SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 3, Page 1320-1347, January 2021. We propose a framework of a reservoir with an inertial manifold, which is a finite-dimensional object that attracts all trajectories. In addition, based on the theoretical results concerning infinite-dimensional dynamical systems, we demonstrate that this reservoir should have some desirable properties called

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 3, Page 1277-1319, January 2021. We explore equivariant dynamics under the symmetric group $S_N$ of all permutations of $N$ elements. Specifically we study one-parameter vector fields, up to cubic order, which commute with the standard real $(N-1)$-dimensional irreducible representation of $S_N$. The parameter is the linearization at the trivial

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 3, Page 1253-1276, January 2021. In this work we study the level of synchronization in stochastic biochemical reaction networks that support stable mean-field limit cycles and are subject to common external switching noise. Synchronization in stochastic limit cycle oscillators due to common noise is usually demonstrated by applying Ito's lemma

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 3, Page 1232-1252, January 2021. This work is a continuation and extension of a recent one by Tang and Chen [Global dynamics of a Lotka--Volterra competition-diffusion system in advective homogeneous environments, J. Differential Equations, 269 (2020), pp. 1465--1483]. In that work, we studied a Lotka--Volterra competition-diffusion model

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 3, Page 1209-1231, January 2021. We propose a mathematical model to measure how multiple repetitions may influence the ultimate proportion of the population never hearing a rumor during a given outbreak. The model is a multidimensional continuous-time Markov chain that can be seen as a generalization of the Maki--Thompson model for the propagation

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 3, Page 1177-1208, January 2021. Threshold-linear networks (TLNs) are recurrent networks where the dynamics are threshold-linear (linearly rectified at zero). Mathematically, they consist of coupled nonsmooth ordinary differential equations. When the nodes in the network are assumed to be neurons or neuronal populations, TLNs represent firing

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 1158-1176, January 2021. The synchronization of different brain regions is widely observed under both normal and pathological conditions, such as epilepsy. However, the relationship between the dynamics of these brain regions, the connectivity between them, and the ability to synchronize remains an open question. We investigate the

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 1135-1157, January 2021. Most countries have started vaccinating people against COVID-19. However, due to limited production capacities and logistical challenges it will take months/years until herd immunity is achieved. Therefore, vaccination and social distancing have to be coordinated. In this paper, we provide some insight on this

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 1104-1134, January 2021. We study a uniform-in-time continuum limit of the lattice Winfree model (LWM) on an inifinite cylinder under restricted initial datum and suitable assumptions on system functions such as natural frequency functions and the coupling strength function. Roughly speaking, the continuum Winfree model (CWM) is an

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 1090-1103, January 2021. To assess which ecosystems are most vulnerable it is necessary to compare the resilience of complex interaction networks in a meaningful way. A fundamental problem for the comparative analysis of ecological stability is that the organisms in ecological networks operate on different time scales. A conventional

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 1053-1089, January 2021. Quasi-stationary states consisting of localized spots in a reaction-diffusion system are considered on the surface of a torus with major radius $R$ and minor radius $r$. Under the assumption that these localized spots persist stably, the evolution equation of the spot cores is derived analytically based on

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 1022-1052, January 2021. Understanding principles of neurolocomotion requires the synthesis of neural activity, sensory feedback, and biomechanics. The nematode C. elegans is an ideal model organism for studying locomotion in an integrated neuromechanical setting because its neural circuit has a well-characterized modular structure

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 997-1021, January 2021. We consider the rough differential equation $dY=f(Y)d\bm \omega$, where $\bm \omega=(\omega,\bbomega)$ is a rough path defined by a Brownian motion $\omega$ on $\mathbb{R}^m$. Under the usual regularity assumption on $f$, namely, $f\in C^3_b (\mathbb{R}^d, \mathbb{R}^{d\times m})$, the rough differential equation

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 981-996, January 2021. We propose a computational approach to estimate the stability domain of quadratic-bilinear reduced-order models (ROMs), which are low-dimensional approximations of large-scale dynamical systems. For nonlinear ROMs, it is important not only to show that the origin is locally asymptotically stable, but also to

