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It has long been known that the excitation of fast motion in certain two-scale dynamical systems is linked to the singularity structure in complex time of the slow variables. We demonstrate that, in the context of a fast harmonic oscillator forced by one component of the Lorenz 1963 model, this principle can be used to construct time-discrete surrogate models by numerically extracting approximate locations and residues of complex poles via adaptive Antoulas--Anderson (AAA) rational interpolation and feeding this information into the known "connection formula"" to compute the resulting fast amplitude. Despite small but nonnegligible local errors, the surrogate model maintains excellent accuracy over very long times. In addition, we observe that the long-time behavior of fast energy offers a continuous-time analogue of Gottwald and Melbourne's 2004 "0-1 test for chaos""; that is, the asymptotic growth rate of the energy in the oscillator can discern whether or not the forcing function is chaotic. [ABSTRACT FROM AUTHOR]

We introduce a novel procedure that, given sparse data generated from a stationary deterministic nonlinear dynamical system, can characterize specific local and/or global dynamic behavior with rigorous probability guarantees. More precisely, the sparse data is used to construct a statistical surrogate model based on a Gaussian process (GP). The dynamics of the surrogate model is interrogated using combinatorial methods and characterized using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the confidence that these topological invariants, and hence the characterized dynamics, apply to the unknown dynamical system (assumed to be a sample path of the GP). The focus of this paper is on explaining the ideas, thus we restrict our examples to one-dimensional systems and show how to capture the existence of fixed points, periodic orbits, connecting orbits, bistability, and chaotic dynamics. [ABSTRACT FROM AUTHOR]

We investigate the effect of bias on the formation and dynamics of opinion clusters in the bounded confidence model. For weak bias, we quantify the change in average opinion and potential dispersion and decrease in cluster size. For nonlinear bias modeling self-incitement, we establish coherent drifting motion of clusters on a background of uniform opinion distribution for biases below a critical threshold where clusters dissolve. Technically, we use geometric singular perturbation theory to derive drift speeds, we rely on a nonlocal center manifold analysis to construct drifting clusters near threshold, and we implement numerical continuation in a forward-backward delay equation to connect asymptotic regimes. [ABSTRACT FROM AUTHOR]

Physics-based and first-principles models pervade the engineering and physical sciences, allowing for the ability to model the dynamics of complex systems with a prescribed accuracy. The approximations used in deriving governing equations often result in discrepancies between the model and sensor-based measurements of the system, revealing the approximate nature of the equations and/or the signal-to-noise ratio of the sensor itself. In modern dynamical systems, such discrepancies between model and measurement can lead to poor quantification, often undermining the ability to produce accurate and precise control algorithms. We introduce a discrepancy modeling framework to identify the missing physics and resolve the model-measurement mismatch with two distinct approaches: (i) by learning a model for the evolution of systematic state-space residual, and (ii) by discovering a model for the deterministic dynamical error. Regardless of approach, a common suite of data-driven model discovery methods can be used. Specifically, we use four fundamentally different methods to demonstrate the mathematical implementations of discrepancy modeling: (i) the sparse identification of nonlinear dynamics, (ii) dynamic mode decomposition, (iii) Gaussian process regression, and (iv) neural networks. The choice of method depends on one's intent (e.g., mechanistic interpretability) for discrepancy modeling, sensor measurement characteristics (e.g., quantity, quality, resolution), and constraints imposed by practical applications (e.g., state- or dynamical-space operability). We demonstrate the utility and suitability for discrepancy modeling using the suite of data-driven modeling methods on four dynamical systems under varying signal-to-noise ratios. Finally, we emphasize structural shortcomings of each discrepancy modeling approach depending on error type. In summary, if the true dynamics are unknown (i.e., an imperfect model), one should learn a discrepancy model of the missing physics in the dynamical space. Yet, if the true dynamics are known yet model-measurement mismatch still exists, one should learn a discrepancy model in the state space. [ABSTRACT FROM AUTHOR]

Generalized mass-action systems are power-law dynamical systems arising from chemical reaction networks. Essentially, every nonnegative ODE model used in chemistry and biology (for example, in ecology and epidemiology) and even in economics and engineering can be written in this form. Previous results have focused on existence and uniqueness of special steady states (complex-balanced equilibria) for all rate constants, thereby ruling out multiple (special) steady states. Recently, necessary conditions for linear stability have been obtained. In this work, we provide sufficient conditions for the linear stability of complex-balanced equilibria for all rate constants (and also for the nonexistence of other steady states). In particular, via sign vector conditions (on the stoichiometric coefficients and kinetic orders), we guarantee that the Jacobian matrix is a P-matrix. Technically, we use a new decomposition of the graph Laplacian which allows us to consider orders of (generalized) monomials. Alternatively, we use cycle decomposition which allows a linear parametrization of all Jacobian matrices. In any case, we guarantee stability without explicit computation of steady states. We illustrate our results in examples from chemistry and biology: generalized Lotka--Volterra systems and SIR models, a two-component signaling system, and an enzymatic futile cycle. [ABSTRACT FROM AUTHOR]

