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Contact defects are time-periodic patterns in one space dimension that resemble spatially homogeneous oscillations with a defect embedded in their core region. For theoretical and numerical purposes, it is important to understand whether these defects persist when the domain is truncated to large spatial intervals, supplemented by appropriate boundary conditions. The present work shows that truncated contact defects exist and are unique on sufficiently large spatial intervals. [ABSTRACT FROM AUTHOR]

We consider the existence and spectral stability of periodic multi-pulse solutions in Hamiltonian systems which are translation invariant and reversible, for which the fifth-order Korteweg-de Vries equation is a prototypical example. We use Lin's method to construct multi-pulses on a periodic domain, and in particular demonstrate a pitchfork bifurcation structure for periodic double pulses. We also use Lin's method to reduce the spectral problem for periodic multi-pulses to computing the determinant of a block matrix, which encodes both eigenvalues resulting from interactions between neighboring pulses and eigenvalues associated with the essential spectrum. We then use this matrix to compute the spectrum associated with periodic single and double pulses. Most notably, we prove that brief instability bubbles form when eigenvalues collide on the imaginary axis as the periodic domain size is altered. These analytical results are all in good agreement with numerical computations, and numerical timestepping experiments demonstrate that these instability bubbles correspond to oscillatory instabilities. [ABSTRACT FROM AUTHOR]

Planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction. In this paper, we investigate the stability of planar wave trains in reaction-diffusion systems. We establish nonlinear diffusive stability against perturbations that are bounded along a line in R 2 and decay exponentially in the distance from this line. Our analysis is the first to treat spatially nonlocalized perturbations that do not originate from a phase modulation. We also consider perturbations that are fully localized and establish nonlinear stability with better decay rates, suggesting a trade-off between spatial localization of perturbations and temporal decay rate. Our stability analysis utilizes pointwise estimates to exploit the spatial structure of the perturbations. The nonlocalization of perturbations prevents the use of damping estimates in the nonlinear iteration scheme; instead, we track the perturbed solution in two different coordinate systems. [ABSTRACT FROM AUTHOR]

Localized planar patterns in spatially extended bistable systems are known to exist along intricate bifurcation diagrams, which are commonly referred to as snaking curves. Their analysis is challenging as techniques such as spatial dynamics that have been used to explain snaking in one space dimension no longer work in the planar case. Here, we consider bistable systems posed on square lattices and provide an analytical explanation of snaking near the anti-continuum limit using Lyapunov–Schmidt reduction. We also establish stability results for localized patterns, discuss bifurcations to asymmetric states, and provide further numerical evidence that the shape of snaking curves changes drastically as the coefficient that reflects the strength of the spatial coupling crosses a finite threshold. [ABSTRACT FROM AUTHOR]

We investigate stationary, spatially localized patterns in lattice dynamical systems that exhibit bistability. The profiles associated with these patterns have a long plateau where the pattern resembles one of the bistable states, while the profile is close to the second bistable state outside this plateau. We show that the existence branches of such patterns generically form either an infinite stack of closed loops (isolas) or intertwined s-shaped curves (snaking). We then use bifurcation theory near the anti-continuum limit, where the coupling between edges in the lattice vanishes, to prove existence of isolas and snaking in a bistable discrete real Ginzburg–Landau equation. We also provide numerical evidence for the existence of snaking diagrams for planar localized patches on square and hexagonal lattices and outline a strategy to analyse them rigorously. [ABSTRACT FROM AUTHOR]

Spiral wave patterns observed in models of cardiac arrhythmias and chemical oscillations develop alternans and stationary line defects, which can both be thought of as period-doubling instabilities. These instabilities are observed on bounded domains and may be caused by the spiral core, far-field asymptotics, or boundary conditions. Here, we introduce a methodology to disentangle the impacts of each region on the instabilities by analyzing spectral properties of spiral waves and boundary sinks on bounded domains with appropriate boundary conditions. We apply our techniques to spirals formed in reaction-diffusion systems to investigate how and why alternans and line defects develop. Our results indicate that the mechanisms driving these instabilities are quite different; alternans are driven by the spiral core, whereas line defects appear from boundary effects. Moreover, we find that the shape of the alternans eigenfunction is due to the interaction of a point eigenvalue with curves of continuous spectra. [ABSTRACT FROM AUTHOR]

