obtained completely separable states by associating rank-1 tensors. In this paper, we further investigate another type of solutions, namely “quadratically separable states" consisting of tensor products of matrices and their component rank-2 tensors are solutions to the double matrix model whose emergent dynamics can be studied using the same methodology of the Lohe matrix model.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-05-26T07:00:00Z
DOI: 10.1137/21M1409664
Issue No: Vol. 21, No. 2 (2022)

• Topological Entropy of Surface Braids and Maximally Efficient Mixing

Authors: Spencer A. Smith, Sierra Dunn
Pages: 1209 - 1244
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1209-1244, June 2022.
The deep connections between braids and dynamics by way of the Nielsen--Thurston classification theorem have led to a wide range of practical applications. Braids have been used to detect coherent structures and mixing regions in oceanic flows, drive the design of industrial mixing machines, contextualize the evolution of taffy pullers, and characterize the chaotic motion of topological defects in active nematics. Mixing plays a central role in each of these examples, and the braids naturally associated with each system come equipped with a useful measure of mixing efficiency, the topological entropy per operation (TEPO). This motivates the following questions. What is the maximum mixing efficiency for braids, and what braids realize this' The answer depends on how we define braids. For the standard Artin presentation, well-known braids with mixing efficiencies related to the golden and silver ratios have been proven to be maximal. However, it is fruitful to consider surface braids, a natural generalization of braids, with presentations constructed from Artin-like braid generators on embedded graphs. In this work, we introduce an efficient and elegant algorithm for finding the topological entropy and TEPO of surface braids on any pairing of orientable surface and planar embeddable graph. Of the myriad possible graphs and surfaces, graphs that can be embedded in $\mathbb{R}^2$ as a lattice are a simple, highly symmetric choice, and the braids that result more naturally model the motion of points on the plane. We extensively search for a maximum mixing efficiency braid on planar lattice graphs and examine a novel candidate braid, which we conjecture to have this maximal property.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-05-26T07:00:00Z
DOI: 10.1137/21M142647X
Issue No: Vol. 21, No. 2 (2022)

• Angular Values of Nonautonomous and Random Linear Dynamical Systems: Part
I---Fundamentals

Authors: Wolf-Jürgen Beyn, Gary Froyland, Thorsten Hüls
Pages: 1245 - 1286
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1245-1286, June 2022.
We introduce the notion of angular values for deterministic linear difference equations and ran- dom linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on long-term averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For one-dimensional subspaces in two-dimensional systems our angular values agree with the classical theory of rotation numbers for orientation-preserving circle homeomorphisms if the matrix has positive determinant and does not rotate vectors by more than $\frac{\pi}{2}$. Because our notion of angular values ignores orientation by looking at subspaces rather than vectors, our results apply to dynamical systems of any dimension and to subspaces of arbitrary dimension. The second part of the paper delves deeper into the theory of the autonomous case. We explore the relation to (generalized) eigenspaces, provide some explicit formulas for angular values, and set up a general numerical algorithm for computing angular values via Schur decompositions.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-05-31T07:00:00Z
DOI: 10.1137/20M1387730
Issue No: Vol. 21, No. 2 (2022)

• Reactivity, Attenuation, and Transients in Metapopulations

Authors: Peter D. Harrington, Mark A. Lewis, P. van den Driessche
Pages: 1287 - 1321
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1287-1321, June 2022.
Transient dynamics often differ drastically from the asymptotic dynamics of systems. In this paper we analyze transient dynamics in birth-jump metapopulations where dispersal occurs immediately after birth (e.g., via larval dispersal). We address the choice of appropriate norms as well as the effect of stage structure on transient dynamics. We advocate the use of the $\ell_1$ norm, because of its biological interpretation, and extend the transient metrics of reactivity and attenuation to birth-jump metapopulations in this norm. By way of examples we compare this norm to the more commonly used $\ell_2$ norm. Our focus is the case where transient dynamics are very different than asymptotic dynamics. We provide simple examples of metapopulations where this is the case and also show how increasing the number of habitat patches can increase this difference. We then connect the reactivity and attenuation of metapopulations to the source-sink classification of habitat patches and demonstrate how to meaningfully measure reactivity when metapopulations are stage-structured, with a focus on marine metapopulations. Our paper makes three primary contributions. First, it provides guidance to readers as to the appropriate norm and scalings for studying transients in birth-jump metapopulations. Second, it provides three examples of transient behavior in metapopulations involving slow-fast systems, crawl-bys, and high dimensionality. Third, it connects the concepts of reactivity and attenuation to the source-sink classification of habitat patches more commonly found in marine metapopulations.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-06-01T07:00:00Z
DOI: 10.1137/21M140451X
Issue No: Vol. 21, No. 2 (2022)

• Dynamics of Nonpolar Solutions to the Discrete Painlevé I Equation

Authors: Nicholas Ercolani, Joceline Lega, Brandon Tippings
Pages: 1322 - 1351
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1322-1351, June 2022.
This manuscript develops a novel understanding of nonpolar solutions of the discrete Painlevé I equation (dP1). As the nonautonomous counterpart of an analytically completely integrable difference equation, this system is endowed with a rich dynamical structure. In addition, its nonpolar solutions, which grow without bounds as the iteration index $n$ increases, are of particular relevance to other areas of mathematics. We combine theory and asymptotics with high-precision numerical simulations to arrive at the following picture: when extended to include backward iterates, known nonpolar solutions of dP1 form a family of heteroclinic connections between two fixed points at infinity. One of these solutions, the Freud orbit of orthogonal polynomial theory, is a singular limit of the other solutions in the family. Near their asymptotic limits, all solutions converge to the Freud orbit, which follows invariant curves of dP1, when written as a three-dimensional autonomous system, and reaches the point at positive infinity along a center manifold. This description leads to two important results. First, the Freud orbit tracks sequences of period-1 and 2 points of the autonomous counterpart of dP1 for large positive and negative values of $n$, respectively. Second, we identify an elegant method to obtain an asymptotic expansion of the iterates on the Freud orbit for large positive values of $n$. The structure of invariant manifolds emerging from this picture contributes to a deeper understanding of the global analysis of an interesting class of discrete dynamical systems.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-06-01T07:00:00Z
DOI: 10.1137/21M1445156
Issue No: Vol. 21, No. 2 (2022)