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 953-980, January 2021. Experimental characterization of neuronal dynamics involves recording both of spontaneous activity patterns and of responses to transient and sustained inputs. While much theoretical attention has been devoted to the spontaneous activity of neurons, less is known about the dynamic mechanisms shaping their responses

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 941-952, January 2021. In this paper, we aim at proving a conjecture proposed in [X. Cen, X. Cheng, Z. Huang, and M. Zhang, SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 1271--1290] on the stability of symmetric periodic solutions of the elliptic Sitnikov problem $(S_e)$. By using the stability criteria there, it is shown that odd $4n\pi$-periodic

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 908-940, January 2021. The paper deals with a nonlinear evolution equation describing the dynamics of a nonhomogeneous multiply hinged beam, subject to a nonlocal restoring force of displacement type. First, a spectral analysis for the associated weighted stationary problem is performed, providing a complete system of eigenfunctions

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 853-907, January 2021. We propose a method to integrate dissipative PDEs rigorously forward in time with the use of the finite element method (FEM). The technique is based on the Galerkin projection on the FEM space and estimates on the residual terms. The proposed approach is illustrated on a periodically forced one-dimensional Burgers

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 826-852, January 2021. In this paper, we investigate a family of coupled excitable cells, using a Kuramoto-type model with either fixed excitability and Gaussian (dynamic) noise or a random distribution of the excitability. In both cases, we reduce the coupled system to a low-dimensional system using mean field approaches such as the

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 784-825, January 2021. We establish versions of Conley's (i) fundamental theorem and (ii) decomposition theorem for a broad class of hybrid dynamical systems. The hybrid version of (i) asserts that a globally defined hybrid complete Lyapunov function exists for every hybrid system in this class. Motivated by mechanics and control settings

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 745-783, January 2021. Moving averages and other functional forecasting models are used to inform policy in pandemic response. In this paper, we analyze an infectious disease model in which the contact rate switches between two levels when the moving average of active cases crosses one of two thresholds. The switching mechanism naturally

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 701-744, January 2021. When dynamical systems that produce rhythmic behaviors operate within hard limits, they may exhibit limit cycles with sliding components, that is, closed isolated periodic orbits that make and break contact with a constraint surface. Examples include heel-ground interaction in locomotion, firing rate rectification

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 671-700, January 2021. Stationary fronts connecting the trivial state and a cellular (distorted) hexagonal pattern in the Swift--Hohenberg equation with a quadratic-cubic nonlinearity are known to undergo a process of infinitely many folds as a parameter is varied, known as homoclinic snaking, where new hexagon cells are added to the

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 636-670, January 2021. For a regular coupled cell network, synchrony subspaces are the polydiagonal subspaces that are invariant under the network adjacency matrix. The complete lattice of synchrony subspaces of an $n$-cell regular network can be seen as an intersection of the partition lattice of $n$ elements and a lattice of invariant

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 608-635, January 2021. For three typical sets of small reaction networks (networks with two reactions, one irreversible and one reversible reaction, or two reversible-reaction pairs), we completely answer the challenging question “What is the smallest subset of all multistable networks such that any multistable network outside of the

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 573-607, January 2021. In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddle-node bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 571-572, January 2021. We are very grateful to Grégoire Nadin for bringing the following issue in the work of Alfaro, Ducrot, and Faye [SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 1291--1311] to our attention: since it lacks regularity, the function $w = w(t, x)$ defined in (3.5) cannot be the solution to the reaction-diffusion equation

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 541-570, January 2021. We study the dynamics of the periodically forced May--Leonard system. We extend previous results on the field, and we identify different dynamical regimes depending on the strength of attraction $\delta$ of the network and the frequency $\omega$ of the periodic forcing. We focus our attention in the case $\delta\gg1$

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 500-540, January 2021. An in-depth analysis of nonautonomous bifurcations of saddle-node type for scalar differential equations $x'=-x^2+q(t)\,x+p(t)$, where $q\colon\mathbb R\to\mathbb R$ and $p\colon\mathbb R\to\mathbb R$ are bounded and uniformly continuous, is fundamental to explaining the absence or occurrence of rate-induced

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 438-499, January 2021. We analyze a class of cell-bulk coupled PDE-ODE models, motivated by quorum-sensing and diffusion-mediated behavior in microbial systems, that characterize communication between localized spatially segregated dynamically active signaling compartments or “cells” that have a permeable boundary. In this model, the

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 403-437, January 2021. Based on the nonlinear time transformation method, in this paper we propose a recursive algorithm for arbitrary order approximation of heteroclinic orbits. This approach works fine for a wide class of systems that are perturbations of non-Hamiltonian integrable planar vector fields. Specifically, our method can

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 371-402, January 2021. We use geometric singular perturbation techniques combined with an action functional approach to study traveling pulse solutions in a three-component FitzHugh--Nagumo model. First, we derive the profile of traveling 1-pulse solutions with undetermined width and propagating speed. Next, we compute the associated

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 333-370, January 2021. Pseudospectral approximation reduces delay differential equations (DDE) to ordinary differential equations (ODE). Next one can use ODE tools to perform a numerical bifurcation analysis. By way of an example we show that this yields an efficient and reliable method to qualitatively as well as quantitatively analyze

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 299-332, January 2021. In this study, we develop model bias estimators based on an asymptotic expansion of the model dynamics for small time scales and small perturbations in a model parameter, and then use the estimators to improve the performance of a data assimilation system. We employ the well-known Lorenz (1963) model so that

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 262-298, January 2021. The one-fluid stellar wind problem for steady, radial outflow is considered, including effects of heat conduction and viscosity. The associated nondimensionalized equations of conservation of mass, momentum, and energy are singularly perturbed in the large Reynolds number limit, and stellar wind profiles are

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 232-261, January 2021. The method of using periodic approximations to compute the spectral decomposition of the Koop- man operator is generalized to the class of measure-preserving flows on compact metric spaces. It is shown that the spectral decomposition of the continuous one-parameter unitary group can be approximated from an intermediate

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 208-231, January 2021. The thermal convection of rotating fluids in spherical geometry is a classical problem with application to many geophysical and astrophysical problems. The study of the transition to periodic solutions from the steady conduction state of a rotating and self-gravitating fluid sphere, heated uniformly from the

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 165-207, January 2021. Replica-mean-field models have been proposed to decipher the activity of otherwise analytically intractable neural networks via a multiply-and-conquer approach. In this approach, one considers limit networks made of infinitely many replicas with the same basic neural structure as that of the network of interest

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 149-164, January 2021. In this paper, we shall prove the stability of the particle advection around a fixed vortex in a two-dimensional ideal fluid under the action of a periodic background flow induced by a polynomial field. The proof relies on the identification of closed invariant curves around the origin by means of Moser's invariant

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 130-148, January 2021. We present a simple proof of asymptotic consensus in the discrete Hegselmann--Krause model and flocking in the discrete Cucker--Smale model with normalization and variable delay. This proof utilizes the convexity of the normalized communication weights and a Gronwall--Halanay-type inequality. The main advantage

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 94-129, January 2021. In this paper we analyze nonlinear dynamics of the fullerene molecule. We prove, under isotypic nonresonant assumptions, the existence of global branches of periodic solutions emerging from an icosahedral equilibrium (nonlinear normal modes). We use equivariant degree theory to prove the existence of several branches

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 65-93, January 2021. The totally asymmetric simple exclusion process (TASEP) is a basic model of statistical mechanics that has found numerous applications. We consider the case of TASEP with a finite chain where particles may enter from the left and leave to the right at prescribed rates. This model can be formulated as a Markov process

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 38-64, January 2021. In this article, we aim to classify the dynamic transitions and bifurcations for a family of axisymmetric geophysical fluid problems of a generic fourth-second order structure. A transition theorem is established by reducing the governing partial differential equations to a complex-valued ordinary differential

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 1-37, January 2021. Deciding whether and where a system of parametrized ordinary differential equations displays bistability, that is, has at least two asymptotically stable steady states for some choice of parameters, is a hard problem. For systems modeling biochemical reaction networks, we introduce a procedure to determine, exclusively

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2847-2886, January 2020. Motivated by models that arise in controlled ship maneuvering, we analyze Hopf bifurcations in systems with piecewise smooth nonlinear part. In particular, we derive explicit formulas for the generalization of the first Lyapunov coefficient to this setting. This generically determines the direction of branching