We consider Reaction-Diffusion systems on R , and prove diffusive mixing of asymptotic states u 0 (k x - ϕ ± , k) , where u0 is a spectrally stable periodic wave. Our analysis is the first to treat arbitrarily large phase-offsets ϕ d = ϕ + - ϕ - , so long as this offset proceeds in a sufficiently regular manner. The offset ϕ d completely determines the size of the asymptotic profiles in any topology, placing our analysis in the large data setting. In addition, the present result is a global stability result, in the sense that the class of initial data considered is not near the asymptotic profile in any sense. We prove the global existence, decay, and asymptotic self-similarity of the associated wavenumber equation. We develop a functional analytic framework to handle the linearized operator around large Burgers profiles via the exact integrability of the underlying Burgers flow. This framework enables us to prove a crucial, new mean-zero coercivity estimate, which we then combine with a nonlinear energy method. [ABSTRACT FROM AUTHOR]

Abstract Spatially localized patterns have been observed in numerous physical contexts, and their bifurcation diagrams often exhibit similar snaking behavior: symmetric solution branches, connected by bifurcating asymmetric solution branches, wind back and forth in an appropriate parameter. Previous papers have addressed existence of such solutions; here we address their stability, taking the necessary first step of unifying existence and uniqueness proofs for symmetric and asymmetric solutions. We then show that, under appropriate assumptions, temporal eigenvalues of the front and back underlying a localized solution are added with multiplicity in the right half plane. In a companion paper, we analyze the behavior of eigenvalues at λ = 0 and inside the essential spectrum. Our results show that localized snaking solutions are stable if, and only if, the underlying fronts and backs are stable: unlike localized non-oscillatory solutions, no interaction eigenvalues are present. We use the planar Swift–Hohenberg system to illustrate our results. [ABSTRACT FROM AUTHOR]

Planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction. In this paper, we investigate the stability of planar wave trains in reaction–diffusion systems. We establish nonlinear diffusive stability against perturbations that are bounded along a line in R 2 and decay exponentially in the distance from this line. Our analysis is the first to treat spatially nonlocalized perturbations that do not originate from a phase modulation. We also consider perturbations that are fully localized and establish nonlinear stability with better decay rates, suggesting a trade-off between spatial localization of perturbations and temporal decay rate. Our stability analysis utilizes pointwise estimates to exploit the spatial structure of the perturbations. The nonlocalization of perturbations prevents the use of damping estimates in the nonlinear iteration scheme; instead, we track the perturbed solution in two different coordinate systems. [ABSTRACT FROM AUTHOR]

The FitzHugh{Nagumo equations are known to admit fast traveling pulse solutions with monotone tails. It is also known that this system admits traveling pulses with exponentially decaying oscillatory tails. Upon numerical continuation in parameter space, it has been observed that the oscillations in the tails of the pulses grow into a secondary excursion resembling a second copy of the primary pulse. In this paper, we outline in detail the geometric mechanism responsible for this single-to-double-pulse transition, and we construct the transition analytically using geometric singular perturbation theory and blow-up techniques. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give a short self-contained proof of the center-manifold theorem for maps and vector fields in finite-dimensional spaces using the graph transform. In particular, regularity of the center manifold is established using a direct argument that is based on the closedness of sets of differentiable functions whose highest derivatives are Lipschitz continuous in the space of continuous functions; this argument avoids the fiber contraction theorem that is commonly used in this context. [ABSTRACT FROM AUTHOR]