• A Generalized Model of Flocking with Steering

Authors: Guy A. Djokam, Muruhan Rathinam
Pages: 1352 - 1381
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1352-1381, June 2022.
We introduce and analyze a model for the dynamics of flocking and steering of a finite number of agents. In this model, each agent's acceleration consists of flocking and steering components. The flocking component is a generalization of many of the existing models and allows for the incorporation of many real world features such as acceleration bounds, partial masking effects, and orientation bias. The steering component is also integral to capture real world phenomena. We provide rigorous sufficient conditions under which the agents flock and steer together. We also provide a formal singular perturbation study of the situation where flocking happens much faster than steering. We end our work by providing some numerical simulations to illustrate our theoretical results.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-06-01T07:00:00Z
DOI: 10.1137/21M1398793
Issue No: Vol. 21, No. 2 (2022)

• Emergent Behaviors of Rotation Matrix Flocks

Authors: Razvan C. Fetecau, Seung-Yeal Ha, Hansol Park
Pages: 1382 - 1425
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1382-1425, June 2022.
We derive an explicit form for the Cucker--Smale (CS) model on the special orthogonal group ${SO}(3)$ by identifying closed form expressions for geometric quantities such as covariant derivative and parallel transport in exponential coordinates. We study the emergent dynamics of the model by using a Lyapunov functional approach and LaSalle's invariance principle. Specifically, we show that velocity alignment emerges from some admissible class of initial data, under suitable assumptions on the communication weight function. We characterize the $\omega$-limit set of the dynamical system and identify a dichotomy in the asymptotic behavior of solutions. Several numerical examples are provided to support the analytical results.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-06-02T07:00:00Z
DOI: 10.1137/21M1404569
Issue No: Vol. 21, No. 2 (2022)

• Multistability of Reaction Networks with One-Dimensional Stoichiometric
Subspaces

Authors: Xiaoxian Tang, Zhishuo Zhang
Pages: 1426 - 1454
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1426-1454, June 2022.
For the reaction networks with one-dimensional stoichiometric subspaces, we show the following results. (1) If the maximum number of positive steady states is an even number $N$, then the maximum number of stable positive steady states is $\frac{N}{2}$. (2) If the maximum number of positive steady states is an odd number $N$, then we provide a condition on the parameters of the network such that the maximum number of stable positive steady states is $\frac{N-1}{2}$ if this condition is satisfied, and this maximum number is $\frac{N+1}{2}$ otherwise.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-06-02T07:00:00Z
DOI: 10.1137/21M1424676
Issue No: Vol. 21, No. 2 (2022)

• Bulk Topological States in a New Collective Dynamics Model

Authors: Pierre Degond, Antoine Diez, Mingye Na
Pages: 1455 - 1494
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1455-1494, June 2022.
In this paper, we demonstrate the existence of topological states in a new collective dynamics model. This individual-based model (IBM) describes self-propelled rigid bodies moving with constant speed and adjusting their rigid-body attitude to that of their neighbors. In previous works, a macroscopic model has been derived from this IBM in a suitable scaling limit. In the present work, we exhibit explicit solutions of the macroscopic model characterized by a nontrivial topology. We show that these solutions are well approximated by the IBM during a certain time but then the IBM transitions toward topologically trivial states. Using a set of appropriately defined topological indicators, we reveal that the breakage of the nontrivial topology requires the system to go through a phase of maximal disorder. We also show that similar but topologically trivial initial conditions result in markedly different dynamics, suggesting that topology plays a key role in the dynamics of this system.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-06-06T07:00:00Z
DOI: 10.1137/21M1393935
Issue No: Vol. 21, No. 2 (2022)

• Modeling Multilane Traffic with Moving Obstacles by Nonlocal Balance Laws

Authors: Alexandre Bayen, Jan Friedrich, Alexander Keimer, Lukas Pflug, Tanya Veeravalli
Pages: 1495 - 1538
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1495-1538, June 2022.
We consider a system of nonlocal balance laws where each balance law is coupled with the remaining balance laws both by a nonlocal velocity function that takes into account the averaged density of all other equations and by a right-hand “semilinear” term. We demonstrate existence and uniqueness of weak solutions for a small time horizon and a maximum principle for additional assumptions on the input data. This maximum principle ensures the applicability of the considered system of nonlocal balance laws to real-world problems. In the case of traffic flow, we show how the nonlocal impact and the coupling via the “semilinear” term can model multilane traffic flow with lane changing. We also demonstrate the applicability of the model to an on-ramp scenario in which the cars have to move from the on-ramp to the adjacent lane within a finite spatial domain. Moreover, we demonstrate how the problem of having obstacles can be modeled with these equations, introducing an additional ODE for the obstacle's dynamic. Several numerical results are presented and their reasonability is discussed.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-06-21T07:00:00Z
DOI: 10.1137/20M1366654
Issue No: Vol. 21, No. 2 (2022)

