SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 2162-2193, January 2020. In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via dynamic mode decomposition, a quantization approach was recently

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 2135-2161, January 2020. The ability of a circadian system to entrain to the 24-hour light-dark cycle is one of its most important properties. A new tool, called the entrainment map, was recently introduced to study this process for a single oscillator. Here we generalize the map to study the effects of light-dark forcing in a hierarchical

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 2103-2134, January 2020. In this paper, we study an agent-based interacting particle system with attractive and singular repulsive forces. We prove the collision avoidance between particles from different groups due to repulsive forces. Moreover, we provide a sufficient condition for the emergence of asymptotic consensus in the same

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 2059-2102, January 2020. In this paper, we revisit the folded node and the bifurcations of secondary canards at resonances $\mu\in \mathbb N$. In particular, we prove for the first time that pitchfork bifurcations occur at all even values of $\mu$. Our approach relies on a time-reversible version of the Melnikov approach in [M. Wechselberger

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 2030-2058, January 2020. Spatially localized two-dimensional spot patterns occur for a wide variety of two component reaction-diffusion systems in the singular limit of a large diffusivity ratio. Such localized, far-from-equilibrium patterns are known to exhibit a wide range of different instabilities such as breathing oscillations

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1993-2029, January 2020. Permanent charge is the major structural quantity of an ion channel. It defines the ion channel and its interaction with boundary conditions plays the predominate role for ionic flow properties or functions of the ion channel. In this work, we investigate effects of large magnitude permanent charges of a simple

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1956-1992, January 2020. This article proposes a framework which allows the study of stability robustness of equilibria of a nonlinear system in the face of parametric uncertainties from the point of view of bifurcation theory. In this context, a branch of equilibria is stable if bifurcations (i.e., qualitative changes of the steady-state

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1920-1955, January 2020. Dynamic mode decomposition (DMD) is a data-driven method that models high-dimensional time series as a sum of spatiotemporal modes, where the temporal modes are constrained by linear dynamics. For nonlinear dynamical systems exhibiting strongly coherent structures, DMD can be a useful approximation to extract

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1884-1919, January 2020. To make decisions we are guided by the evidence we collect and the opinions of friends and neighbors. How do we combine our private beliefs with information we obtain from our social network? To understand the strategies humans use to do so, it is useful to compare them to observers that optimally integrate

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1865-1883, January 2020. We study a thermomechanical system comprising an alpha Stirling engine and a flywheel from the perspective of dynamical systems theory. Thermodynamics establishes a static relation between the flywheel's angle and the forces exerted by the two power pistons that constitute the engine. Mechanics, in turn, provides

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1823-1864, January 2020. We study a convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs). This framework bounds from above the largest value of an observable along trajectories that start from a chosen set and evolve over a finite

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1798-1822, January 2020. An important dynamical property of biological interaction networks is persistence, which intuitively means that “no species goes extinct." It has been conjectured that dynamical system models of weakly reversible networks (i.e., networks for which each reaction is part of a cycle) are persistent. The property

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1758-1797, January 2020. We consider a linearly unstable standing wave solution of a parabolic partial differential equation (PDE) on the real line and develop a high order method for polynomial approximation of the local unstable manifold. The unstable manifold describes the breakdown of the nonlinear wave after the loss of stability

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1736-1757, January 2020. Many civilizations have risen and then collapsed. There can be many causes. A major influence can be how Elite (wealthy or ruling) populations interact with the Commoners (workers) and with the environment. Each population's size can fluctuate. We say a model is “Elite-dominated" when the Elites' per capita

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1701-1735, January 2020. Early afterdepolarizations (EADs) are pathological voltage fluctuations that can occur in cardiac cells and are a potent source of potentially fatal arrhythmias. Recent works examining the mechanisms underlying EADs in minimal computational cardiac models have revealed that voltage-driven EADs are canard-induced