Numerical studies indicate that the FitzHugh-Nagumo system exhibits stable traveling pulses with oscillatory tails. In this paper, the existence of such pulses is proved analytically in the singular perturbation limit near parameter values where the FitzHugh-Nagumo system exhibits folds. In addition, the stability of these pulses is investigated numerically, and a mechanism is proposed that explains the transition from single to double pulses that was observed in earlier numerical studies. The existence proof utilizes geometric blow-up techniques combined with the exchange lemma: the main challenge is to understand the passage near two fold points on the slow manifold where normal hyperbolicity fails. [ABSTRACT FROM AUTHOR]

Recent advances in sequencing technologies have allowed us to capture various aspects of the genome at single-cell resolution. However, with the exception of a few of co-assaying technologies, it is not possible to simultaneously apply different sequencing assays on the same single cell. In this scenario, computational integration of multi-omic measurements is crucial to enable joint analyses. This integration task is particularly challenging due to the lack of sample-wise or feature-wise correspondences. We present single-cell alignment with optimal transport (SCOT), an unsupervised algorithm that uses the Gromov–Wasserstein optimal transport to align single-cell multi-omics data sets. SCOT performs on par with the current state-of-the-art unsupervised alignment methods, is faster, and requires tuning of fewer hyperparameters. More importantly, SCOT uses a self-tuning heuristic to guide hyperparameter selection based on the Gromov–Wasserstein distance. Thus, in the fully unsupervised setting, SCOT aligns single-cell data sets better than the existing methods without requiring any orthogonal correspondence information. [ABSTRACT FROM AUTHOR]

Although the availability of various sequencing technologies allows us to capture different genome properties at single-cell resolution, with the exception of a few co-assaying technologies, applying different sequencing assays on the same single cell is impossible. Single-cell alignment using optimal transport (SCOT) is an unsupervised algorithm that addresses this limitation by using optimal transport to align single-cell multiomics data. First, it preserves the local geometry by constructing a k-nearest neighbor (k-NN) graph for each data set (or domain) to capture the intra-domain distances. SCOT then finds a probabilistic coupling matrix that minimizes the discrepancy between the intra-domain distance matrices. Finally, it uses the coupling matrix to project one single-cell data set onto another through barycentric projection, thus aligning them. SCOT requires tuning only two hyperparameters and is robust to the choice of one. Furthermore, the Gromov-Wasserstein distance in the algorithm can guide SCOT's hyperparameter tuning in a fully unsupervised setting when no orthogonal alignment information is available. Thus, SCOT is a fast and accurate alignment method that provides a heuristic for hyperparameter selection in a real-world unsupervised single-cell data alignment scenario. We provide a tutorial for SCOT and make its source code publicly available on GitHub. [ABSTRACT FROM AUTHOR]

Identifying unique parameters for mathematical models describing biological data can be challenging and often impossible. Parameter identifiability for partial differential equations models in cell biology is especially difficult given that many established in vivo measurements of protein dynamics average out the spatial dimensions. Here, we are motivated by recent experiments on the binding dynamics of the RNA-binding protein PTBP3 in RNP granules of frog oocytes based on fluorescence recovery after photobleaching (FRAP) measurements. FRAP is a widely-used experimental technique for probing protein dynamics in living cells, and is often modeled using simple reaction-diffusion models of the protein dynamics. We show that current methods of structural and practical parameter identifiability provide limited insights into identifiability of kinetic parameters for these PDE models and spatially-averaged FRAP data. We thus propose a pipeline for assessing parameter identifiability and for learning parameter combinations based on re-parametrization and profile likelihoods analysis. We show that this method is able to recover parameter combinations for synthetic FRAP datasets and investigate its application to real experimental data. [ABSTRACT FROM AUTHOR]

Equation-free modeling aims at extracting low-dimensional macroscopic dynamics from complex high-dimensional systems that govern the evolution of microscopic states. This algorithm relies on lifting and restriction operators that map macroscopic states to microscopic states and vice versa. Combined with simulations of the microscopic state, this algorithm can be used to apply Newton solvers to the implicitly defined low-dimensional macroscopic system or solve it more efficiently using direct numerical simulations. The key challenge is the construction of the lifting and restrictions operators that usually require a priori insight into the underlying application. In this paper, we design an application-independent algorithm that uses diffusion maps to construct these operators from simulation data. Code is available at https://doi.org/10.5281/zenodo.5793299. [ABSTRACT FROM AUTHOR]