• Ensemble Inference Methods for Models With Noisy and Expensive Likelihoods

Authors: Oliver R. A. Dunbar, Andrew B. Duncan, Andrew M. Stuart, Marie-Therese Wolfram
Pages: 1539 - 1572
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1539-1572, June 2022.
The increasing availability of data presents an opportunity to calibrate unknown parameters which appear in complex models of phenomena in the biomedical, physical, and social sciences. However, model complexity often leads to parameter-to-data maps which are expensive to evaluate and are only available through noisy approximations. This paper is concerned with the use of interacting particle systems for the solution of the resulting inverse problems for parameters. Of particular interest is the case where the available forward model evaluations are subject to rapid fluctuations, in parameter space, superimposed on the smoothly varying large-scale parametric structure of interest. A motivating example from climate science is presented, and ensemble Kalman methods (which do not use the derivative of the parameter-to-data map) are shown, empirically, to perform well. Multiscale analysis is then used to analyze the behavior of interacting particle system algorithms when rapid fluctuations, which we refer to as noise, pollute the large-scale parametric dependence of the parameter-to-data map. Ensemble Kalman methods and Langevin-based methods (the latter use the derivative of the parameter-to-data map) are compared in this light. The ensemble Kalman methods are shown to behave favorably in the presence of noise in the parameter-to-data map, whereas Langevin methods are adversely affected. On the other hand, Langevin methods have the correct equilibrium distribution in the setting of noise-free forward models, while ensemble Kalman methods only provide an uncontrolled approximation, except in the linear case. Therefore a new class of algorithms, ensemble Gaussian process samplers, which combine the benefits of both ensemble Kalman and Langevin methods, are introduced and shown to perform favorably.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-06-21T07:00:00Z
DOI: 10.1137/21M1410853
Issue No: Vol. 21, No. 2 (2022)

• A Geometric Approach for Analyzing Parametric Biological Systems by
Exploiting Block Triangular Structure

Authors: Changbo Chen, Wenyuan Wu
Pages: 1573 - 1596
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1573-1596, June 2022.
We present an efficient geometric approach for computing the steady states of biparametric biological systems modeled by autonomous ordinary differential equations by taking advantage of their potential block triangular structures. While the parametric steady states of a given system are initially described by an implicit real algebraic surface in high-dimensional space, this approach computes a border curve, approximated by polygonal chains, which separates a bounded box of the parametric space into finitely many open cells such that the parametric steady states are continuous functions of the parameters with disjoint graphs above each cell. A block triangular structure of the system is discovered by combining both Tarjan's algorithm for computing strongly connected components of a graph defined for the system and Gauss--Jordan elimination. This particular structure enables one to greatly reduce the size of polynomial systems defining border curves. The effectiveness of the approach is demonstrated by analyzing two biological systems with, respectively, 17 and 20 unknowns.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-06-23T07:00:00Z
DOI: 10.1137/21M1436373
Issue No: Vol. 21, No. 2 (2022)

• Sequential Attractors in Combinatorial Threshold-Linear Networks

Authors: Caitlyn Parmelee, Juliana Londono Alvarez, Carina Curto, Katherine Morrison
Pages: 1597 - 1630
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 2, Page 1597-1630, June 2022.
Sequences of neural activity arise in many brain areas, including cortex, hippocampus, and central pattern generator circuits that underlie rhythmic behaviors like locomotion. While network architectures supporting sequence generation vary considerably, a common feature is an abundance of inhibition. In this work, we focus on architectures that support sequential activity in recurrently connected networks with inhibition-dominated dynamics. Specifically, we study emergent sequences in a special family of threshold-linear networks, called combinatorial threshold-linear networks (CTLNs), whose connectivity matrices are defined from directed graphs. Such networks naturally give rise to an abundance of sequences whose dynamics are tightly connected to the underlying graph. We find that architectures based on generalizations of cycle graphs produce limit cycle attractors that can be activated to generate transient or persistent (repeating) sequences. Each architecture type gives rise to an infinite family of graphs that can be built from arbitrary component subgraphs. Moreover, we prove a number of graph rules for the corresponding CTLNs in each family. The graph rules allow us to strongly constrain, and in some cases fully determine, the fixed points of the network in terms of the fixed points of the component subnetworks. Finally, we also show how the structure of certain architectures gives insight into the sequential dynamics of the corresponding attractor.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-06-24T07:00:00Z
DOI: 10.1137/21M1445120
Issue No: Vol. 21, No. 2 (2022)

• A Bounded-Confidence Model of Opinion Dynamics on Hypergraphs

Authors: Abigail Hickok, Yacoub Kureh, Heather Z. Brooks, Michelle Feng, Mason A. Porter
Pages: 1 - 32
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 1-32, January 2022.
People's opinions evolve with time as they interact with their friends, family, colleagues, and others. In the study of opinion dynamics on networks, one often encodes interactions between people in the form of dyadic relationships, but many social interactions in real life are polyadic (i.e., they involve three or more people). In this paper, we extend an asynchronous bounded-confidence model (BCM) on graphs, in which nodes are connected pairwise by edges, to an asynchronous BCM on hypergraphs, in which arbitrarily many nodes can be connected by a single hyperedge. We show that our hypergraph BCM converges to consensus for a wide range of initial conditions for the opinions of the nodes, including for nonuniform and asymmetric initial opinion distributions. We also show that, under suitable conditions, echo chambers can form on hypergraphs with community structure. We demonstrate that the opinions of nodes can sometimes jump from one opinion cluster to another in a single time step; this phenomenon (which we call “opinion jumping'') is not possible in standard dyadic BCMs. Additionally, we observe a phase transition in the convergence time of our BCM on a complete hypergraph when the variance $\sigma^2$ of the initial opinion distribution equals the confidence bound $c$. We prove that the convergence time grows at least exponentially fast with the number of nodes when $\sigma^2> c$ and the initial opinions are normally distributed. Therefore, to determine the convergence properties of our hypergraph BCM when the variance and the number of hyperedges are both large, it is necessary to use analytical methods instead of relying only on Monte Carlo simulations.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-04T08:00:00Z
DOI: 10.1137/21M1399427
Issue No: Vol. 21, No. 1 (2022)

• A Nonautonomous Equation Discovery Method for Time Signal Classification

Authors: Ryeongkyung Yoon, Harish S. Bhat, Braxton Osting
Pages: 33 - 59
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 33-59, January 2022.
Certain neural network architectures, in the infinite-depth limit, lead to systems of nonlinear differential equations. Motivated by this idea, we develop a framework for analyzing time signals based on nonautonomous dynamical systems. We view the time signal as a forcing function for a dynamical system that governs a time-evolving hidden variable. As in equation discovery, the dynamical system is represented using a dictionary of functions and the coefficients are learned from data. This framework is applied to the time signal classification problem. We show how gradients can be efficiently computed using the adjoint method, and we apply methods from dynamical systems to establish stability of the classifier. Through a variety of experiments, on both synthetic and real datasets, we show that the proposed method uses orders of magnitude fewer parameters than competing methods, while achieving comparable accuracy. We created the synthetic datasets using dynamical systems of increasing complexity; though the ground truth vector fields are often polynomials, we find consistently that a Fourier dictionary yields the best results. We also demonstrate how the proposed method yields graphical interpretability in the form of phase portraits.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-04T08:00:00Z
DOI: 10.1137/21M1405216
Issue No: Vol. 21, No. 1 (2022)