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1659-1700, January 2020. Understanding the macroscopic behavior of dynamical systems is an important tool to unravel transport mechanisms in complex flows. A decomposition of the state space into coherent sets is a popular way to reveal this essential macroscopic evolution. To compute coherent sets from an aperiodic time-dependent

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1633-1658, January 2020. We propose a computational method (with acronym ALDI) for sampling from a given target distribution based on first-order (overdamped) Langevin dynamics which satisfies the property of affine invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance serving as a

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1609-1632, January 2020. The problems of (i) maximizing or minimizing Lagrangian mixing in fluids via the introduction of a spatiotemporally varying control velocity and (ii) globally controlling the finite-time location of trajectories beginning at all initial conditions in a chaotic system are considered. A particular form of solution

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1575-1608, January 2020. Delays are important phenomena arising in a wide variety of real-world systems, including biological ones, because of diffusion/propagation effects or as simplifying modeling elements. We propose here to consider delayed stochastic reaction networks, a class of networks that has received little attention until

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1540-1573, January 2020. We study a model of the cell cycles of a large ensemble of cells that was conceived to explain the coupling between metabolism and the cell cycle observed in cell cycle related, metabolic oscillations in yeast continuous culture. An essential feature of the model is that a negative feedback mechanism robustly

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1525-1539, January 2020. This work deals with the full reduction of the spatial Kepler system for bounded and unbounded motions. Precisely, we consider the four-dimensional oscillator associated to the Kepler system and carry out our program in three stages: axial-axial-energy rather than energy-axial-axial as is customary. This approach

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1496-1524, January 2020. We propose an approach for the synthesis of robust and optimal feedback controllers for nonlinear PDEs. Our approach considers the approximation of infinite-dimensional control systems by a pseudospectral collocation method, leading to high-dimensional nonlinear dynamics. For the reduced-order model, we construct

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1472-1495, January 2020. Pulses in an actively mode-locked laser can occasionally slip relative to the timing signal, leading to fluctuations in the pulse repetition rate. Such events happen rarely, however, making it infeasible to use traditional methods to determine the slip rate. Here, in a model of a soliton-based mode-locked laser

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1438-1471, January 2020. In this paper, we investigate a time-delayed differential model of the transmission dynamics of schistosomiasis with seasonality. In order to study the influence of water temperature on egg hatching into miracidia and the development from miracidia to cercariae, we incorporate time-dependent delays into the

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1392-1437, January 2020. Dengue, zika, and chikungunya are viruses transmitted to humans by Aedes aegypti mosquitoes. In the absence of medical treatments and efficient vaccines, one of the control methods is to introduce Aedes aegypti mosquitoes infected by the bacterium Wolbachia into a population of wild (uninfected) mosquitoes

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1372-1391, January 2020. The aim of this paper is to solve an inverse problem which regards a mass moving in a bounded domain. We assume that the mass moves following an unknown velocity field and that the evolution of the mass density can be described by a partial differential equation, which is also unknown. The input data of the

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1343-1371, January 2020. We study the existence of periodic solutions in a class of planar Filippov systems obtained from non-autonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a simple twofold cycle, which is characterized by a closed trajectory

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1312-1342, January 2020. We study emergent behaviors of the Lohe hermitian sphere (LHS) model which is an aggregation model on ${\mathbb C}^d$. The LHS model is a complex analogue of the Lohe sphere model on ${\mathbb R}^d$, and hermitian spheres are invariant sets for the LHS dynamics. For the derivation of the LHS model, we use a

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1291-1311, January 2020. We focus on the (sharp) threshold phenomena arising in some reaction-diffusion equations supplemented with some compactly supported initial data. In the so-called ignition and bistable cases, we prove the first sharp quantitative estimate on the (sharp) threshold values. Furthermore, numerical explorations

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1271-1290, January 2020. Motivated by the recent works on the stability of symmetric periodic orbits of the elliptic Sitnikov problem, for time-periodic Newtonian equations with symmetries, we will study symmetric periodic solutions which emanated from nonconstant periodic solutions of autonomous equations. By using the theory of Hill's