We study the structure of stationary patterns in bistable lattice dynamical systems posed on rings with a symmetric coupling structure in the regime of small coupling strength. We show that sparse coupling (for instance, nearest-neighbour or next-nearest-neighbour coupling) and all-to-all coupling lead to significantly different solution branches. In particular, sparse coupling leads to snaking branches with many saddle-node bifurcations, while all-to-all coupling leads to branches with six saddle nodes, regardless of the size of the number of nodes in the graph. [ABSTRACT FROM AUTHOR]

Motivated by pulse-replication phenomena observed in the FitzHugh--Nagumo equation, we investigate traveling pulses whose slow/fast profiles exhibit canard-like transitions. We show that the spectra of the PDE linearization about such pulses may contain many point eigenvalues that accumulate onto a union of curves as the slow scale parameter approaches zero. The limit sets are related to the absolute spectrum of the homogeneous rest states involved in the canard-like transitions. Our results are formulated for general systems that admit an appropriate slow/fast structure. [ABSTRACT FROM AUTHOR]

In its original formulation the Krein matrix was used to locate the spectrum of first-order star-even polynomial operators where both operator coefficients are nonsingular. Such operators naturally arise when considering first-order-in-time Hamiltonian PDEs. Herein the matrix is reformulated to allow for operator coefficients with nontrivi al kernel. Moreover, it is extended to allow for the study of the spectral problem associated with quadratic star-even operators, which arise when considering the spectral problem associated with second-order-in-time Hamiltonian PDEs. In conjunction with the Hamiltonian-Krein index (HKI) the Krein matrix is used to study two problems: conditions leading to Hamiltonian-Hopf bifurcations for small spatially periodic waves, and the location and Krein signature of small eigenvalues associated with, e.g., n-pulse problems. For the first case we consider in detail a first-order-in-time fifth-order KdV-like equation. In the latter case we use a combination of Lin's method, the HKI, and the Krein matrix to study the spectrum associated with n-pulses for a second-order-in-time Hamiltonian system which is used to model the dynamics of a suspension bridge. [ABSTRACT FROM AUTHOR]

Emerging large-scale biobanks pairing genotype data with phenotype data present new opportunities to prioritize shared genetic associations across multiple phenotypes for molecular validation. Past research, by our group and others, has shown gene-level tests of association produce biologically interpretable characterization of the genetic architecture of a given phenotype. Here, we present a new method, Ward clustering to identify Internal Node branch length outliers using Gene Scores (WINGS), for identifying shared genetic architecture among multiple phenotypes. The objective of WINGS is to identify groups of phenotypes, or "clusters," sharing a core set of genes enriched for mutations in cases. We validate WINGS using extensive simulation studies and then combine gene-level association tests with WINGS to identify shared genetic architecture among 81 case-control and seven quantitative phenotypes in 349,468 European-ancestry individuals from the UK Biobank. We identify eight prioritized phenotype clusters and recover multiple published gene-level associations within prioritized clusters. [ABSTRACT FROM AUTHOR]