• Robust and Scalable Methods for the Dynamic Mode Decomposition

Authors: Travis Askham, Peng Zheng, Aleksandr Aravkin, J. Nathan Kutz
Pages: 60 - 79
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 60-79, January 2022.
The dynamic mode decomposition (DMD) is a broadly applicable dimensionality reduction algorithm that decomposes a matrix of time-series data into a product of a matrix of exponentials, representing Fourier-like time dynamics, and a matrix of coefficients, representing spatial structures. This interpretable spatio-temporal decomposition is classically formulated as a nonlinear least squares problem and solved within the variable projection framework. When the data contains outliers, or other features that are not well represented by exponentials in time, the standard Frobenius norm misfit penalty creates significant biases in the recovered time dynamics. As a result, practitioners are left to clean such defects from the data manually or to use a black-box cleaning approach like robust principal component analysis (PCA). As an alternative, we propose a robust statistical framework for the optimization used to compute the DMD itself. We also develop variable projection algorithms for these new formulations, which allow for regularizers and constraints on the decomposition parameters. Finally, we develop a scalable version of the algorithm by combining the structure of the variable projection framework with the stochastic variance reduction (SVRG) paradigm. The approach is tested on a range of synthetic examples, and the methods are implemented in an open source software package RobustDMD.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-04T08:00:00Z
DOI: 10.1137/21M1417405
Issue No: Vol. 21, No. 1 (2022)

• Mechanistic Modeling of Longitudinal Shape Changes: Equations of Motion
and Inverse Problems

Authors: Dai-Ni Hsieh, Sylvain Arguillère, Nicolas Charon, Laurent Younes
Pages: 80 - 101
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 80-101, January 2022.
This paper examines a longitudinal shape evolution model in which a three-dimensional volume progresses through a family of elastic equilibria in response to the time-derivative of an internal force, or yank, with an additional regularization to ensure diffeomorphic transformations. We consider two different models of yank and address the long time existence and uniqueness of solutions for the equations of motion in both models. In addition, we derive sufficient conditions for the existence of an optimal yank that best describes the change from an observed initial volume to an observed volume at a later time. The main motivation for this work is the understanding of processes such as growth and atrophy in anatomical structures, where the yank could be roughly interpreted as a metabolic event triggering morphological changes. We provide preliminary results on simple examples to illustrate, under this model, the retrievability of some attributes of such events.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-04T08:00:00Z
DOI: 10.1137/21M1423099
Issue No: Vol. 21, No. 1 (2022)

• Hebbian Network of Kuramoto Oscillators with Second-Order Couplings for
Binary Pattern Retrieve: II. Nonorthogonal Standard Patterns and
Structural Stability

Authors: Zhuchun Li, Xiaoxue Zhao, Xiaoping Xue
Pages: 102 - 136
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 102-136, January 2022.
We continue the study on the Hebbian network of Kuramoto oscillators with a second-order Fourier term, aiming to apply the system to the binary pattern retrieve task with nonorthogonal standard binary patterns. In [SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 1124--1159] the authors considered the system and studied the stability/instability by introducing the so-called $\varepsilon$-independent stability; this theory deals with the scenario where the memorized patterns coincide with the standard ones and the key assumption is the mutual orthogonality in standard patterns (or memorized ones). The key idea was to memorize these mutually orthogonal binary patterns in the network and retrieve one of them for a given defective pattern. However, in practice the orthogonality usually fails. When the orthogonality in the standard patterns fails, we find it is not a proper use of the system memorizing these standard patterns. In this paper we propose a new strategy which converts the problem to a case with orthogonality, and the standard patterns are $\varepsilon$-independently asymptotically stable. The structural stability is also considered, and sharp quantitative estimates on the strength of perturbations for the stability of binary patterns are presented. Numerical simulations illustrate the approach and the results.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-05T08:00:00Z
DOI: 10.1137/21M1393224
Issue No: Vol. 21, No. 1 (2022)

• Dynamical Analysis and Optimization of a Generalized Resource Allocation
Model of Microbial Growth

Authors: Agustín G. Yabo, Jean-Baptiste Caillau, Jean-Luc Gouzé, Hidde de Jong, Francis Mairet
Pages: 137 - 165
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 137-165, January 2022.
Gaining a better comprehension of the growth of microorganisms is a major scientific challenge, which has often been approached from a resource allocation perspective. Simple mathematical self-replicator models based on resource allocation principles have been surprisingly effective in accounting for experimental observations of the growth of microorganisms. Previous work, using a three-variable resource allocation model, predicted an optimal resource allocation scheme for the adaptation of microbial cells to a sudden nutrient change in the environment. We here propose an extended version of this model considering also proteins responsible for basic housekeeping functions, and we study their impact on predicted optimal strategies for resource allocation following changes in the environment. A full dynamical analysis of the system shows there is a single globally attractive equilibrium, which can be related to steady-state growth conditions of bacteria observed in experiments. We then explore the optimal allocation strategies using optimization and optimal control theory. We show that the solutions to this dynamical problem have a complicated structure that includes a second-order singular arc given in feedback form and characterized by (i) Fuller's phenomenon and (ii) the turnpike effect, producing a very particular asymptotic behavior towards the solution of the static optimization problem. Our work thus provides a generalized perspective on the analysis of microbial growth by means of simple self-replicator models.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-10T08:00:00Z
DOI: 10.1137/21M141097X
Issue No: Vol. 21, No. 1 (2022)