Recent advances in high-resolution experimental methods have highlighted the significance of cell signal pathway crosstalk and localized signaling activity in the development and disease of numerous biological systems. The investigation of multiple signal pathways often introduces different methods of cell-cell communication, i.e., contact-based or diffusive signaling, which generates both a spatial and a temporal dependence on cell behaviors. Motivated by cellular mechanisms that control cell-fate decisions in developing bilayer tissues, we use dynamical systems coupled with multilayer graphs to analyze the role of signaling polarity and pathway crosstalk in fine-grain pattern formation of protein activity. Specifically, we study how multilayer graph edge structures and weights influence the layerwise (laminar) patterning of cells in bilayer structures, which are commonly found in glandular tissues. We present sufficient conditions for the existence, uniqueness, and instability of homogeneous cell states in the large-scale spatially discrete dynamical system. Using methods of pattern templating by graph partitioning to generate quotient systems, in combination with concepts from monotone dynamical systems, we exploit the extensive dimensionality reduction to provide existence conditions for the polarity required to induce fine-grain laminar patterns with multiple spatially dependent intracellular components. We then explore the spectral links between the quotient and large-scale dynamical systems to extend the laminar patterning criteria from existence to convergence for sufficiently large amounts of cellular polarity in the large-scale dynamical system, independent of spatial dimension and number of cells in the tissue. [ABSTRACT FROM AUTHOR]

We propose to adopt statistical regression as the projection operator to enable data-driven learning of the operators in the Mori--Zwanzig formalism. We present a principled method to extract the Markov and memory operators for any regression models. We show that the choice of linear regression results in a recently proposed data-driven learning algorithm based on Mori's projection operator, which is a higher-order approximate Koopman learning method. We show that more expressive nonlinear regression models naturally fill in the gap between the highly idealized and computationally efficient Mori's projection operator and the most optimal yet computationally infeasible Zwanzig's projection operator. We performed numerical experiments and extracted the operators for an array of regression-based projections, including linear, polynomial, spline, and neural network--based regressions, showing a progressive improvement as the complexity of the regression model increased. Our proposition provides a general framework to extract memory-dependent corrections and can be readily applied to an array of data-driven learning methods for stationary dynamical systems in the literature. [ABSTRACT FROM AUTHOR]

The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space H with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace X of H and its dual space X'. The three topological spaces X ⊂ H ⊂ X' are called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron--Frobenius operators of the one-sided and twosided shifts of finite type (symbolic dynamical systems) are determined. A one-sided subshift of finite type which is conjugate to the multiplication with the golden ratio on [0, 1] modulo 1 is also considered. A new construction of the Gelfand triplet for the generalized spectrum of symbolic dynamical systems is proposed by means of an algebraic procedure. The asymptotic formula of the iteration of Perron--Frobenius operators is also given. The iteration converges to the mixing state whose rate of convergence is determined by the generalized spectrum. [ABSTRACT FROM AUTHOR]

This paper considers a deterministic susceptible-infected-susceptible (SIS) networked bivirus epidemic model (termed the bivirus model for short), in which two competing viruses spread through a set of populations (nodes) connected by two graphs, which may be different if the two viruses have different transmission pathways. The networked dynamics can give rise to complex equilibria patterns, and most current results identify conditions on the model parameters for convergence to the healthy equilibrium (where both viruses are extinct) or a boundary equilibrium (where one virus is endemic and the other is extinct). However, there are only limited results on coexistence equilibria (where both viruses are endemic). This paper establishes a set of "counting" results which provide lower bounds on the number of coexistence equilibria and, perhaps more importantly, establish properties on the local stability/instability properties of these equilibria. In order to do this, we employ the Poincaré--Hopf theorem but with significant modifications to overcome several challenges arising from the bivirus system model, such as the fact that the system dynamics do not evolve on a manifold in the typical sense required to apply Poincaré--Hopf theory. Subsequently, Morse inequalities are used to tighten the counting results, under the reasonable assumption that the bivirus system is a Morse--Smale dynamical system. Numerical examples are provided which demonstrate the presence of multiple attractor equilibria and multiple coexistence equilibria. [ABSTRACT FROM AUTHOR]

Dynamical systems have a wide range of applications in mechanics, electrical engineering, chemistry, and so on. In this work, we propose the adaptive spectral Koopman (ASK) method to solve nonlinear autonomous dynamical systems. This novel numerical method leverages the spectral-collocation (i.e., pseudospectral) method and properties of the Koopman operator to obtain the solution of a dynamical system. Specifically, this solution is represented as a linear combination of the multiplication of the Koopman operator's eigenfunctions and eigenvalues, and these eigenpairs are approximated by the spectral method. Unlike conventional time evolution algorithms such as Euler's scheme and the Runge--Kutta scheme, ASK is mesh free and hence is more flexible when evaluating the solution. Numerical experiments demonstrate high accuracy of ASK for solving one-, two-, and three-dimensional dynamical systems. [ABSTRACT FROM AUTHOR]

Kernel analog forecasting (KAF) is a methodology for data-driven, nonparametric forecasting of dynamically generated time series data. This approach has a rigorous foundation in Koopman operator theory and it produces good forecasts in practice, but it suffers from the heavy computational costs common to kernel methods. This paper proposes a streaming algorithm for KAF that only requires a single pass over the training data. This algorithm dramatically reduces the costs of training and prediction without sacrificing forecasting skill. Computational experiments demonstrate that the streaming KAF method can successfully forecast several classes of dynamical systems (periodic, quasi-periodic, and chaotic) in both data-scarce and data-rich regimes. The overall methodology may have wider interest as a new template for streaming kernel regression. [ABSTRACT FROM AUTHOR]

We develop a probabilistic characterization of trajectorial divergence rates in nonautonomous stochastic dynamical systems that are systematically shown to be suitable for uncertainty quantification in Lagrangian (trajectory-based) predictions. We construct scalar fields of finite-time divergence rates which are rooted in information theory/geometry, and we show their existence and space-time continuity for general stochastic flows. Combining these divergence rate fields with information inequalities derived in [M. Branicki and K. Uda, SIAM/ASA J. Uncertain. Quantif ., 9 (2021), pp. 1242-1313] allows for a principled quantification of Lagrangian uncertainty in a given dynamics, as well as a mitigation of the uncertainty in path-based observables estimated from simplified models of the truth/reference dynamics in a way that is amenable to algorithmic implementations. This approach can be utilized in information-geometric analysis of statistical estimation and inference, as well as in a data-driven machine learning of reduced-order models. We also derive a link between the divergence rates and finite-time Lyapunov exponents for probability measures and for path-based observables. [ABSTRACT FROM AUTHOR]

Dynamic mode decomposition (DMD) is an efficient tool for decomposing spatio-temporal data into a set of low-dimensional modes, yielding the oscillation frequencies and the growth rates of physically significant modes. In this paper, we propose a novel DMD model that can be used for dynamical systems affected by multiplicative noise. We first derive a maximum a posteriori (MAP) estimator for the data-based model decomposition of a linear dynamical system corrupted by certain multiplicative noise. Applying penalty relaxation to the MAP estimator, we obtain the proposed DMD model whose epigraphical limits are the MAP estimator and the conventional optimized DMD model. We also propose an efficient alternating gradient descent method for solving the proposed DMD model and analyze its convergence behavior. The proposed model is demonstrated on both the synthetic data and the numerically generated one-dimensional combustor data and is shown to have superior reconstruction properties compared to state-of-the-art DMD models. Considering that multiplicative noise is ubiquitous in numerous dynamical systems, the proposed DMD model opens up new possibilities for accurate data-based modal decomposition. [ABSTRACT FROM AUTHOR]

This work focuses on angular values of nonautonomous dynamical systems which have been introduced for general random and (non)autonomous dynamical systems in a previous publication [W.-J. Beyn, G. Froyland, and T. H\"uls, SIAM J. Appl. Dyn. Syst., 21 (2022), pp. 1245-1286]. The angular value of dimension s measures the maximal average rotation which an s-dimensional subspace of the phase space experiences through the dynamics of a discrete-time linear system. Our main results relate the notion of angular value to the well-known dichotomy (or Sacker--Sell) spectrum and its associated spectral bundles. In particular, we prove a reduction theorem which shows that instead of maximizing over all subspaces, it suffices to maximize over so-called trace spaces which have their basis in the spectral fibers. The reduction leads to an algorithm for computing angular values of dimensions one and two. We apply the algorithm to several systems of dimension up to 4 and demonstrate its efficiency to detect the fastest rotating subspace even if it is not dominant under the forward dynamics. [ABSTRACT FROM AUTHOR]

We study an optimal transportation approach for recovering parameters in dynamical systems with a single smoothly varying attractor. We assume that the data are not sufficient for estimating time derivatives of state variables but enough to approximate the long-time behavior of the system through an approximation of its physical measure. Thus, we fit physical measures by taking the Wasserstein distance from optimal transportation as a misfit function between two probability distributions. In particular, we analyze the regularity of the resulting loss function for general transportation costs and derive gradient formulas. Physical measures are approximated as fixed points of suitable PDE-based Perron--Frobenius operators. Test cases discussed in the paper include common low-dimensional dynamical systems. [ABSTRACT FROM AUTHOR]

One of the most important problems arising in time series analysis is that of classifying the states of a dynamical system. That is, given a collection of time series, is it possible to perform twostate classification (chaotic versus periodic) of the underlying system? For this task, we turn to the field of topological data analysis, which encodes information about the shape and structure of data. In this paper, we investigate a more recent method for encoding the structure of the attractor as a weighted graph, known as the ordinal partition network, representing information about when the dynamical system has passed between certain regions of state space. We provide methods to incorporate the weighting information and show that this framework provides more resilience to noise or perturbations in the system as well as improving the accuracy of dynamic state identification. [ABSTRACT FROM AUTHOR]

We consider the incompressible hyperdissipative Navier--Stokes equations on a two- or three-dimensional periodic torus, where the power α ξ 3/2 and the forcing function f is time-dependent. We intend to reveal how the fractional dissipation and the time-dependent force affect long-time dynamics of weak solutions. More precisely, we prove that with certain conditions on f, there exists a finite-dimensional Lipschitz manifold in the L2-space of divergent-free vector fields with zero mean. The manifold is locally forward invariant and pullback exponentially attracting. Moreover, the compact uniform attractor is contained in the union of all fibers of the manifold. In our result, no large viscosity Έ is assumed. It is also significant that in the three-dimensional case the spectrum of the fractional Laplacian (κ)3/2 does not have arbitrarily large gaps. [ABSTRACT FROM AUTHOR]

We propose a new method for computing Dynamic Mode Decomposition (DMD)evolution matrices, which we use to analyze dynamical systems. Unlike themajority of existing methods, our approach is based on a variationalformulation consisting of data alignment penalty terms and constitutiveorthogonality constraints. Our method does not make any assumptions on thestructure of the data or their size, and thus it is applicable to a wide rangeof problems including non-linear scenarios or extremely small observation sets.In addition, our technique is robust to noise that is independent of thedynamics and it does not require input data to be sequential. Our key idea isto introduce a regularization term for the forward and backward dynamics. Theobtained minimization problem is solved efficiently using the AlternatingMethod of Multipliers (ADMM) which requires two Sylvester equation solves periteration. Our numerical scheme converges empirically and is similar to aprovably convergent ADMM scheme. We compare our approach to variousstate-of-the-art methods on several benchmark dynamical systems.

We analyze a family of clustered excitatory-inhibitory neural networks and the underlying bifurcation structures that arise because of permutation symmetries in the network as the global coupling strength g is varied. We primarily consider two network topologies: an all-to-all connected network which excludes self-connections, and a network in which the excitatory cells are broken into clusters of equal size. Although in both cases the bifurcation structure is determined by symmetries in the system, the behavior of the two systems is qualitatively different. In the all-to-all connected network, the system undergoes Hopf bifurcations leading to periodic orbit solutions; notably, for large g, there is a single, stable periodic orbit solution and no stable fixed points. By contrast, in the clustered network, there are no Hopf bifurcations, and there is a family of stable fixed points for large g. [ABSTRACT FROM AUTHOR]

In this paper, we extend a recently established subgradient method for the computation of Riemannian metrics that optimizes certain singular value functions associated with dynamical systems. This extension is threefold. First, we introduce a projected subgradient method which results in Riemannian metrics whose parameters are confined to a compact convex set and we can thus prove that a minimizer exists; second, we allow inexact subgradients and study the effect of the errors on the computed metrics; and third, we analyze the subgradient algorithm for three different choices of step sizes: constant, exogenous, and Polyak. The new methods are illustrated by application to dimension and entropy estimation of the H\'enon map. [ABSTRACT FROM AUTHOR]

We present a decomposition of the Koopman and Perron-Frobenius operator based on the sparse structure of the underlying dynamical system, allowing one to consider the system as a family of subsystems interconnected by a graph. Using the intrinsic properties of the Koopman operator, we show that eigenfunctions for the subsystems induce eigenfunctions for the whole system. The use of principal eigenfunctions allows us to reverse this result. Similarly for the adjoint operator, the Perron-Frobenius operator, invariant measures for the dynamical system induce invariant measures of the subsystems, while constructing invariant measures from invariant measures of the subsystems is less straightforward. We address this question and show that under necessary compatibility assumptions such an invariant measure exists. Based on these results we demonstrate that the a priori knowledge of a decomposition of a dynamical system allows for a reduction of the computational cost on the examples of the dynamic mode decomposition and invariant measure computation. [ABSTRACT FROM AUTHOR]

An extended dynamical system is considered that shows some striking, very complex spatio-temporal patterns. Specifically, we consider superhighway patterns that appear in binary fluid convection in slightly inclined, shallow cylindrical containers. These patterns show a number of parallel thermal lanes, each containing aligned coherent structures that counterpropagate in adjacent lanes. Several types of superhighway convection states have been obtained by direct numerical simulation. The numerical outcomes are analyzed using a recent data processing tool, known as higher order dynamic mode decomposition, which efficiently identifies relevant spatio-temporal patterns in numerically computed data. [ABSTRACT FROM AUTHOR]

This manuscript develops a novel understanding of nonpolar solutions of the discrete Painlev\'e I equation (dP1). As the nonautonomous counterpart of an analytically completely integrable difference equation, this system is endowed with a rich dynamical structure. In addition, its nonpolar solutions, which grow without bounds as the iteration index n increases, are of particular relevance to other areas of mathematics. We combine theory and asymptotics with high-precision numerical simulations to arrive at the following picture: when extended to include backward iterates, known nonpolar solutions of dP1 form a family of heteroclinic connections between two fixed points at infinity. One of these solutions, the Freud orbit of orthogonal polynomial theory, is a singular limit of the other solutions in the family. Near their asymptotic limits, all solutions converge to the Freud orbit, which follows invariant curves of dP1, when written as a three-dimensional autonomous system, and reaches the point at positive infinity along a center manifold. This description leads to two important results. First, the Freud orbit tracks sequences of period-1 and 2 points of the autonomous counterpart of dP1 for large positive and negative values of n, respectively. Second, we identify an elegant method to obtain an asymptotic expansion of the iterates on the Freud orbit for large positive values of n. The structure of invariant manifolds emerging from this picture contributes to a deeper understanding of the global analysis of an interesting class of discrete dynamical systems. [ABSTRACT FROM AUTHOR]

An analytical study of dynamical properties of a semiconductor laser with optical injection of arbitrary polarization is presented. It is shown that if the injected field is sufficiently weak, then the laser has nine equilibrium points; however, only one of them is stable. Even if the injected field is linearly polarized, six of the equilibrium points have a state of polarization that is elliptical. Dependence of the equilibrium points on the injected field is described, and it is shown that as the intensity of the injected field increases, the number of equilibrium points decreases, with only a single equilibrium point remaining for strong enough injected fields. As an application, a complex-valued optical neural network with working principle based on injection locking is proposed. [ABSTRACT FROM AUTHOR]

The sensitivity of time averages in a chaotic system to an infinitesimal parameter perturbation grows exponentially with the averaging time. However, long-term averages or ensemble statistics often vary differentiably with system parameters. Ruelle's response theory gives a rigorous formula for these parametric derivatives of statistics or linear response. But the direct evaluation of this formula is ill-conditioned, and hence linear response and downstream applications of sensitivity analysis, such as optimization and uncertainty quantification, have been a computational challenge in chaotic dynamical systems. This paper presents the space-split sensitivity (S3) algorithm to transform Ruelle's formula into a well-conditioned ergodic-averaging computation. We prove a decomposition of Ruelle's formula that is differentiable on the unstable manifold, which we assume to be one-dimensional. This decomposition of Ruelle's formula ensures that one of the resulting terms, the stable contribution, can be computed using a regularized tangent equation, similarly as in a nonchaotic system. The remaining term, known as the unstable contribution, is regularized and converted into an efficiently computable ergodic average. In this process, we develop new algorithms, which may be useful beyond linear response, to compute the unstable derivatives of the regularized tangent vector field and the unstable direction. We prove that the S3 algorithm, which combines these computational ingredients that enter the stable and unstable contributions, converges like a Monte Carlo approximation of Ruelle's formula. The algorithm presented here is hence a first step toward full-fledged applications of sensitivity analysis in chaotic systems, wherever such applications have been limited due to lack of availability of long-term sensitivities. [ABSTRACT FROM AUTHOR]

We derive an explicit form for the Cucker--Smale (CS) model on the special orthogonal group SOp3q by identifying closed form expressions for geometric quantities such as covariant derivative and parallel transport in exponential coordinates. We study the emergent dynamics of the model by using a Lyapunov functional approach and LaSalle's invariance principle. Specifically, we show that velocity alignment emerges from some admissible class of initial data, under suitable assumptions on the communication weight function. We characterize the \omega -limit set of the dynamical system and identify a dichotomy in the asymptotic behavior of solutions. Several numerical examples are provided to support the analytical results. [ABSTRACT FROM AUTHOR]

Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The first captures changes in the Conley index, while the second captures changes in the Morse decomposition. However, such approaches have limitations. The former approach only describes how the Conley index changes across a selected isolated invariant set though the dynamics can be much more complicated than the behavior of a single isolated invariant set. Likewise, considering a Morse decomposition omits much information about the individual Morse sets. In this paper, we propose a method to summarize changes in combinatorial dynamical systems by capturing changes in the so-called Conley--Morse graphs. A Conley--Morse graph contains information about both the structure of a selected Morse decomposition and about the Conley index at each Morse set in the decomposition. Hence, our method summarizes the changing structure of a sequence of dynamical systems at a finer granularity than previous approaches. [ABSTRACT FROM AUTHOR]

We introduce the notion of angular values for deterministic linear difference equations and random linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on long-term averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For one-dimensional subspaces in two-dimensional systems our angular values agree with the classical theory of rotation numbers for orientation-preserving circle homeomorphisms if the matrix has positive determinant and does not rotate vectors by more than π 2 . Because our notion of angular values ignores orientation by looking at subspaces rather than vectors, our results apply to dynamical systems of any dimension and to subspaces of arbitrary dimension. The second part of the paper delves deeper into the theory of the autonomous case. We explore the relation to (generalized) eigenspaces, provide some explicit formulas for angular values, and set up a general numerical algorithm for computing angular values via Schur decompositions. [ABSTRACT FROM AUTHOR]

In this article we develop further an algorithm for data assimilation based upon a shadowing refinement technique [de Leeuw et al., SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 2446--2477] to take partial observations into account. Our method is based on a regularized Gauss--Newton method. We prove local convergence to the solution manifold and provide a lower bound on the algorithmic time step. We use numerical experiments with the Lorenz 63 and Lorenz 96 models to illustrate convergence of the algorithm and show that the results compare favorably with a variational technique---weakconstraint four-dimensional variational method---and a shadowing technique--pseudo-orbit data assimilation. Numerical experiments show that a preconditioner chosen based on a cost function allows the algorithm to find an orbit of the dynamical system in the vicinity of the true solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider an N-dimensional queueing model where customers join according to a multinomial logit choice model. However, we assume that the information that the customer receives about the queue length is delayed by a constant\Delta. We review how a large customer scaling of the stochastic model yields a system of delay differential equations. We also show how these limiting delay differential equations approximate the stochastic mean dynamics of the original queueing model. To gain insight about the queue length dynamics such as the amplitude, the frequency, and the value of the critical delay we use the harmonic balance method and the method of multiple scales. We show that the method of multiple scales is more accurate but is less tractable from a computational perspective than the harmonic balance method. Using the method of multiple scales, we also prove that the only stable mode is where all N queues have the same amplitude and frequency; however, each queue is shifted by 2\π/N from its neighbor. This analysis provides great insights for queues with delayed information and how the oscillations manifest themselves. [ABSTRACT FROM AUTHOR]

Microchannels are well known in microfluidic applications for the control and separation of microdroplets and cells. Often the objects in the flow experience inertial effects, resulting in dynamics that is a departure from the underlying channel flow dynamics. This paper considers small neutrally buoyant spherical particles suspended in flow through a curved duct having a square cross-section. The particle experiences a combination of inertial lift force induced by the disturbance from the primary flow along the duct, and drag from the secondary vortices in the cross-section, which drive migration of the particle within the cross-section. We construct a simplified model that preserves the core topology of the force field yet depends on a single parameter κ, quantifying the relative strength of the two forces. We show that κ is a bifurcation parameter for the dynamical system that describes motion of the particle in the cross-section of the duct. At large values of κ there exists an attracting limit cycle in each of the upper and lower halves of the duct. At small κ we find that particles migrate to one of four stable foci. Between these extremes, there is an intermediate range of κ for which all particles migrate to a single stable focus. Noting that the positions of the limit cycles and foci vary with the value of κ, this behavior indicates that, for a suitable particle mixture, duct bend radius might be chosen to segregate particles by size. We evaluate the time and axial distance required to focus particles near the unique stable node, which determines the duct length required for particle segregation. [ABSTRACT FROM AUTHOR]

We investigate the nonlinear mechanisms of generation of preferred frequency response patterns to oscillatory inputs in a phenomenological (caricature) model that captures the type of nonlinearities present in electrochemical (EC) cells and is simple enough to begin to develop the dynamical systems tools necessary for the understanding of the relationship between the model nonlinearities and the response patterns. Previous work has shown that linearized EC models exhibit resonance (preferred frequency response to oscillatory inputs at a nonzero, resonant, input frequency) in parameter regimes where the stable equilibrium is either a node (no intrinsic oscillations) or a focus (damped oscillations). We use a combination of numerical simulations and dynamical systems tools to understand how the model nonlinearities, partially captured by the geometry of the nullclines in the phase-space diagram, shape the response patterns to oscillatory inputs away from the validity of the corresponding linearization. We develop and adapt an extended version of the classical phase-plane diagram that allows us to track the evolution of the response trajectories and their interaction with the so-called moving nullclines (in response to the oscillatory inputs). We use this approach to explain the mechanisms of generation of nonlinear resonant patterns, the nonlinear amplification/attenuation of these patterns by increasing input amplitudes, and the mechanisms of generation of more complex patterns of mixed-mode--type, where the stationary amplitude response is not uniform across cycles. [ABSTRACT FROM AUTHOR]

We consider a harvesting strategy that allows constant catches if the population size is above a certain threshold value (to obtain predictable yield) and no catches if the population size is below the threshold (to protect the population). We refer to this strategy as threshold constant-catch (TCC) harvesting. We provide analytical and numerical results when applying TCC to monotone population growth models. TCC remedies the tendency to fishery collapse of pure constant-catch harvesting and provides a buffer for quotas larger than the maximum sustainable yield. From a dynamical systems point of view, TCC gives rise to a piecewise-smooth map with a discontinuity at the threshold population size. The dynamical behavior includes border-collision bifurcations, basin boundary metamorphoses, and boundary-collision bifurcation. We further find Farey trees, a slightly modified truncated skew tent map scenario, and the bandcount incrementing scenario. Our results underline, on the one hand, the protective function of thresholds in harvest control rules. On the other hand, they highlight the dynamical complexities due to discontinuities that can arise naturally in threshold-based harvesting strategies. [ABSTRACT FROM AUTHOR]

We consider a broad class of second-order dynamical systems and study the role of damping as a system parameter in the stability, hyperbolicity, and bifurcation in such systems. We prove a monotonic effect of damping on the hyperbolicity of the equilibrium points of the corresponding firstorder system. This provides a rigorous formulation and theoretical justification for some intuitive notions that damping increases stability. To establish this result, we prove a matrix perturbation result for complex symmetric matrices with positive semidefinite perturbations to their imaginary parts, which may be of independent interest. Furthermore, we establish necessary and sufficient conditions for the breakdown of hyperbolicity of the first-order system under damping variations in terms of observability of a pair of matrices relating damping, inertia, and Jacobian matrices, and propose sufficient conditions for Hopf bifurcation resulting from such hyperbolicity breakdown. The developed theory has significant applications in the stability of electric power systems, which are one of the most complex and important engineering systems. In particular, we characterize the impact of damping on the hyperbolicity of swing equations, which is the fundamental dynamical model of power systems, and demonstrate Hopf bifurcations resulting from damping variations. [ABSTRACT FROM AUTHOR]

Certain neural network architectures, in the infinite-depth limit, lead to systems of nonlinear differential equations. Motivated by this idea, we develop a framework for analyzing time signals based on nonautonomous dynamical systems. We view the time signal as a forcing function for a dynamical system that governs a time-evolving hidden variable. As in equation discovery, the dynamical system is represented using a dictionary of functions and the coefficients are learned from data. This framework is applied to the time signal classification problem. We show how gradients can be efficiently computed using the adjoint method, and we apply methods from dynamical systems to establish stability of the classifier. Through a variety of experiments, on both synthetic and real datasets, we show that the proposed method uses orders of magnitude fewer parameters than competing methods, while achieving comparable accuracy. We created the synthetic datasets using dynamical systems of increasing complexity; though the ground truth vector fields are often polynomials, we find consistently that a Fourier dictionary yields the best results. We also demonstrate how the proposed method yields graphical interpretability in the form of phase portraits. [ABSTRACT FROM AUTHOR]

Neuronal networks vary dramatically in size, connectivity structure, and functionality across downstream layers of the brain. This raises the question of whether information is lost as it is re-encoded along compressive and expansive pathways. In this work, we develop a potential data-driven mechanism for the preservation of information in the activity of neuronal networks across downstream layers, which uses the widespread linearity of individual neuronal responses to sufficiently strong ramped artificial inputs to fit a linear input-output mapping across the network. We analyze the dynamics of several families of two-layer neuronal network models, where the input components far outnumber the downstream neurons, as in compressive pathways, and apply the fitted mapping in conjunction with compressive sensing theory to reconstruct stimuli with sparse structure. The input-output mapping facilitates stimulus reconstructions that only use measurements of downstream neuronal firing rates in response to inputs over a short time duration, furnishing stimulus recovery even when theoretical analysis is intractable or the governing equations of the dynamical system are unknown as in experiment. Similarly accurate stimulus reconstructions are obtained across different single-neuron models, network coupling functions, and image classes. Improved reconstructions are yielded when uniformly random feedforward connectivity is replaced by spatially localized feedforward connectivity akin to receptive fields. We expect that similar principles could be leveraged experimentally in prosthetics as well as in the reconstruction of large-scale network connectivity. [ABSTRACT FROM AUTHOR]

A theoretical framework which unifies the conventional Mori--Zwanzig formalism and the approximate Koopman learning of deterministic dynamical systems from noiseless observation is presented. In this framework, the Mori--Zwanzig formalism, developed in statistical mechanics to tackle the hard problem of construction of reduced-order dynamics for high-dimensional dynamical systems, can be considered as a natural generalization of the Koopman description of the dynamical system. We next show that, similar to the approximate Koopman learning methods, data-driven methods can be developed for the Mori--Zwanzig formalism with Mori's linear projection operator. We have developed two algorithms to extract the key operators, the Markov and the memory kernel, using time series of a reduced set of observables in a dynamical system. We have adopted the Lorenz '96 system as a test problem and solved for the above operators. These operators exhibit complex behaviors, which are unlikely to be captured by traditional modeling approaches in Mori--Zwanzig analysis. The nontrivial generalized fluctuation-dissipation relationship, which relates the memory kernel with the two-time correlation statistics of the orthogonal dynamics, was numerically verified as a validation of the solved operators. We present numerical evidence that the generalized Langevin equation, a key construct in the Mori--Zwanzig formalism, is more advantageous in predicting the evolution of the reduced set of observables than the conventional approximate Koopman operators. [ABSTRACT FROM AUTHOR]

Work on standard piecewise-smooth (PWS) dynamical systems, with codimension-1 discontinuity sets, relies on the Filippov framework, which does not always readily generalize to systems with higher codimension discontinuities. These higher order degeneracies occur in applications, with spatial Coulomb friction being the prominent example. In this work, we consider PWS systems with isolated codimension-2 discontinuity sets using regularization and blowup to study the dynamics. We present a general framework and give specific results for the local dynamics in a class of problems, and generalizing Filippov sliding, crossing, and sliding vector fields. [ABSTRACT FROM AUTHOR]

The study of motifs can help researchers uncover links between the structure and function of networks in biology, sociology, economics, and many other areas. Empirical studies of networks have identified feedback loops, feedforward loops, and several other small structures as "motifs"" that occur frequently in real-world networks and may contribute by various mechanisms to important functions in these systems. However, these mechanisms are unknown for many of these motifs. We propose to distinguish between "structure motifs"" (i.e., weakly connected graphlets) in networks and "process motifs"" (which we define as structured sets of walks) on networks and consider process motifs as building blocks of processes on networks. Using steady-state covariance and steady-state correlation in a multivariate Ornstein--Uhlenbeck process on a network as examples, we demonstrate that distinguishing between structure motifs and process motifs makes it possible to gain quantitative insights into mechanisms that contribute to important functions of dynamical systems on networks. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a mass-spring-friction oscillator with the friction modeled by a regularized stiction model. We do so in the limit where the ratio of the natural spring frequency and the forcing frequency is on the same order of magnitude as the scale associated with the regularized stiction model. The motivation for studying this special parameter regime (which can be interpreted as a rigid body limit) comes from [E. Bossolini, M. Br{\e}ns, and K. U. Kristiansen, SIAM J. Appl. Dyn. Syst., 16, (2017), pp. 2233-2258] which demonstrated new friction phenomena in this regime. The results of this paper led to some open problems; see also [E. Bossolini, M. Br{\e}ns, and K. U. Kristiansen, SIAM Rev., 62 (2020), pp. 869-897], that we resolve in this paper. In particular, using GSPT and blowup [C. K. R. T. Jones, in Dynamical Systems, Lecture Notes in Math. 1609, Springer, Berlin, 1995, pp. 44-118; M. Krupa and P. Szmolyan, SIAM J. Math. Anal., 33, (2001), pp. 286-314], we provide a simple geometric description of the bifurcation of stick-slip limit cycles through a combination of a canard and a global return mechanism. We also show that this combination leads to a canard-based horseshoe and are therefore able to prove existence of chaos in this fundamental oscillator system. [ABSTRACT FROM AUTHOR]

For signals driven by multiple separate and decoupled dynamical systems, traditional methods in signal analysis often fail to decompose the signal into the parts driven by the different dynamics. Two signals from different dynamical systems may have similar energy at every frequency, so techniques such as the Fourier transform will be unable to separate a time series if it contains multiple such signals. This is especially true when the dynamical system is chaotic, as the produced signals are inherently broadband. We present how an algorithm called shadow manifold interpolation (SMI), inspired by recent advances in applied dynamical systems theory, can succeed in this task. SMI takes two causally related signals and reconstructs one from the other, producing a reconstruction that only captures the shared dynamics between the two signals, i.e., the portion of the signals driven by some underlying dynamical systems that contribute to both signals. From this we produce a decomposition of the signal into different signals representing separate dynamics. Furthermore, we demonstrate the effectiveness of SMI at decomposing signals in a variety of test cases. We consider three ways in which two signals may be causally related. Through testing the algorithm on simulated composite dynamical systems, we demonstrate that SMI succeeds in separating out the constituent systems in two schemes, failing only when the two signals share multiple decoupled dynamics. The main limitation of this algorithm is that a second causally related reference signal is needed in order to decompose a given signal. This reference signal must share at least one dynamic of interest with the given signal. [ABSTRACT FROM AUTHOR]

In this work we characterize the set of persistent species in dynamical systems related to chemical reaction networks. Chemical reaction networks consist of a set of species and a set of reaction rules describing the interactions between the species. From a reaction network, by differentiation with respect to time and space, a reaction-diffusion system (RDS) can be derived describing the dynamics of the concentrations of the species. We show how double integration with respect to time and space of the solutions of the RDS conversely leads back to the reaction network and reveals all the possibly persistent subsets of species. We show that organizations as defined in chemical organization theory (COT) are strongly related to the persistent subsets of species. Organizations are subsets of species that have two properties. First, they are closed, that is, there is no reaction running on them that produces new species which are not contained in the organizations. Second, organizations are selfmaintaining. By additionally allowing for the distribution of species, we generalize organizations towards distributed organizations (DOs). After introducing our concept of persistence as the first main result of this study, we prove that for a given reaction network the set of DOs is always a lattice. The second main result is that the set of persistent species of a solution of an RDS is always a DO. By linking these two results we achieve a connection between persistence concerning a single solution of an RDS and persistence with regard to all solutions of all RDSs having one and the same underlying reaction network. We show how this strongly benefits reaction network analysis. By presenting simulation results performed with MATLAB, we illustrate the discussed phenomena. [ABSTRACT FROM AUTHOR]

We consider a planar dynamical system generated by two stable linear vector fields with distinct fixed points and random switching between them. We characterize singularities of the invariant density in terms of the switching rates and contraction rates. We prove boundedness away from those singularities. We also discuss some motivating biological examples. [ABSTRACT FROM AUTHOR]