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1225-1270, January 2020. We study emergent behaviors of the swarm sphere model under attractive-repulsive couplings and present several sufficient frameworks leading to the complete and practical bicluster aggregations using two key ingredients (two-point correlation function and order parameter). From the modeling perspective, we

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1200-1224, January 2020. We develop a definition of rate-induced tipping (R-tipping) in discrete-time dynamical systems (maps) and prove results giving conditions under which R-tipping will or will not happen. Specifically, we study (possibly noninvertible) maps with a time-varying parameter subject to a parameter shift. We show that

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1160-1199, January 2020. The dynamic mode decomposition (DMD) extracted dynamic modes are the nonorthogonal eigenvectors of the matrix that best approximates the one-step temporal evolution of the multivariate samples. In the context of dynamical system analysis, the extracted dynamic modes are a generalization of global stability

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1124-1159, January 2020. We study the stability of an oscillatory associative memory network consisting of $N$ coupled Kuramoto oscillators with applications in binary pattern retrieve. In this model, the coupling function consists of a Hebbian term and a second-order Fourier term with nonnegative strength $\varepsilon$. In [Phys.

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1080-1123, January 2020. We study the emergent behavior of the matrix ensemble on the unitary group in which a state of each oscillator is influenced by the relative distances. Thus, the coupling strength becomes a time-dependent function and its temporal evolution is determined by a feedback rule incorporating the linear damping and

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1057-1079, January 2020. In this work, we introduce a new Laplacian matrix, referred to as the hubs-attracting Laplacian, accounting for diffusion processes on networks where the hopping of a particle occurs with higher probability from low to high degree nodes. This notion complements the one of the hubs-repelling Laplacian discussed

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1029-1056, January 2020. Lyapunov functions are an important tool to determine the basin of attraction of equilibria. In particular, the connected component of a sublevel set, which contains the equilibrium, is a forward invariant subset of the basin of attraction. One method to compute a Lyapunov function for a general nonlinear autonomous

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 994-1028, January 2020. The computational singular perturbation (CSP) method is an algorithm which iteratively approximates slow manifolds and fast fibers in multiple-timescale dynamical systems. Since its inception due to Lam and Goussis [Twenty-Second Symposium (International) on Combustion, Combustion Institute, Pittsburgh, PA,

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 964-993, January 2020. A synchrony subspace of $\mathbb{R}^{n}$ is defined by setting certain components of the vectors equal according to an equivalence relation. Synchrony subspaces invariant under a given set of square matrices ordered by inclusion form a lattice. Applications of these invariant synchrony subspaces include equitable

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 918-963, January 2020. In this paper, we study intralayer synchronization of multiplex networks where nodes in each layer interact through diverse types of coupling functions associated with different time-varying network topologies, referred to as multiplex hypernetworks. Here, the intralayer connections are evolving with respect

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 886-917, January 2020. Nonlinear dynamical systems are ubiquitous in science and engineering, yet analysis and prediction of these systems remains a challenge. Koopman operator theory circumvents some of these issues by considering the dynamics in the space of observable functions on the state, in which the dynamics are intrinsically

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 860-885, January 2020. A systematic mathematical framework for the study of numerical algorithms would allow comparisons, facilitate conjugacy arguments, and enable the discovery of improved, accelerated, data-driven algorithms. Over the course of the past century, the Koopman operator has provided a mathematical framework for the

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 828-859, January 2020. The present paper provides exact mathematical expressions for the high-order moments of spiking activity in a recurrently connected network of linear Hawkes processes. It extends previous studies that have explored the case of a (linear) Hawkes network driven by deterministic intensity functions to the case of

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 788-827, January 2020. We investigate the dynamics of a limit of interacting FitzHugh--Nagumo neurons in the regime of large interaction coefficients. We consider the dynamics described by a mean-field model given by a nonlinear evolution partial differential equation representing the probability distribution of one given neuron in

