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

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2403-2427, January 2020. Using the Klein--Majda--Damodaran model of nearly parallel vortex filaments, we construct vortex knots and links on a torus involving periodic boundary conditions and analyze their stability. For a special class of vortex knots---toroidal knots---we give a full characterization of both their energetic and dynamical

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2371-2402, January 2020. We discuss tipping phenomena in nonautonomous systems using an example of a bistable ecosystem model with environmental changes represented by time-varying parameters [Scheffer et al., Ecosystems, 11 (2008), pp. 275--279]. We give simple testable criteria for the occurrence of nonautonomous tipping from the

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2343-2370, January 2020. Given $p,q\in\mathbb{Z}_{\geq 2}$ with $p\neq q$, we study generalized Abel differential equations $\frac{dx}{d\theta}=A(\theta)x^p+B(\theta)x^q,$ where $A$ and $B$ are trigonometric polynomials of degrees $n, m\ge 1,$ respectively, and we are interested in the number of limit cycles (i.e., isolated periodic

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2322-2342, January 2020. The Kuramoto model is a system of nonlinear differential equations that is often used to model synchronization between coupled oscillators in a network. A particular form of synchronization is phase-locking whereby the oscillators rotate at a common frequency with fixed angle differences. It has been observed

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2286-2321, January 2020. In this paper, we consider a class of quasi-periodically forced reversible systems, obtained as perturbations of a set of harmonic oscillators, and study Stoker's problem (the existence of response solutions) of such systems in the case of Liouvillean frequency. This is based on a finite dimensional Kolmogorov--Arnold--Moser

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2268-2285, January 2020. This note studies 1 and 2 degrees of freedom Hamiltonian systems that are thermostated by a single-variable thermostat. It provides concrete examples of integrable Hamiltonians and single thermostats, including the Nosé--Hoover thermostat, where the existence of a horseshoe in the flow of the thermostated system

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2233-2267, January 2020. We study the classical planar spin-orbit model from an analytical point of view, with no requirements of the smallness of the orbital eccentricity and taking into account dissipative forces. The problem depends on $e$, the eccentricity of the orbit, and on $\Lambda$, the oblateness of the spinning body. Our

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 2194-2231, January 2020. We rigorously prove the existence of traveling fronts in neural field models with lateral inhibition coupling types and smooth sigmoidal firing rates. With Heaviside firing rates as our base point (where unique traveling fronts exist), we repeatedly apply the implicit function theorem in Banach spaces to provide

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