• Stable Synchronized Periodic Solutions in a Pair of Mutually Inhibitory
Model Neurons with Conduction Delays

Authors: Euiwoo Lee
Pages: 166 - 203
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 166-203, January 2022.
We investigate the synchronizing properties of two synaptically coupled model neurons in the presence of conduction delays. The neurons are modeled as relaxation oscillators, and they interact through inhibitory synapses that activate rapidly but decay slowly. We derive a reduced system through a slow/fast decomposition process, and special coordinates are introduced into the system. Then we construct a map, for which a stable fixed point corresponds to a stable synchronized periodic (SP) solution. The functional benefit of these coordinates is that, on the stability of an SP solution, the map can be decomposed into one-dimensional submaps. By analyzing these submaps, we examine how conduction delays affect the stability of the SP solution, depending on the period and duty cycle of an intrinsic oscillator and the synaptic decay rate.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-10T08:00:00Z
DOI: 10.1137/21M140064X
Issue No: Vol. 21, No. 1 (2022)

• An Adaptive Phase-Amplitude Reduction Framework without
$\mathcal{O}(\epsilon)$ Constraints on Inputs

Authors: Dan Wilson
Pages: 204 - 230
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 204-230, January 2022.
Phase reduction is a well-established technique used to analyze the timing of oscillations in response to weak external inputs. In the preceding decades, a wide variety of results have been obtained for weakly perturbed oscillators that place restrictive limits on the magnitude of the inputs or on the magnitude of the time derivatives of the inputs. By contrast, no general reduction techniques currently exist to analyze oscillatory dynamics in response to arbitrary, large magnitude inputs, and comparatively very little is understood about these strongly perturbed limit cycle oscillators. In this work, the theory of isostable reduction is leveraged to develop an adaptive phase-amplitude transformation that does not place any restrictions on the allowable input. Additionally, provided some of the Floquet multipliers of the underlying periodic orbits are near zero, the proposed method yields a reduction in dimension comparable to that of other phase-amplitude reduction frameworks. Numerical illustrations show that the proposed method accurately reflects synchronization and entrainment of coupled oscillators in regimes where a variety of other phase-amplitude reductions fail.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-11T08:00:00Z
DOI: 10.1137/21M1391791
Issue No: Vol. 21, No. 1 (2022)

• Bifurcation Analysis of a Modified Cardiac Cell Model

Authors: André H. Erhardt, Susanne Solem
Pages: 231 - 247
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 231-247, January 2022.
The 19-dimensional TP06 cardiac muscle cell model is reduced to a 17-dimensional version, which satisfies the required conditions for performing an analysis of its dynamics by means of bifurcation theory. The reformulated model is shown to be a good approximation of the original one. As a consequence, one can extract fairly precise predictions of the behavior of the original model from the bifurcation analysis of the modified model. Thus, the findings of bifurcations linked to complex dynamics in the modified model---like early afterdepolarizations (EADs), which can be precursors to cardiac death---predicts the occurrence of the same dynamics in the original model. It is shown that bifurcations linked to EADs in the modified model accurately predict EADs in the original model at the single cell level. Finally, these bifurcations are linked to wave break-up leading to cardiac death at the tissue level.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-11T08:00:00Z
DOI: 10.1137/21M1425359
Issue No: Vol. 21, No. 1 (2022)

• Graphop Mean-Field Limits for Kuramoto-Type Models

Authors: Marios Antonios Gkogkas, Christian Kuehn
Pages: 248 - 283
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 248-283, January 2022.
Originally arising in the context of interacting particle systems in statistical physics, dynamical systems and differential equations on networks/graphs have permeated into a broad number of mathematical areas as well as into many applications. One central problem in the field is to find suitable approximations of the dynamics as the number of nodes/vertices tends to infinity, i.e., in the large graph limit. A cornerstone in this context are Vlasov equations (VEs) describing a particle density on a mean-field level. For all-to-all coupled systems, it is quite classical to prove the rigorous approximation by VEs for many classes of particle systems. For dense graphs converging to graphon limits, one also knows that mean-field approximation holds for certain classes of models, e.g., for the Kuramoto model on graphs. Yet, the space of intermediate density and sparse graphs is clearly extremely relevant. Here we prove that the Kuramoto model can be be approximated in the mean-field limit by far more general graph limits than graphons. In particular, our contributions are as follows. (I) We show how to introduce operator theory more abstractly into VEs by considering graphops. Graphops have recently been proposed as a unifying approach to graph limit theory, and here we show that they can be used for differential equations on graphs. (II) For the Kuramoto model on graphs we rigorously prove that there is a VE approximating it in the mean-field sense. (III) This mean-field VE involves a graphop, and we prove the existence, uniqueness, and continuous graphop dependence of weak solutions. (IV) On a technical level, our results rely on designing a new suitable metric of graphop convergence and on employing Fourier analysis on compact abelian groups to approximate graphops using summability kernels.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-18T08:00:00Z
DOI: 10.1137/21M1391523
Issue No: Vol. 21, No. 1 (2022)

• Derivative-Free Bayesian Inversion Using Multiscale Dynamics

Authors: G. A. Pavliotis, A. M. Stuart, U. Vaes
Pages: 284 - 326
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 284-326, January 2022.
Inverse problems are ubiquitous because they formalize the integration of data with mathematical models. In many scientific applications the forward model is expensive to evaluate, and adjoint computations are difficult to employ; in this setting derivative-free methods which involve a small number of forward model evaluations are an attractive proposition. Ensemble Kalman-based interacting particle systems (and variants such as consensus-based and unscented Kalman approaches) have proven empirically successful in this context, but suffer from the fact that they cannot be systematically refined to return the true solution, except in the setting of linear forward models [A. Garbuno-Inigo et al., SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 412--441]. In this paper, we propose a new derivative-free approach to Bayesian inversion, which may be employed for posterior sampling or for maximum a posteriori estimation, and may be systematically refined. The method relies on a fast/slow system of stochastic differential equations for the local approximation of the gradient of the log-likelihood appearing in a Langevin diffusion. Furthermore the method may be preconditioned by use of information from ensemble Kalman--based methods (and variants), providing a methodology which leverages the documented advantages of those methods, while also being provably refinable. We define the methodology, highlighting its flexibility and many variants, provide a theoretical analysis of the proposed approach, and demonstrate its efficacy by means of numerical experiments.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-24T08:00:00Z
DOI: 10.1137/21M1397416
Issue No: Vol. 21, No. 1 (2022)

