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

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 233-251, January 2020. We characterize the synchronization phenomenon in discrete-time, discrete-state random dynamical systems, with random and probabilistic Boolean networks as particular examples. In terms of multiplicative ergodic properties of the induced linear cocycle, we show such a random dynamical system with finite state

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 208-232, January 2020. We present an analysis of two Haken--Kelso--Bunz (HKB) oscillators coupled by a neurologically motivated function. We study the effect of time delay and weighted self-feedback and mutual feedback on the synchronization behavior of the model. We focus on identifying parameter regimes supporting experimentally

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 181-207, January 2020. We derive the exact equations of motion of the Keplerian dumbbell (KD) system, which consists of two point masses connected by a rigid massless rod, moving under the gravitational influence of a third point mass. We find several classes of continuous families of periodic orbits (principal periodic orbits) of

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 151-180, January 2020. A common method for analyzing the effects of molecular noise in chemical reaction networks is to approximate the underlying chemical master equation by a Fokker--Planck equation and to study the statistics of the associated chemical Langevin equation. This so-called system-size expansion involves performing a

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 124-150, January 2020. It has been recognized that many complex dynamical systems in the real world require a description in terms of multiplex networks, where a set of common, mutually connected nodes belong to distinct network layers and play a different role in each layer. In spite of recent progress toward data based inference

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 85-123, January 2020. Our recent work identifies material surfaces in incompressible flows that extremize the transport of an arbitrary, weakly diffusive scalar field relative to neighboring surfaces. Such barriers and enhancers of transport can be located directly from the deterministic component of the velocity field without diffusive

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 58-84, January 2020. A recently introduced technique, local orthogonal rectification (LOR), provides a way to derive coordinate systems that are tailored to locating objects of interest within flows in arbitrary dimensions. In this work, we apply LOR to identify periodic orbits and study the transient dynamics nearby. In the LOR method

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 1-57, January 2020. Motivated by its application in ecology, we consider an extended Klausmeier model, a singularly perturbed reaction-advection-diffusion equation with spatially varying coefficients. We rigorously establish existence of stationary pulse solutions by blending techniques from geometric singular perturbation theory with

We show how a graph algorithm for finding matching labeled paths in pairs of labeled directed graphs can be used to perform model invalidation for a class of dynamical systems including regulatory network models of relevance to systems biology. In particular, given a partial order of events describing local minima and local maxima of observed quantities from experimental time series data, we produce

We describe the theoretical and computational framework for the Dynamic Signatures for Genetic Regulatory Network ( DSGRN) database. The motivation stems from urgent need to understand the global dynamics of biologically relevant signal transduction/gene regulatory networks that have at least 5 to 10 nodes, involve multiple interactions, and decades of parameters. The input to the database computations

Delays in gene networks result from the sequential nature of protein assembly. However, it is unclear how models of gene networks that use delays should be modified when considering time-dependent changes in temperature. This is important, as delay is often used in models of genetic oscillators that can be entrained by periodic fluctuations in temperature. Here, we analytically derive the time dependence

Detecting and explaining the relationships among interacting components has long been a focal point of dynamical systems research. In this paper, we extend these types of data-driven analyses to the realm of public policy, whereby individual legislative entities interact to produce changes in their legal and political environments. We focus on the U.S. public health policy landscape, whose complexity

The dynamics of systems with stochastically varying time delays are investigated in this paper. It is shown that the mean dynamics can be used to derive necessary conditions for the stability of equilibria of the stochastic system. Moreover, the second moment dynamics can be used to derive sufficient conditions for almost sure stability of equilibria. The results are summarized using stability charts

Pancreatic islets exhibit bursting oscillations in response to elevated blood glucose. These oscillations are accompanied by oscillations in the free cytosolic Ca2+ concentration (Cac ), which drives pulses of insulin secretion. Both islet Ca2+ and metabolism oscillate, but there is some debate about their interrelationship. Recent experimental data show that metabolic oscillations in some cases persist

We describe methods to identify cylinder sets inside a basin of attraction for Boolean dynamics of biological networks. Such sets are used for designing regulatory interventions that make the system evolve towards a chosen attractor, for example initiating apoptosis in a cancer cell. We describe two algebraic methods for identifying cylinders inside a basin of attraction, one based on the Groebner

Rhythmic activity which underlies motor output is often initiated and controlled by descending modulatory projection pathways onto central pattern generator (CPG) networks. In turn, these descending pathways receive synaptic feedback from their target CPG network, which can influence the CPG output. However, the mechanisms underlying such bi-directional synaptic interactions are mostly unexplored.

Integrate-and-fire models of biological neurons combine differential equations with discrete spike events. In the simplest case, the reset of the neuronal voltage to its resting value is the only spike event. The response of such a model to constant input injection is limited to tonic spiking. We here study a generalized model in which two simple spike-induced currents are added. We show that this

We analyze spatial patterns on networks of cells where adjacent cells inhibit each other through contact signaling. We represent the network as a graph where each vertex represents the dynamics of identical individual cells and where graph edges represent cell-to-cell signaling. To predict steady-state patterns we find equitable partitions of the graph vertices and assign them into disjoint classes

Previous models of neuromodulation in cortical circuits have used either physiologically based networks of spiking neurons or simplified gain adjustments in low-dimensional connectionist models. Here we reduce a high-dimensional spiking neuronal network model, first to a four-population mean-field model and then to a two-population model. This provides a realistic implementation of neuromodulation