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2829-2846, January 2020. Progression through different synchronized and desynchronized regimes in brain networks has been reported to reflect physiological and behavioral states, such as working memory and attention. Moreover, intracranial recordings of epileptic seizures show a progression towards synchronization as brain regions

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2803-2828, January 2020. A Koopman decomposition of a complex system leads to a representation in which nonlinear dynamics appear to be linear. The existence of a linear framework with which to analyze nonlinear dynamical systems brings new strategies for prediction and control, while the approach is straightforward to apply to large

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2783-2802, January 2020. We study the effect of noise on dynamics of a single spike for the classical Gierer--Meinhardt model on a finite interval. When spatio-temporal noise is introduced in the equation for the activator, we derive a stochastic ODE that describes the motion of a single spike on a slow time scale. The steady state

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2749-2782, January 2020. We study a nonlinear wave equation appearing as a model for a membrane (without viscous effects) under the presence of an electrostatic potential with strength $\lambda$. The membrane has a unique stable branch of steady states $u_{\lambda}$ for $\lambda\in\lbrack0,\lambda_{\ast}]$. We prove that the branch

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2737-2748, January 2020. This article studies ordinary differential equations modeling incompressible flow in rigid pipes that connect two distensible vessels, one of which is periodically forced. The forcing controls either the pressure or the volume of the excited vessel and---in part of the period---can be replaced by free relaxation

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2720-2736, January 2020. In this paper we introduce a new approach to computing hidden features of sampled vector fields. The basic idea is to convert the vector field data into a graph structure and use tools designed for automatic, unsupervised analysis of graphs. Using a few data sets, we show that the collected features of the

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2682-2719, January 2020. Deterministic models of vegetation often summarize, at a macroscopic scale, a multitude of intrinsically random events occurring at a microscopic scale. We bridge the gap between these scales by demonstrating convergence to a mean-field limit for a general class of stochastic models representing each individual

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2658-2681, January 2020. This work considers the gravitational $N$-body problem and introduces global time-renormalization functions that allow the efficient numerical integration with fixed time-steps. First, a lower bound of the radius of convergence of the solution to the original equations is derived, which suggests an appropriate

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2628-2657, January 2020. We study distributional solutions to the radially symmetric aggregation equation for power-law potentials. We show that distributions containing spherical shells form part of a basin of attraction in the space of solutions in the sense of “shifting stability." For spherical shell initial data, we prove the

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2594-2627, January 2020. In this paper, we provide a graphic formulation of nonisothermal reaction systems and show that a nonisothermal detailed balanced network system converges (locally) asymptotically to the unique equilibrium within the invariant manifold determined by the initial condition. To model thermal effects, the proposed

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2567-2593, January 2020. We study the dynamics and equilibria induced by training an artificial neural network for regression based on the gradient conjugate prior (GCP) updates. We show that contaminating the training data set by outliers leads to bifurcation of a stable equilibrium from infinity. Furthermore, using the outputs of

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2530-2566, January 2020. We study fast-slow maps obtained by discretization of planar fast-slow systems in continuous time. We focus on describing the so-called delayed loss of stability induced by the slow passage through a singularity in fast-slow systems. This delayed loss of stability can be related to the presence of canard solutions

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2500-2529, January 2020. Motivated by the model proposed by Gandhi et al. in [J. R. Soc. Interface, 15 (2018), 20180508], we propose a two-component reaction-advection-diffusion model for vegetation density and soil water concentration on a curved terrain with elevation given by $z = \zeta(x,y)$ and metric tensor $g(x,y)$. It accounts

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2469-2499, January 2020. In this paper we establish the meta-stability of traveling waves for a class of reaction-diffusion equations forced by a multiplicative noise term. In particular, we show that the phase-tracking technique developed in [C. H. S. Hamster and H. J. Hupkes, SIAM J. Appl. Dyn. Syst., 18, pp. 205--278; Phys. D, 401

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2428-2468, January 2020. Dynamical systems often admit geometric properties that must be taken into account when studying their behavior. We show that many such properties can be encoded by means of quiver representations. These properties include classical symmetry, hidden symmetry, and feedforward structure, as well as subnetwork