• Hausdorff Continuity of Region of Attraction Boundary Under Parameter
Variation with Application to Disturbance Recovery

Authors: Michael W. Fisher, Ian A. Hiskens
Pages: 327 - 365
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 327-365, January 2022.
Consider a parameter dependent vector field on either Euclidean space or a compact Riemannian manifold. Suppose that it possesses a parameter dependent initial condition and a parameter dependent stable hyperbolic equilibrium point. It is valuable to determine the set of parameter values, which we call the recovery set, whose corresponding initial conditions lie within the region of attraction of the corresponding stable equilibrium point. A boundary parameter value is a parameter value whose corresponding initial condition lies in the boundary of the region of attraction of the corresponding stable equilibrium point. Prior algorithms numerically estimated the recovery set by estimating its boundary via computation of boundary parameter values. The primary purpose of this work is to provide theoretical justification for those algorithms for a large class of parameter dependent vector fields. This includes proving that, for these vector fields, the boundary of the recovery set consists of boundary parameter values, and that the properties exploited by the algorithms to compute these desired boundary parameters will be satisfied. The main technical result which these proofs rely on is establishing that the region of attraction boundary varies continuously in an appropriate sense with respect to small variation in parameter value for this class of vector fields. Hence, the majority of this work is devoted to proving this result, which may be of independent interest. The proof of continuity proceeds by proving that, for this class of vector fields, the region of attraction permits a decomposition into a union of the stable manifolds of the equilibrium points and periodic orbits it contains, and this decomposition persists under small perturbations to the vector field.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-25T08:00:00Z
DOI: 10.1137/20M1371944
Issue No: Vol. 21, No. 1 (2022)

• Validated Spectral Stability via Conjugate Points

Authors: Margaret Beck, Jonathan Jaquette
Pages: 366 - 404
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 366-404, January 2022.
Classical results from Sturm--Liouville theory state that the number of unstable eigenvalues of a scalar, second-order linear operator is equal to the number of associated conjugate points. Recent work has extended these results to a much more general setting, thus allowing for spectral stability of nonlinear waves in a variety of contexts to be determined by counting conjugate points. However, in practice, it is not yet clear whether it is easier to compute conjugate points than to just directly count unstable eigenvalues. We address this issue by developing a framework for the computation of conjugate points using validated numerics. Moreover, we apply our method to a parameter-dependent system of bistable equations and show that there exist both stable and unstable standing fronts. This application can be seen as complementary to the classical result via Sturm--Liouville theory that in scalar reaction-diffusion equations pulses are unstable whereas fronts are stable and to the more recent result of Beck et al. [Philos. Trans. Roy. Soc. A, 376 (2018), 20170187] that symmetric pulses in reaction-diffusion systems with gradient nonlinearity are also necessarily unstable.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-25T08:00:00Z
DOI: 10.1137/21M1420095
Issue No: Vol. 21, No. 1 (2022)

• The Impact of Damping in Second-Order Dynamical Systems with Applications
to Power Grid Stability

Authors: Amin Gholami, Xu A. Sun
Pages: 405 - 437
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 405-437, January 2022.
We consider a broad class of second-order dynamical systems and study the role of damping as a system parameter in the stability, hyperbolicity, and bifurcation in such systems. We prove a monotonic effect of damping on the hyperbolicity of the equilibrium points of the corresponding first-order system. This provides a rigorous formulation and theoretical justification for some intuitive notions that damping increases stability. To establish this result, we prove a matrix perturbation result for complex symmetric matrices with positive semidefinite perturbations to their imaginary parts, which may be of independent interest. Furthermore, we establish necessary and sufficient conditions for the breakdown of hyperbolicity of the first-order system under damping variations in terms of observability of a pair of matrices relating damping, inertia, and Jacobian matrices, and propose sufficient conditions for Hopf bifurcation resulting from such hyperbolicity breakdown. The developed theory has significant applications in the stability of electric power systems, which are one of the most complex and important engineering systems. In particular, we characterize the impact of damping on the hyperbolicity of swing equations, which is the fundamental dynamical model of power systems, and demonstrate Hopf bifurcations resulting from damping variations.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-01-31T08:00:00Z
DOI: 10.1137/20M1370392
Issue No: Vol. 21, No. 1 (2022)

• On a Competition-Diffusion-Advection System from River Ecology:
Mathematical Analysis and Numerical Study

Authors: Xiao Yan, Hua Nie, Peng Zhou
Pages: 438 - 469
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 438-469, March 2022.
This paper is mainly concerned with a two-species competition model in open advective environments, where individuals cannot pass through the upstream boundary and do not return to the habitat after leaving the downstream boundary. By the theory of principal eigenvalue, we first obtain two critical curves ($\Gamma_1$ and $\Gamma_2$) in the plane of bifurcation parameters that sharply determine the local stability of the two semitrivial steady states. Then under various conditions on given parameters, we discuss the global dynamics via different techniques, including the comparison principle for eigenvalues and perturbation and compactness arguments, and show that both competitive exclusion and coexistence are possible. For general values of parameters, we take both analytic and numerical approaches to further understand the potential behaviors of $\Gamma_1$ and $\Gamma_2$, and we numerically observe that in addition to the competitive exclusion and coexistence, the bistable phenomenon is also possible, which is different from the known results of competitive ODE and reaction-diffusion systems (where bistability is impossible). The implication of our numerical results on future work is also discussed.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-02-07T08:00:00Z
DOI: 10.1137/20M1387924
Issue No: Vol. 21, No. 1 (2022)

• Bifurcation Sequences in a Discontinuous Piecewise-Smooth Map Combining
Constant-Catch and Threshold-Based Harvesting Strategies

Authors: Cristina Lois-Prados, Frank M. Hilker
Pages: 470 - 499
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 470-499, March 2022.
We consider a harvesting strategy that allows constant catches if the population size is above a certain threshold value (to obtain predictable yield) and no catches if the population size is below the threshold (to protect the population). We refer to this strategy as threshold constant-catch (TCC) harvesting. We provide analytical and numerical results when applying TCC to monotone population growth models. TCC remedies the tendency to fishery collapse of pure constant-catch harvesting and provides a buffer for quotas larger than the maximum sustainable yield. From a dynamical systems point of view, TCC gives rise to a piecewise-smooth map with a discontinuity at the threshold population size. The dynamical behavior includes border-collision bifurcations, basin boundary metamorphoses, and boundary-collision bifurcation. We further find Farey trees, a slightly modified truncated skew tent map scenario, and the bandcount incrementing scenario. Our results underline, on the one hand, the protective function of thresholds in harvest control rules. On the other hand, they highlight the dynamical complexities due to discontinuities that can arise naturally in threshold-based harvesting strategies.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-02-10T08:00:00Z
DOI: 10.1137/21M1416515
Issue No: Vol. 21, No. 1 (2022)

• Nonlinearities Shape the Response Patterns to Oscillatory Inputs in a
Caricature Electrochemical Cell Model

Authors: Horacio G. Rotstein
Pages: 500 - 522
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 500-522, March 2022.
We investigate the nonlinear mechanisms of generation of preferred frequency response patterns to oscillatory inputs in a phenomenological (caricature) model that captures the type of nonlinearities present in electrochemical (EC) cells and is simple enough to begin to develop the dynamical systems tools necessary for the understanding of the relationship between the model nonlinearities and the response patterns. Previous work has shown that linearized EC models exhibit resonance (preferred frequency response to oscillatory inputs at a nonzero, resonant, input frequency) in parameter regimes where the stable equilibrium is either a node (no intrinsic oscillations) or a focus (damped oscillations). We use a combination of numerical simulations and dynamical systems tools to understand how the model nonlinearities, partially captured by the geometry of the nullclines in the phase-space diagram, shape the response patterns to oscillatory inputs away from the validity of the corresponding linearization. We develop and adapt an extended version of the classical phase-plane diagram that allows us to track the evolution of the response trajectories and their interaction with the so-called moving nullclines (in response to the oscillatory inputs). We use this approach to explain the mechanisms of generation of nonlinear resonant patterns, the nonlinear amplification/attenuation of these patterns by increasing input amplitudes, and the mechanisms of generation of more complex patterns of mixed-mode--type, where the stationary amplitude response is not uniform across cycles.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-02-14T08:00:00Z
DOI: 10.1137/21M1402121
Issue No: Vol. 21, No. 1 (2022)

• Dynamical Organization of Recollisions by a Family of Invariant Tori

Authors: J. Dubois, M. Jorba-Cuscó, À. Jorba, C. Chandre
Pages: 523 - 541
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 523-541, March 2022.
We consider the motion of an electron in an atom subjected to a strong linearly polarized laser field. We identify the invariant structures organizing a very specific subset of trajectories, namely, recollisions. Recollisions are trajectories which first escape the ionic core (i.e., ionize) and later return to this ionic core, for instance, to transfer the energy gained during the large excursion away from the core to bound electrons. We consider the role played by the directions transverse to the polarization direction in the recollision process. We compute the family of two-dimensional invariant tori associated with a specific hyperbolic-elliptic periodic orbit and their stable and unstable manifolds. We show that these manifolds organize recollisions in phase space.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-02-22T08:00:00Z
DOI: 10.1137/21M1400912
Issue No: Vol. 21, No. 1 (2022)

• A Dynamical Theory for Singular Stochastic Delay Differential Equations I:
Linear Equations and a Multiplicative Ergodic Theorem on Fields of Banach
Spaces

Authors: M. Ghani Varzaneh, S. Riedel, M. Scheutzow
Pages: 542 - 587
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 542-587, March 2022.
We investigate singular stochastic delay differential equations (SDDEs) in view of their long-time behavior. Using Lyons's rough path theory, we show that SDDEs can be solved pathwise and induce a continuous stochastic flow on the space of (Gubinelli's) controlled paths. In the language of random dynamical systems, this result shows that SDDEs induce a continuous cocycle on random fibers, or, more precisely, on a measurable field of Banach spaces. We furthermore prove a multiplicative ergodic theorem (MET) on measurable fields of Banach spaces that applies under significantly weaker structural and measurability assumptions than preceding METs. Applying it to linear SDDEs shows that the induced cocycle possesses a discrete Lyapunov spectrum that can be used to describe the long-time behavior.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-02-24T08:00:00Z
DOI: 10.1137/21M1433435
Issue No: Vol. 21, No. 1 (2022)

• An Algebraic Approach to Product-form Stationary Distributions for Some
Reaction Networks

Authors: Beatriz Pascual-Escudero, Linard Hoessly
Pages: 588 - 615
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 588-615, March 2022.
Exact results for product-form stationary distributions of Markov chains are of interest in different fields. In stochastic reaction networks (CRNs), stationary distributions are mostly known in special cases where they are of product-form. However, there is no full characterization of the classes of networks whose stationary distributions have product-form. We develop an algebraic approach to product-form stationary distributions in the framework of CRNs. Under certain hypotheses on linearity and decomposition of the state space for conservative CRNs, this gives sufficient and necessary algebraic conditions for product-form stationary distributions. Correspondingly, we obtain a semialgebraic subset of the parameter space that captures rates where, under the corresponding hypotheses, CRNs have product-form. We employ the developed theory to CRNs and some models of statistical mechanics, besides sketching the pertinence in other models from applied probability.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-03-07T08:00:00Z
DOI: 10.1137/21M1401498
Issue No: Vol. 21, No. 1 (2022)