Self-organized pattern behavior is ubiquitous throughout nature, from fish schooling to collective cell dynamics during organism development. Qualitatively these patterns display impressive consistency, yet variability inevitably exists within pattern-forming systems on both microscopic and macroscopic scales. Quantifying variability and measuring pattern features can inform the underlying agent interactions and allow for predictive analyses. Nevertheless, current methods for analyzing patterns that arise from collective behavior capture only macroscopic features or rely on either manual inspection or smoothing algorithms that lose the underlying agent-based nature of the data. Here we introduce methods based on topological data analysis and interpretable machine learning for quantifying both agent-level features and global pattern attributes on a large scale. Because the zebrafish is a model organism for skin pattern formation, we focus specifically on analyzing its skin patterns as a means of illustrating our approach. Using a recent agent-based model, we simulate thousands of wild-type and mutant zebrafish patterns and apply our methodology to better understand pattern variability in zebrafish. Our methodology is able to quantify the differential impact of stochasticity in cell interactions on wild-type and mutant patterns, and we use our methods to predict stripe and spot statistics as a function of varying cellular communication. Our work provides an approach to automatically quantifying biological patterns and analyzing agent-based dynamics so that we can now answer critical questions in pattern formation at a much larger scale. [ABSTRACT FROM AUTHOR]

Different cell types aggregate and sort into hierarchical architectures during the formation of animal tissues. The resulting spatial organization depends (in part) on the strength of adhesion of one cell type to itself relative to other cell types. However, automated and unsupervised classification of these multicellular spatial patterns remains challenging, particularly given their structural diversity and biological variability. Recent developments based on topological data analysis are intriguing to reveal similarities in tissue architecture, but these methods remain computationally expensive. In this article, we show that multicellular patterns organized from two interacting cell types can be efficiently represented through persistence images. Our optimized combination of dimensionality reduction via autoencoders, combined with hierarchical clustering, achieved high classification accuracy for simulations with constant cell numbers. We further demonstrate that persistence images can be normalized to improve classification for simulations with varying cell numbers due to proliferation. Finally, we systematically consider the importance of incorporating different topological features as well as information about each cell type to improve classification accuracy. We envision that topological machine learning based on persistence images will enable versatile and robust classification of complex tissue architectures that occur in development and disease. [ABSTRACT FROM AUTHOR]

The Evans function has been used extensively to study spectral stability of travelling-wave solutions in spatially extended partial differential equations. To compute Evans functions numerically, several shooting methods have been developed. In this paper, an alternative scheme for the numerical computation of Evans functions is presented that relies on an appropriate boundary-value problem formulation. Convergence of the algorithm is proved, and several examples, including the computation of eigenvalues for a multi-dimensional problem, are given. The main advantage of the scheme proposed here compared with earlier methods is that the scheme is linear and scalable to large problems. [ABSTRACT FROM AUTHOR]

The FitzHugh-Nagumo equations are known to admit fast traveling pulses that have monotone tails and arise as the concatenation of Nagumo fronts and backs in an appropriate singular limit, where a parameter $$\varepsilon $$ goes to zero. These pulses are known to be nonlinearly stable with respect to the underlying PDE. Recently, the existence of fast pulses with oscillatory tails was proved for the FitzHugh-Nagumo equations. In this paper, we prove that the fast pulses with oscillatory tails are also nonlinearly stable. Similar to the case of monotone tails, stability is decided by the location of a nontrivial eigenvalue near the origin of the PDE linearization about the traveling pulse. We prove that this real eigenvalue is always negative. However, the expression that governs the sign of this eigenvalue for oscillatory pulses differs from that for monotone pulses, and we show indeed that the nontrivial eigenvalue in the monotone case scales with $$\varepsilon $$ , while the relevant scaling in the oscillatory case is $$\varepsilon ^{2/3}$$ . [ABSTRACT FROM AUTHOR]

In the present work, we consider the existence and spectral stability of multi-pulse solutions in Hamiltonian lattice systems which are invariant under a one-parameter unitary group of symmetries. We provide a general framework for the study of such wave patterns based on a discrete analogue of Lin's method, previously used in the continuum realm. We develop explicit conditions for the existence of multi-pulse standing wave structures and subsequently develop a reduced matrix allowing us to address their spectral stability. As a prototypical example, we consider the discrete nonlinear Schrödinger equation (DNLS). Using Lin's method, we extend existence and linear stability results of multi-pulse solutions beyond the anti-continuum and continuum limits. Different families of 2- and 3-pulse solitary waves are discussed, and analytical expressions for the corresponding stability eigenvalues are obtained which are in very good agreement with numerical results. • Multi-pulse solutions to DNLS exist for intermediate values of the coupling parameter. • Spectral stability of these multi-pulse solutions depends on the underlying pulse configuration. • Lin's method leads to analytical expressions for the eigenvalues of DNLS for 2- and 3-pulses. [ABSTRACT FROM AUTHOR]

We analyze the bifurcation diagrams of spatially localized stationary patterns that exhibit a long spatially periodic interior plateau (referred to as localized rolls). In a wide variety of contexts, these bifurcation diagrams consist of isolas or of intertwined s-shaped curves that are commonly referred to as snaking branches. These diagrams have been rigorously analyzed by connecting the existence curves of localized rolls with the bifurcation structure of fronts that connect the rolls to the trivial state. Previous work assumed that the stable and unstable manifolds of rolls were orientable. Here, we extend these results to the nonorientable case and also discuss topological barriers that prevent snaking, thus allowing only isolas to occur. The results are applied to the Swift–Hohenberg system for which we show that nonorientable roll patterns cannot snake. [ABSTRACT FROM AUTHOR]

Data assimilation of traffic flow remains a challenging problem. One difficulty is that data come from different sources ranging from stationary sensors and camera data to GPS and cell phone data from moving cars. Sensors and cameras give information about traffic density, while GPS data provide information about the positions and velocities of individual cars. Previous methods for assimilating Lagrangian data collected from individual cars relied on specific properties of the underlying computational model or its reformulation in Lagrangian coordinates. These approaches make it hard to assimilate both Eulerian density and Lagrangian positional data simultaneously. In this paper, we propose an alternative approach that allows us to assimilate both Eulerian and Lagrangian data. We show that the proposed algorithm is accurate and works well in different traffic scenarios and regardless of whether ensemble Kalman or particle filters are used. We also show that the algorithm is capable of estimating parameters and assimilating real traffic observations and synthetic observations obtained from microscopic models. [ABSTRACT FROM AUTHOR]

Invariant manifolds are key objects in describing how trajectories partition the phase spaces of a dynamical system. Examples include stable, unstable, and center manifolds of equilibria and periodic orbits, quasiperiodic invariant tori, and slow manifolds of systems with multiple timescales. Changes in these objects and their intersections with variation of system parameters give rise to global bifurcations. Bifurcation manifolds in the parameter spaces of multi-parameter families of dynamical systems also play a prominent role in dynamical systems theory. Much progress has been made in developing theory and computational methods for invariant manifolds during the past 25 years. This article highlights some of these achievements and remaining open problems. [ABSTRACT FROM AUTHOR]

Lagrangian measurements from passive ocean instruments provide a useful source of data for estimating and forecasting the ocean's state (velocity field, salinity field, etc.). However, trajectories from these instruments are often highly nonlinear, leading to difficulties with widely used data assimilation algorithms such as the ensemble Kalman filter (EnKF). Additionally, the velocity field is often modeled as a high-dimensional variable, which precludes the use of more accurate methods such as the particle filter (PF). Here, a hybrid particle-ensemble Kalman filter is developed that applies the EnKF update to the potentially high-dimensional velocity variables, and the PF update to the relatively low-dimensional, highly nonlinear drifter position variable. This algorithm is tested with twin experiments on the linear shallow water equations. In experiments with infrequent observations, the hybrid filter consistently outperformed the EnKF, both by better capturing the Bayesian posterior and by better tracking the truth. [ABSTRACT FROM AUTHOR]

The optimal-velocity follow-the-leader model is augmented with an equation that allows each driver to adjust their target headway according to the velocity difference between the driver and the car in front. In this more detailed model, which is investigated on a ring, stable and unstable multipulse or multijam solutions emerge. Analytical investigations using truncated Fourier analysis are confirmed and complemented by a detailed numerical bifurcation analysis. In addition to standard rotating waves, time-modulated waves are found. [ABSTRACT FROM AUTHOR]