Given the high dimensionality and underlying complexity of many oscillatory dynamical systems, phase reduction is often an imperative first step in control applications where oscillation timing and entrainment are of interest. Unfortunately, most phase reduction frameworks place restrictive constraints on the magnitude of allowable inputs, limiting the practical utility of the resulting phase reduced models in many situations. In this work, motivated by the search for control strategies to hasten recovery from jet-lag caused by rapid travel through multiple time zones, the efficacy of the recently developed adaptive phase-amplitude reduction is considered for manipulating oscillation timing in the presence of a large magnitude entraining stimulus. The adaptive phase-amplitude reduced equations allow for a numerically tractable optimal control formulation, and the associated optimal stimuli significantly outperform those resulting from previously proposed optimal control formulations. Additionally, a data-driven technique to identify the necessary terms of the adaptive phase-amplitude reduction is proposed and validated using a model describing the aggregate oscillations of a large population of coupled limit cycle oscillators. Such data-driven model reduction algorithms are essential in situations where the underlying model equations are either unreliable or unavailable. [ABSTRACT FROM AUTHOR]

We use the theory of Koopman (or composition) operators to study the dynamical system on N∗ emerging from the well-known 3x+1 (or Collatz) problem. The advantage of this approach is that the conjecture can be described via bounded linear operators. We study the properties of these operators and their connection with the problem and the trajectories of the dynamical system. Furthermore, in order to investigate the trajectories, we utilize Fourier transform techniques to reveal the frequency content of the sequences of signs emerging from these trajectories. This method also enables us to define an isometry on a Hilbert space. Finally, the C∗-algebra generated by this isometry is used in order to study how the sequences of signs correlate with each other. [ABSTRACT FROM AUTHOR]

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 of a rumor within a homogeneously mixing population. In the well-known basic model, the population is made up of "spreaders,"" "ignorants,"" and "stiflers,"" and any spreader attempts to transmit the rumor to the other individuals via directed contacts. In case the contacted individual is an ignorant, it becomes a spreader, while in the other two cases the initiating spreader turns into a stifler. The process in a finite population will eventually reach an equilibrium situation, where individuals are either stiflers or ignorants. We generalize the model by assuming that each ignorant becomes a spreader only after hearing the rumor a predetermined number of times. We identify and analyze a suitable limiting dynamical system of the model, and we prove limit theorems that characterize the ultimate proportion of individuals in the different classes of the population. [ABSTRACT FROM AUTHOR]

The understanding of how spatio-temporal patterns of neural activity may arise in the cortex of the brain has advanced with the development and analysis of neural field models. Replicating this success for subcortical tissues, such as the thalamus, requires an extension to include relevant ionic currents that can further shape firing response. Here we advocate for one such approach that can accommodate slow currents. By way of illustration we focus on incorporating a T-type calcium current into the standard neural field framework. Direct numerical simulations are used to show that the resulting tissue model has many of the properties seen in more biophysically detailed model studies and, most important, the generation of oscillations, waves, and patterns that arise from rebound firing. To explore the emergence of such solutions we focus on one- and two-dimensional spatial models and show that exact solutions describing homogeneous oscillations can be constructed in the limit that the firing rate nonlinearity is a Heaviside function. A linear stability analysis, using techniques from nonsmooth dynamical systems, is used to determine the points at which bifurcations from synchrony can occur. Furthermore, we construct periodic traveling waves and investigate their stability with the use of an appropriate Evans function. The stable branches of the dispersion curve for periodic traveling waves are found to be in excellent agreement with simulations initiated from an unstable branch of the synchronous solution. [ABSTRACT FROM AUTHOR]

This paper investigates fundamental properties of a new class of dynamical systems, which are everywhere smooth except for a codimension-n discontinuity manifold with an arbitrary positive integer n. Such systems emerge naturally in modeling the motion of bodies with spatial point contacts as well as with finite contact surfaces under dry friction. As a special case, the investigated class includes Filippov systems (n = 1) as well as the recently introduced extended Filippov systems (n = 2). Trajectories reaching the discontinuity manifold are studied in detail, and new types of pathological behavior are uncovered, in systems where the local dynamics around the discontinuity manifold involves polycycles or strange attractors. The concept of crossing and sliding dynamics is extended for this type of system. The results are illustrated by several examples. [ABSTRACT FROM AUTHOR]

We consider weighted coupled cell networks, that is networks where the interactions between any two cells have an associated weight that is a real valued number. Weighted networks are ubiquitous in real-world applications. We consider a dynamical systems perspective by associating with each network a set of continuous dynamical systems, the ones that respect the graph structure of the network. For weighted networks it is natural for the admissible coupled cell systems to have an additive input structure. We present a characterization of the synchrony subspaces and the antisynchrony subspaces for a weighted network depending on the restrictions that are imposed in their admissible input-additive coupled cell systems. These subspaces are flow invariant by those systems and are generalized polydiagonal subspaces, that is, are characterized by conditions on the cell coordinates of the types xi = xj and/or xk = xl and/or xm = 0. The existence and identification of the synchrony and antisynchrony subspaces for a weighted network are deeply relevant from the applications and dynamics points of view. Our characterization of the synchrony and antisynchrony subspaces of a weighted network follows from our results where we give necessary and sufficient conditions for a generalized polydiagonal to be invariant by the adjacency matrix and/or the Laplacian matrix of the network. [ABSTRACT FROM AUTHOR]

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 infinitedimensional dynamical systems, we demonstrate that this reservoir should have some desirable properties called the echo state property and the separation property. By considering the projection of the system onto the inertial manifold, we describe the behavior of the reservoir as a system of finite-dimensional ordinary differential equations called an inertial form. Next, we present a finitedimensional model that approximates the inertial form. Finally, we propose a temporally discretized model, which is more suitable for computer computation. We give the theoretical proofs of our theorems herein. [ABSTRACT FROM AUTHOR]