• Asymptotic Behavior of Fronts and Pulses of the Bidomain Model

Authors: Hiroshi Matano, Yoichiro Mori, Mitsunori Nara, Koya Sakakibara
Pages: 616 - 649
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 616-649, March 2022.
The bidomain model is the standard model for cardiac electrophysiology. This paper investigates the instability and asymptotic behavior of planar fronts and planar pulses of the bidomain Allen--Cahn equation and the bidomain FitzHugh--Nagumo equation in two spatial dimensions. Previous work showed that planar fronts of the bidomain Allen--Cahn equation could become unstable in contrast to the classical Allen--Cahn equation. After the planar front is destabilized, a rotating zigzag front develops whose shape can be explained by simple geometric arguments using a suitable Frank diagram. We also show that the Hopf bifurcation through which the front becomes unstable can be either supercritical or subcritical by demonstrating a parameter regime in which a stable planar front and zigzag front can coexist. Our computational studies of the bidomain FitzHugh--Nagumo pulse solution show that the pulses can also become unstable, like the bidomain Allen--Cahn fronts. However, unlike the bidomain Allen--Cahn case, the destabilized pulse does not necessarily develop into a zigzag pulse. For certain choices of parameters, the destabilized pulse can disintegrate entirely. These studies are made possible by developing a numerical scheme that allows for the accurate computation of the bidomain equation in a two-dimensional strip domain of an infinite extent.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-03-07T08:00:00Z
DOI: 10.1137/21M1416904
Issue No: Vol. 21, No. 1 (2022)

• Anti-integrability for Three-Dimensional Quadratic Maps

Authors: Amanda E. Hampton, James D. Meiss
Pages: 650 - 675
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 650-675, March 2022.
We study the dynamics of the three-dimensional quadratic diffeomorphism using a concept first introduced 30 years ago for the Frenkel--Kontorova model of condensed matter physics: the anti-integrable (AI) limit. At the traditional AI limit, orbits of a map degenerate to sequences of symbols and the dynamics is reduced to the shift operator, a pure form of chaos. Under nondegeneracy conditions, a contraction mapping argument can show that infinitely many AI states continue to orbits of the deterministic map. For the 3D quadratic map, the AI limit that we study is a quadratic correspondence whose branches, a pair of one-dimensional maps, introduce symbolic dynamics on two symbols. The AI states, however, are nontrivial orbits of this correspondence. The character of these orbits depends on whether the quadratic takes the form of an ellipse, a hyperbola, or a pair of lines. Using contraction arguments, we find parameter domains for each case such that each symbol sequence corresponds to a unique AI state. In some parameter domains, sufficient conditions are then found for each such AI state to continue away from the limit becoming an orbit of the original 3D map. Numerical continuation methods extend these results, allowing computation of bifurcations to obtain orbits with horseshoe-like structures and intriguing self-similarity. We conjecture that pairs of periodic orbits in saddle-node and period-doubling bifurcations have symbol sequences that differ in exactly one position.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-03-14T07:00:00Z
DOI: 10.1137/21M1433289
Issue No: Vol. 21, No. 1 (2022)

• Queues with Delayed Information: A Dynamical Systems Perspective

Authors: Faouzi Lakrad, Jamol Pender, Richard Rand
Pages: 676 - 713
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 676-713, March 2022.
In this paper, we consider an $N$-dimensional queueing model where customers join according to a multinomial logit choice model. However, we assume that the information that the customer receives about the queue length is delayed by a constant $\Delta$. We review how a large customer scaling of the stochastic model yields a system of delay differential equations. We also show how these limiting delay differential equations approximate the stochastic mean dynamics of the original queueing model. To gain insight about the queue length dynamics such as the amplitude, the frequency, and the value of the critical delay we use the harmonic balance method and the method of multiple scales. We show that the method of multiple scales is more accurate but is less tractable from a computational perspective than the harmonic balance method. Using the method of multiple scales, we also prove that the only stable mode is where all $N$ queues have the same amplitude and frequency; however, each queue is shifted by $\frac{2\pi}{N}$ from its neighbor. This analysis provides great insights for queues with delayed information and how the oscillations manifest themselves.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-03-14T07:00:00Z
DOI: 10.1137/20M1334358
Issue No: Vol. 21, No. 1 (2022)

• Dynamics of Small Particle Inertial Migration in Curved Square Ducts

Authors: Kyung Ha, Brendan Harding, Andrea L. Bertozzi, Yvonne M. Stokes
Pages: 714 - 734
Abstract: SIAM Journal on Applied Dynamical Systems, Volume 21, Issue 1, Page 714-734, March 2022.
Microchannels are well known in microfluidic applications for the control and separation of microdroplets and cells. Often the objects in the flow experience inertial effects, resulting in dynamics that is a departure from the underlying channel flow dynamics. This paper considers small neutrally buoyant spherical particles suspended in flow through a curved duct having a square cross-section. The particle experiences a combination of inertial lift force induced by the disturbance from the primary flow along the duct, and drag from the secondary vortices in the cross-section, which drive migration of the particle within the cross-section. We construct a simplified model that preserves the core topology of the force field yet depends on a single parameter $\kappa$, quantifying the relative strength of the two forces. We show that $\kappa$ is a bifurcation parameter for the dynamical system that describes motion of the particle in the cross-section of the duct. At large values of $\kappa$ there exists an attracting limit cycle in each of the upper and lower halves of the duct. At small $\kappa$ we find that particles migrate to one of four stable foci. Between these extremes, there is an intermediate range of $\kappa$ for which all particles migrate to a single stable focus. Noting that the positions of the limit cycles and foci vary with the value of $\kappa$, this behavior indicates that, for a suitable particle mixture, duct bend radius might be chosen to segregate particles by size. We evaluate the time and axial distance required to focus particles near the unique stable node, which determines the duct length required for particle segregation.
Citation: SIAM Journal on Applied Dynamical Systems
PubDate: 2022-03-24T07:00:00Z
DOI: 10.1137/21M1430935
Issue No: Vol. 21, No. 1 (2022)

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