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 763-787, January 2020. Tobasco, Goluskin, and Doering [https://doi.org/10.1016/j.physleta.2017.12.023 Phys. Lett. A, 382 (2018), pp. 382--386}] recently suggested that trajectories of ODE systems that optimize the infinite-time average of a certain observable can be localized using sublevel sets of a function that arise when bounding

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 725-762, January 2020. We consider a model of two competing species of Lotka--Volterra type with diffusion (migration), where the spatial domain is an arbitrary finite graph (network). Depending on the parameters of the model, we describe the spatially homogeneous stationary states and their stability, discuss the existence and number

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 705-723, January 2020. In this work we present a set-oriented path following method for the computation of relative global attractors of parameter-dependent dynamical systems. We start with an initial approximation of the relative global attractor for a fixed parameter $\lambda_0$ computed by a set-oriented subdivision method. By using

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 665-704, January 2020. The topological method for the reconstruction of dynamics from time series [K. Mischaikow et al., Phys. Rev. Lett., 82 (1999), pp. 1144--1147] is reshaped to improve its range of applicability, particularly in the presence of sparse data and strong expansion. The improvement is based on a multivalued map representation

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 619-664, January 2020. The integral control of positive systems using nonnegative control input is an important problem arising in among others, biochemistry, epidemiology, and ecology. An immediate solution is to use an ON-OFF nonlinearity between the controller and the system. However, this solution is available only when controllers

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 577-618, January 2020. Differential equations with random parameters have gained significant prominence in recent years due to their importance in mathematical modeling and data assimilation. In many cases, random ordinary differential equations (RODEs) are studied by using Monte Carlo methods or by direct numerical simulation techniques

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 541-576, January 2020. We introduce and analyze two general dynamical models for unidirectional movement of particles along a circular chain and an open chain of sites. The models include a soft version of the simple exclusion principle, that is, as the density in a site increases the effective entry rate into this site decreases.

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 510-540, January 2020. This work applies a continuous data assimilation scheme---a framework for reconciling sparse and potentially noisy observations to a mathematical model---to Rayleigh--Bénard convection at infinite or large Prandtl numbers using only the temperature field as observables. These Prandtl numbers are applicable to

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 480-509, January 2020. The Koopman operator has emerged as a powerful tool for the analysis of nonlinear dynamical systems as it provides coordinate transformations to globally linearize the dynamics. While recent deep learning approaches have been useful in extracting the Koopman operator from a data-driven perspective, several challenges

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 442-479, January 2020. Moser derived a normal form for the family of four-dimensional (4D), quadratic, symplectic maps in 1994. This six-parameter family generalizes Hénon's ubiquitous 2D map and provides a local approximation for the dynamics of more general 4D maps. We show that the bounded dynamics of Moser's family is organized

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 412-441, January 2020. Solving inverse problems without the use of derivatives or adjoints of the forward model is highly desirable in many applications arising in science and engineering. In this paper we propose a new version of such a methodology, a framework for its analysis, and numerical evidence of the practicality of the method

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 366-411, January 2020. Center manifold reduction is a standard technique in bifurcation theory, reducing the essential features of local bifurcations to equations in a small number of variables corresponding to critical eigenvalues. This method can be applied to admissible differential equations for a network, but it bears no obvious

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 352-365, January 2020. We give a new proof of the fact that each weakly reversible mass-action system with a single linkage class is permanent.

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 329-351, January 2020. This paper addresses structures of state space in quasiperiodically forced dynamical systems. We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps. The ergodic partition result

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 304-328, January 2020. In this paper, we study the interplay of time-delayed interactions and network structure on the collective behaviors of Kuramoto oscillators. For the continuous and discrete Kuramoto models with time delay on a general digraph, we provide two frameworks leading to the complete synchronization in terms of system

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 252-303, January 2020. In this paper, we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf, and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating