Showing papers in "Siam Journal on Applied Dynamical Systems in 2016"
TL;DR: This work develops a new method which extends dynamic mode decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models.
Abstract: We develop a new method which extends dynamic mode decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems. DMD finds spatial-temporal coherent modes, connects local-linear analysis to nonlinear operator theory, and provides an equation-free architecture which is compatible with compressive sensing. In actuated systems, DMD is incapable of producing an input-output model; moreover, the dynamics and the modes will be corrupted by external forcing. Our new method, dynamic mode decomposition with control (DMDc), capitalizes on all of the advantages of DMD and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models. The method is data-driven in that it does not require knowledge of the underlying governing equations---only snapshots in time of observables and actuation data from historical, experimental, or black-box simulations. W...
665 citations
TL;DR: It is demonstrated that the integration of the recently developed dynamic mode decomposition (DMD) with a multiresolution analysis allows for a decomposition method capable of robustly separating complex systems into a hierarchy ofMultiresolution time-scale components.
Abstract: We demonstrate that the integration of the recently developed dynamic mode decomposition (DMD) with a multiresolution analysis allows for a decomposition method capable of robustly separating complex systems into a hierarchy of multiresolution time-scale components. A one-level separation allows for background (low-rank) and foreground (sparse) separation of dynamical data, or robust principal component analysis. The multiresolution DMD (mrDMD) is capable of characterizing nonlinear dynamical systems in an equation-free manner by recursively decomposing the state of the system into low-rank terms whose temporal coefficients in time are known. DMD modes with temporal frequencies near the origin (zero-modes) are interpreted as background (low-rank) portions of the given dynamics, and the terms with temporal frequencies bounded away from the origin are their sparse counterparts. The mrDMD method is demonstrated on several examples involving multiscale dynamical data, showing excellent decomposition results, ...
315 citations
TL;DR: An asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements is obtained, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors.
Abstract: We adapt a previously introduced continuous in time data assimilation (downscaling) algorithm for the two-dimensional Navier--Stokes equations to the more realistic case when the measurements are obtained discretely in time and may be contaminated by systematic errors. Our algorithm is designed to work with a general class of observables, such as low Fourier modes and local spatial averages over finite volume elements. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution, and the time step between successive measurements, we obtain an asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors. A stationary statistical analysis of our discrete data assimilation algorithm is also provided.
88 citations
TL;DR: It is proved that the transition probabilities converge in total variation norm to the invariant measure and it is shown that the rate is not too far from exponential in that the convergence speed is of the form of a polynomial of any degree.
Abstract: Focusing on asymptotic behavior of a stochastic SIR epidemic model represented by a system of stochastic differential equations with a degenerate diffusion, this paper provides sufficient conditions that are very close to the necessary ones for the permanence. In addition, this paper develops ergodicity of the underlying system. It is proved that the transition probabilities converge in total variation norm to the invariant measure. Our result gives a precise characterization of the support of the invariant measure. Rates of convergence are also ascertained. It is shown that the rate is not too far from exponential in that the convergence speed is of the form of a polynomial of any degree.
78 citations
TL;DR: Theoretical and numerical results confirm the effectiveness and the versatility of the pseudospectral discretization approach, opening a new perspective for the bifurcation analysis of delay equations, in particular coupled renewal and delay differential equations.
Abstract: We apply the pseudospectral discretization approach to nonlinear delay models described by delay differential equations, renewal equations, or systems of coupled renewal equations and delay differential equations. The aim is to derive ordinary differential equations and to investigate the stability and bifurcation of equilibria of the original model by available software packages for continuation and bifurcation for ordinary differential equations. Theoretical and numerical results confirm the effectiveness and the versatility of the approach, opening a new perspective for the bifurcation analysis of delay equations, in particular coupled renewal and delay differential equations.
55 citations
TL;DR: This work presents an approach which utilizes both the local geometry and the local noise dynamics within the data set through a metric which is both insensitive to the fast variables and more general than simple statistical averaging.
Abstract: Multi-time-scale stochastic dynamical systems are ubiquitous in science and engineering, and the reduction of such systems and their models to only their slow components is often essential for scientific computation and further analysis. Rather than being available in the form of an explicit analytical model, often such systems can only be observed as a data set which embodies dynamics on several time scales. We focus on applying and adapting data-mining and manifold learning techniques to detect the slow components in a class of such multiscale data. Traditional data-mining methods are based on metrics (and thus, geometries) which are not informed of the multiscale nature of the underlying system dynamics; such methods cannot successfully recover the slow variables. Here, we present an approach which utilizes both the local geometry and the local noise dynamics within the data set through a metric which is both insensitive to the fast variables and more general than simple statistical averaging. Our anal...
52 citations
TL;DR: In this paper, the authors present necessary and sufficient criteria for injectivity and multistationarity in vector fields associated with a chemical reaction network under a variety of more or less general assumptions on the nature of the network and the reaction rates.
Abstract: The goal of this paper is to gather and develop some necessary and sufficient criteria for injectivity and multistationarity in vector fields associated with a chemical reaction network under a variety of more or less general assumptions on the nature of the network and the reaction rates. The results are primarily linear algebraic or matrix-theoretic, with some graph-theoretic results also mentioned. Several results appear in, or are close to, results in the literature. Here, we emphasize the connections between the results and, where possible, present elementary proofs which rely solely on basic linear algebra and calculus. A number of examples are provided to illustrate the variety of subtly different conclusions which can be reached via different computations. In addition, many of the computations are implemented in a web-based open source platform, allowing the reader to test examples including and beyond those analyzed in the paper.
50 citations
TL;DR: This work presents frameworks that guarantee the emergence of synchronization for various coupling feedback laws in interacting networks by using two adaptive coupling laws for the pairwise coupling strength.
Abstract: We study the synchronization of Kuramoto oscillators with adaptive coupling in interacting networks. Network dynamics preserves the sum of all incoming pairwise coupling strengths and is designed to adaptively interact with system dynamics. For adaptive couplings, we use two adaptive coupling laws for the pairwise coupling strength. Kuramoto oscillators are assumed to be on the nodes of the networks. We present frameworks that guarantee the emergence of synchronization for various coupling feedback laws. Our results generalize earlier work on the synchronization of Kuramoto oscillators in fixed and symmetric networks.
46 citations
TL;DR: In this paper, the authors describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems, where the dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables.
Abstract: We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit c...
43 citations
TL;DR: A process-oriented benthic-drift model is developed that links changes in the flow regime and habitat availability with population dynamics and develops new theory to calculate these quantities and use them to investigate how the various flow regimes, population birth rate, individual transfer rates between zones, and river heterogeneity affect populati...
Abstract: One key issue for theory in stream ecology is how much stream flow can be changed while still maintaining an intact stream ecology, instream flow needs (IFNs); the study of determining IFNs is challenging due to the complex and dynamic nature of the interaction between the stream environment and the biological community. We develop a process-oriented benthic-drift model that links changes in the flow regime and habitat availability with population dynamics. In the model, the stream is divided into two zones, drift zone and benthic zone, and the population is divided into two interacting compartments, individuals residing in the benthic zone and individuals dispersing in the drift zone. We study the population persistence criteria, based on the net reproductive rate $R_0$ and on related measures. We develop new theory to calculate these quantities and use them to investigate how the various flow regimes, population birth rate, individual transfer rates between zones, and river heterogeneity affect populati...
42 citations
TL;DR: It is shown that the above equation generates a random dynamical system because of the existence of a unique pathwise global solution to the stochastic evolution equation.
Abstract: We consider the stochastic evolution equation $du=Audt+G(u)d\omega,\quad u(0)=u_0$ in a separable Hilbert space $V$. Here $G$ is supposed to be three times Frechet-differentiable and $\omega$ is a trace class fractional Brownian motion with Hurst parameter $H\in (1/3,1/2]$. We prove the existence of a unique pathwise global solution, and, since the considered stochastic integral does not produce exceptional sets, we are able to show that the above equation generates a random dynamical system.
TL;DR: In this paper, a general framework for rigorously studying the effect of spatio-temporal noise on traveling waves and stationary patterns is presented, which can incorporate versions of the stochastic neural field equation that may exhibit traveling fronts, pulses, or stationary patterns.
Abstract: In this paper we present a general framework in which to rigorously study the effect of spatio-temporal noise on traveling waves and stationary patterns. In particular, the framework can incorporate versions of the stochastic neural field equation that may exhibit traveling fronts, pulses, or stationary patterns. To do this, we first formulate a local SDE that describes the position of the stochastic wave up until a discontinuity time, at which point the position of the wave may jump. We then study the local stability of this stochastic front, obtaining a result that recovers a well-known deterministic result in the small-noise limit. We finish with a study of the long-time behavior of the stochastic wave.
TL;DR: This paper uses the construction of a natural computable finite decomposition of parameter space into domains where the annotated Morse graph description of dynamics is constant to construct an SQL database that can be effectively searched for dynamical signatures such as bistability, stable or unstable oscillations, and stable equilibria.
Abstract: We describe the theoretical and computational framework for the Dynamic Signatures Generated by Regulatory Networks ($\mathsf{DSGRN}$) database. The motivation stems from an urgent need to understand the global dynamics of biologically relevant signal transduction/gene regulatory networks that have at least 5 to 10 nodes, involve multiple interactions, and have decades of parameters. The input to the database computations is a regulatory network, i.e., a directed graph with edges indicating up or down regulation. A computational model based on switching networks is generated from the regulatory network. The phase space dimension of this model equals the number of nodes and the associated parameter space consists of one parameter for each node (a decay rate) and three parameters for each edge (low level of expression, high level of expression, and threshold at which expression levels change). Since the nonlinearities of switching systems are piecewise constant, there is a natural decomposition of phase spa...
TL;DR: In this paper, the authors study a class of discontinuous vector fields brought to our attention by multilegged animal locomotion and show that the flow is locally conjugate (via a piecewise-differentiable homeomorphism) to a f...
Abstract: We study a class of discontinuous vector fields brought to our attention by multilegged animal locomotion. Such vector fields arise not only in biomechanics, but also in robotics, neuroscience, and electrical engineering, to name a few domains of application. Under the conditions that (i) the vector field's discontinuities are locally confined to a finite number of smooth submanifolds and (ii) the vector field is transverse to these surfaces in an appropriate sense, it is known that the vector field yields a well--defined flow that is Lipschitz continuous. We extend these results by showing this flow is piecewise--differentiable, so that it admits a first--order approximation (known as a Bouligand derivative) that is piecewise--linear and continuous at every point. We exploit this first--order approximation to infer existence of piecewise--differentiable impact maps (including Poincare maps for periodic orbits), show that the flow is locally conjugate (via a piecewise--differentiable homeomorphism) to a f...
TL;DR: Under general conditions, it is proved that the moments of a solution to a randomly switching PDE satisfy a hierarchy of BVPs with lower order moments coupling to higher order moments at the boundaries.
Abstract: Driven by diverse applications, several recent models impose randomly switching boundary conditions on either a PDE or SDE. The purpose of this paper is to provide tools for calculating statistics of these models and establish a connection between these two perspectives on diffusion in a random environment. Under general conditions, we prove that the moments of a solution to a randomly switching PDE satisfy a hierarchy of BVPs with lower order moments coupling to higher order moments at the boundaries. Further, we prove that joint exit statistics for a set of particles following a randomly switching SDE satisfy a corresponding hierarchy of BVPs. In particular, the $M$th moment of a solution to a switching PDE corresponds to exit statistics for $M$ particles following a switching SDE. We note that though the particles are noninteracting, they are nonetheless correlated because they all follow the same switching SDE. Finally, we give several examples of how our theorems reveal the sometimes surprising dynam...
TL;DR: Using bifurcation theory, the authors in this article studied the secular resonances induced by the Sun and Moon on space debris orbits around the Earth, and they focused on a special class of resonances which depend only on the debris' orbital inclination.
Abstract: Using bifurcation theory, we study the secular resonances induced by the Sun and Moon on space debris orbits around the Earth. In particular, we concentrate on a special class of secular resonances, which depend only on the debris' orbital inclination. This class is typically subdivided into three distinct types of secular resonances: those occurring at the critical inclination, those corresponding to polar orbits, and a third type resulting from a linear combination of the rates of variation of the argument of perigee and the longitude of the ascending node. The model describing the dynamics of space debris includes the effects of the geopotential, as well as the Sun's and Moon's attractions, and it is defined in terms of suitable action-angle variables. We consider the system averaged over both the mean anomaly of the debris and those of the Sun and Moon. Such a multiply-averaged Hamiltonian is used to study the lunisolar resonances which depend just on the inclination. Borrowing the technique from the ...
TL;DR: It is shown that the coexistence of two invariant sets with different nature and heteroclinic connections between them give rise to long hyperchaotic transient behavior, and therefore it provides a mechanism for noisy simulations.
Abstract: It has recently been reported [P. C. Reich, Neurocomputing, 74 (2011), pp. 3361--3364] that it is quite difficult to distinguish between chaos and hyperchaos in numerical simulations which are frequently “noisy.” For the classical four-dimensional (4D) Rossler model [O. E. Rossler, Phys. Lett. A, 71 (1979), pp. 155--157] we show that the coexistence of two invariant sets with different nature (a global hyperchaotic invariant set and a chaotic attractor) and heteroclinic connections between them give rise to long hyperchaotic transient behavior, and therefore it provides a mechanism for noisy simulations. The same phenomena is expected in other 4D and higher-dimensional systems. The proof combines topological and smooth methods with rigorous numerical computations. The existence of (hyper)chaotic sets is proved by the method of covering relations [P. Zgliczynski and M. Gidea, J. Differential Equations, 202 (2004), pp. 32--58]. We extend this method to the case of a nonincreasing number of unstable directio...
TL;DR: This paper introduces universal asymptotic unfolding normal forms for nonlinear singular systems, and designs an effective multiple-parametric quadratic state feedback controller for a singular system on a three dimensional central manifold with two imaginary uncontrollable modes.
Abstract: In this paper we introduce universal asymptotic unfolding normal forms for nonlinear singular systems. Next, we propose an approach to finding the parameters of a parametric singular system that play the role of the universal unfolding parameters. These parameters effectively influence the local dynamics of the system. We propose a systematic approach to locating local bifurcations in terms of these parameters. Here we apply the proposed approach on Hopf-zero singularities whose first few low degree terms are incompressible. In this direction, we obtain novel orbital and parametric normal form results for such families by assuming a nonzero quadratic condition. Moreover, we give a truncated universal asymptotic unfolding normal form and prove the finite determinacy of the steady-state bifurcations for the two most generic subfamilies of the associated amplitude systems. We analyze the local primary bifurcations of equilibria and limit cycles, as well as the secondary Hopf bifurcations of invariant tori. T...
TL;DR: In this article, the effects of noise on the traveling wave dynamics in neural fields were analyzed in a weighted $L^2$-space, and the authors derived an expansion of the stochastic wave, describing the influence of noise to different orders of the noise strength.
Abstract: We analyze the effects of noise on the traveling wave dynamics in neural fields. The noise influences the dynamics on two scales: first, it causes fluctuations in the wave profile, and second, it causes a random shift in the phase of the wave. We formulate the problem in a weighted $L^2$-space, allowing us to separate the two spatial scales. By tracking the stochastic solution with a reference wave we obtain an expression for the stochastic phase. We derive an expansion of the stochastic wave, describing the influence of the noise to different orders of the noise strength. To first order of the noise strength the phase shift is roughly diffusive and the fluctuations are given by an Ornstein--Uhlenbeck process of bounded variance that is orthogonal to the direction of movement. This also expresses the stability of the wave under noise.
TL;DR: The dynamics of systems with stochastically varying time delays are investigated and it is shown that the mean dynamics can be used to derive necessary conditions for the stability of equilibria of the stochastic system.
Abstract: The dynamics of systems with stochastically varying time delays are investigated in this paper. It is shown that the mean dynamics can be used to derive necessary conditions for the stability of equilibria of the stochastic system. Moreover, the second moment dynamics can be used to derive sufficient conditions for almost sure stability of equilibria. The results are summarized using stability charts that are obtained via semidiscretization. The theoretical methods are applied to simple gene regulatory networks where it is demonstrated that stochasticity in the delay can improve the stability of steady protein production.
TL;DR: Through Mathieu equations, this work considers a class of second order Hamiltonian systems where one component initially holds almost all the energy of the system and shows that if the total energy is sufficiently small then it remains on this component, whereas if thetotal energy is larger it may transfer to the other components.
Abstract: Motivated by the instability of suspension bridges, we consider a class of second order Hamiltonian systems where one component initially holds almost all the energy of the system. We show that if the total energy is sufficiently small then it remains on this component, whereas if the total energy is larger it may transfer to the other components. Through Mathieu equations we explain the precise mechanism which governs the energy transfer.
TL;DR: This paper proposes a model-free, information-theoretical method to measure the existence and direction of influence of one state's policy or legal activity on others, and tailor a popular notion of causality to handle the slow time-scale of policy adoption dynamics and unravel relationships among states from their recent law enactment histories.
Abstract: Detecting and explaining the relationships among interacting components has long been a focal point of dynamical systems research. In this paper, we extend these types of data-driven analyses to the realm of public policy, whereby individual legislative entities interact to produce changes in their legal and political environments. We focus on the U.S. public health policy landscape, whose complexity determines our capacity as a society to effectively tackle pressing health issues. It has long been thought that some U.S. states innovate and enact new policies, while others mimic successful or competing states. However, the extent to which states learn from others, and the state characteristics that lead two states to influence one another, are not fully understood. Here, we propose a model-free, information-theoretical method to measure the existence and direction of influence of one state's policy or legal activity on others. Specifically, we tailor a popular notion of causality to handle the slow time-scale of policy adoption dynamics and unravel relationships among states from their recent law enactment histories. The method is validated using surrogate data generated from a new stochastic model of policy activity. Through the analysis of real data in alcohol, driving safety, and impaired driving policy, we provide evidence for the role of geography, political ideology, risk factors, and demographic and economic indicators on a state's tendency to learn from others when shaping its approach to public health regulation. Our method offers a new model-free approach to uncover interactions and establish cause-and-effect in slowly-evolving complex dynamical systems.
TL;DR: This analysis analyzes an Arctic energy balance model in the limit as a smoothing parameter associated with ice-albedo feedback tends to zero, which introduces a discontinuity boundary to the dynamical systems model.
Abstract: As Arctic sea ice extent decreases with increasing greenhouse gases, there is a growing interest in whether there could be a bifurcation associated with its loss, and whether there is significant hysteresis associated with that bifurcation. A challenge in answering this question is that the bifurcation behavior of certain Arctic energy balance models have been shown to be sensitive to how ice-albedo feedback is parameterized. We analyze an Arctic energy balance model in the limit as a smoothing parameter associated with ice-albedo feedback tends to zero, which introduces a discontinuity boundary to the dynamical systems model. Our analysis provides a case study where we use the system in this limit to guide the investigation of bifurcation behavior of the original albedo-smoothed system. In this case study, we demonstrate that certain qualitative bifurcation behaviors of the albedo-smoothed system can have counterparts in the limit with no albedo smoothing. We use this perspective to systematically explor...
TL;DR: In this article, the authors derived the minimal well-posed long-wave approximation, which is a degenerate Cahn-Hilliard equation, for a particular nonlocal aggregation model.
Abstract: Biological aggregations such as insect swarms and bird flocks may arise from a combination of social interactions and environmental cues. We focus on nonlocal continuum equations, which are often used to model aggregations, and yet which pose significant analytical and computational challenges. Beginning with a particular nonlocal aggregation model [C. M. Topaz, A. L. Bertozzi, and M. A. Lewis, Bull. Math. Bio., 68 (2006), pp. 1601--1623], we derive the minimal well-posed long-wave approximation, which is a degenerate Cahn--Hilliard equation. Energy minimizers of this reduced, local model retain many salient features of those of the nonlocal model, especially for large populations and away from an aggregation's boundaries. Using the Cahn--Hilliard model as a testbed, we investigate the degree to which an external potential modeling food sources can be used to suppress peak population density, which is essential for controlling locust outbreaks. A random distribution of food sources tends to increase peak ...
TL;DR: A variety of results are presented analyzing the behavior of a class of stochastic processes referred to as piecewise deterministic Markov processes (PDMPs) for the infinite-time interval and determining general conditions on when the moments of such processes will, or will not, be well-behaved.
Abstract: We present a variety of results analyzing the behavior of a class of stochastic processes---referred to as piecewise deterministic Markov processes (PDMPs)---for the infinite-time interval and determine general conditions on when the moments of such processes will, or will not, be well-behaved. We also characterize the types of finite-time blowups that are possible for these processes, and obtain bounds on their probabilities.
TL;DR: An algorithm is presented for numerical computation of choreographies in the plane in a Newtonian potential and on the sphere in a cotangent potential based on stereographic projection, approximation by trigonometric polynomials, and quasi-Newton and Newton optimization methods with exact gradient and exact Hessian matrix.
Abstract: An algorithm is presented for numerical computation of choreographies in the plane in a Newtonian potential and on the sphere in a cotangent potential. It is based on stereographic projection, approximation by trigonometric polynomials, and quasi-Newton and Newton optimization methods with exact gradient and exact Hessian matrix. New choreographies on the sphere are presented.
TL;DR: In this paper, the authors consider the impact of small jump-type spatial heterogeneity on the existence of stationary localized patterns in a system of partial differential equations in one spatial dimension, and define the (pinned) defect solution $\Gamma_\varepsilon$ as a heteroclinic solution to the perturbed system such that
Abstract: In this manuscript, we consider the impact of a small jump-type spatial heterogeneity on the existence of stationary localized patterns in a system of partial differential equations in one spatial dimension, i.e., defined on $\mathbb{R}$. This problem corresponds to analyzing a discontinuous and non-autonomous $n$-dimensional system, $\scriptsize\dot{u}=\left\{ \begin{array}{ll} f(u),& t\leq0,\\ f(u)+\varepsilon g(u),& t>0, \end{array}\right.$ under the assumption that the unperturbed system, i.e., the $\varepsilon \to 0$ limit system, possesses a heteroclinic orbit $\Gamma$ that connects two hyperbolic equilibrium points (plus several additional nondegeneracy conditions). The unperturbed orbit $\Gamma$ represents a localized structure in the PDE setting. We define the (pinned) defect solution $\Gamma_\varepsilon$ as a heteroclinic solution to the perturbed system such that $\lim_{\varepsilon \to 0} \Gamma_\varepsilon = \Gamma$ (as graphs). We distinguish between three types of defect solutions: trivial, ...
TL;DR: It is shown that when a small amount of noise is added, the heavier concentration "leaks" its mass towards the lighter concentration over a very long time scale, eventually resulting in the equilibration of the two masses.
Abstract: We study the long-time effect of noise on pattern formation for the aggregation model. We consider aggregation kernels that generate patterns consisting of two delta-concentrations. Without noise, there is a one-parameter family of admissible equilibria that consist of two concentrations whose mass is not necessarily equal. We show that when a small amount of noise is added, the heavier concentration “leaks” its mass toward the lighter concentration over a very long timescale, eventually resulting in the equilibration of the two masses. We use exponentially small asymptotics to derive the long-time ODEs that quantify this mass exchange. Our theory is validated using full numerical simulations of the original model---of both the original stochastic particle system and its PDE limit. Our formal computations show that adding noise destroys the degeneracy in the equilibrium solution and leads to a unique symmetric steady state.
TL;DR: A computer assisted argument is developed and implemented for proving the existence of some connecting orbits for a nonlinear dynamical system which appears in mathematical ecology as a model of a spatially distributed ecosystem with population dispersion.
Abstract: We develop and implement a computer assisted argument for proving the existence of heteroclinic/homoclinic connecting orbits for compact infinite dimensional maps. The argument is based on a posteriori analysis of a certain “discrete time boundary value problem,” and a key ingredient is representing the local stable/unstable manifolds of the fixed points. For a compact mapping the stable manifold is infinite dimensional, and an important component of the present work is the development of computer assisted error bounds for numerical approximation of infinite dimensional stable manifolds. As an illustration of the utility of our method we prove the existence of some connecting orbits for a nonlinear dynamical system which appears in mathematical ecology as a model of a spatially distributed ecosystem with population dispersion.
TL;DR: It is demonstrated how the technique of Conley--Morse dynamical databases allows us to quickly survey and prove theorems about the global dynamics of the parameterized family of angular dynamics maps.
Abstract: In this paper we showcase the technique of Conley--Morse databases for studying a parameterized family of dynamical systems. The dynamical system of interest arises from considering the limiting behavior of Newton's root-finding method applied to functions $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ when the iterates converge to the origin. Considering the progression of angular orientations gives rise to a self-map of the unit circle we call the angular dynamics map. We demonstrate how the technique of Conley--Morse dynamical databases allows us to quickly survey and prove theorems about the global dynamics of the parameterized family of angular dynamics maps.