Showing papers in "Siam Journal on Applied Dynamical Systems in 2017"

Higher Order Dynamic Mode Decomposition

Abstract: This paper deals with an extension of dynamic mode decomposition (DMD), which is appropriate to treat general periodic and quasi-periodic dynamics, and transients decaying to periodic and quasi-periodic attractors, including cases (not accessible to standard DMD) that show limited spatial complexity but a very large number of involved frequencies. The extension, labeled as higher order dynamic mode decomposition, uses time-lagged snapshots and can be seen as superimposed DMD in a sliding window. The new method is illustrated and clarified using some toy model dynamics, the Stuart--Landau equation, and the Lorenz system. In addition, the new method is applied to (and its robustness is tested in) some permanent and transient dynamics resulting from the complex Ginzburg--Landau equation (a paradigm of pattern forming systems), for which standard DMD is seen to only uncover trivial dynamics, and the thermal convection in a rotating spherical shell subject to a radial gravity field.

Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator

Abstract: We establish the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables. We use...

Spatiotemporal Dynamics of the Diffusive Mussel-Algae Model Near Turing-Hopf Bifurcation

Abstract: Intertidal mussels can self-organize into periodic spot, stripe, labyrinth, and gap patterns ranging from centimeter to meter scales. The leading mathematical explanations for these phenomena are the reaction-diffusion-advection model and the phase separation model. This paper continues the series studies on analytically understanding the existence of pattern solutions in the reaction-diffusion mussel-algae model. The stability of the positive constant steady state and the existence of Hopf and steady-state bifurcations are studied by analyzing the corresponding characteristic equation. Furthermore, we focus on the Turing-Hopf (TH) bifurcation and obtain the explicit dynamical classification in its neighborhood by calculating and investigating the normal form on the center manifold. Using theoretical and numerical simulations, we demonstrates that this TH interaction would significantly enhance the diversity of spatial patterns and trigger the alternative paths for the pattern development.

Sparse Sensing and DMD-Based Identification of Flow Regimes and Bifurcations in Complex Flows

Abstract: We present a sparse sensing framework based on dynamic mode decomposition (DMD) to identify flow regimes and bifurcations in large-scale thermofluid systems. Motivated by real-time sensing and control of thermal-fluid flows in buildings and equipment, we apply this method to a direct numerical simulation (DNS) data set of a two-dimensional laterally heated cavity. The resulting flow solutions can be divided into several regimes, ranging from steady to chaotic flow. The DMD modes and eigenvalues capture the main temporal and spatial scales in the dynamics belonging to different regimes. Our proposed classification method is data driven, robust w.r.t. measurement noise, and exploits the dynamics extracted from the DMD method. Namely, we construct an augmented DMD basis, with “built-in” dynamics, given by the DMD eigenvalues. This allows us to employ a short time series of data from sensors, to more robustly classify flow regimes, particularly in the presence of measurement noise. We also exploit the incoher...

Which Small Reaction Networks Are Multistationary

Abstract: Reaction networks taken with mass-action kinetics arise in many settings, such as epidemiology, population biology, and systems of chemical reactions. Bistable reaction networks are posited to underlie biochemical switches, which motivates the following question: Which reaction networks have the capacity for multiple steady states? Mathematically, this asks, from among certain parametrized families of polynomial systems, which admit multiple positive roots? No complete answer is known. This work analyzes the smallest networks, i.e., those with only a few chemical species or reactions. For these “smallest” networks, we completely answer the question of multistationarity and, in some cases, multistability, thereby extending related work of Boros. Our results highlight the role played by the Newton polytope of a network (the convex hull of the reactant vectors). Also, our work is motivated by recent results that explain how a given network's capacity for multistationarity arises from that of certain related ...

Fenichel Theory for Multiple Time Scale Singular Perturbation Problems

Abstract: This paper is concerned with a geometric study of singularly perturbed systems of ordinary differential equations expressed by ($n-1$)-parameter families of smooth vector fields on $\mathbb{R}^l$, ...

A Posteriori Verification of Invariant Objects of Evolution Equations: Periodic Orbits in the Kuramoto--Sivashinsky PDE

Abstract: In this paper, a method for computing periodic orbits of the Kuramoto--Sivashinsky PDE via rigorous numerics is presented. This is an application and an implementation of the theoretical method introduced in [J.-L. Figueras, M. Gameiro, J.-P. Lessard, and R. de la Llave, “A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations,” SIAM J. Appl. Dyn. Syst., to appear]. Using a Newton--Kantorovich-type argument (the radii polynomial approach), existence of solutions is obtained in a weighted $\ell^\infty$ Banach space of Fourier coefficients. Once a proof of a periodic orbit is done, an associated eigenvalue problem is solved and Floquet exponents are rigorously computed, yielding proofs that some periodic orbits are unstable. Finally, a predictor-corrector continuation method is introduced to rigorously compute global smooth branches of periodic orbits. An alternative approach and independent implementation of [J.-L. Figueras, M. Gameiro, J.-P. Less...

Dynamics in Chemotaxis Models of Parabolic-Elliptic Type on Bounded Domain with Time and Space Dependent Logistic Sources

Abstract: This paper considers the dynamics of a chemotaxis system of parabolic-elliptic type with local as well as nonlocal time and space dependent logistic source on bounded domains. We first prove the local existence and uniqueness of classical solutions for various initial functions. Next, under some explicit conditions on the coefficients, the chemotaxis sensitivity $\chi$ and the space dimension $n$, we prove the global existence and boundedness of classical solutions with certain given integrable or uniformly continuous nonnegative initial functions. Then, under the same conditions for the global existence, we show that the system has an entire positive classical solution. Moreover, if the coefficients are periodic in $t$ with period $T$ or are independent of $t$, then the system has a time periodic positive solution with periodic $T$ or a steady state positive solution. If the coefficients are independent of $x$ (i.e., are spatially homogeneous), then the system has a spatially homogeneous entire positive ...

Spatially Periodic Multipulse Patterns in a Generalized Klausmeier--Gray--Scott Model

Abstract: Semiarid ecosystems form the stage for a plethora of vegetation patterns, a feature that has been captured in terms of mathematical models since the beginning of this millennium. To study these patterns, we use a reaction-advection-diffusion model that describes the interaction of vegetation and water supply on gentle slopes. As water diffuses much faster than vegetation, this model operates on multiple timescales. While many types of patters are observed in the field, our focus is on two-dimensional stripe patterns. These patterns are typically observed on sloped terrains. In the present idealized setting they correspond to solutions of the model that are spatially periodic in one direction---the $x$-direction---and are extended trivially in the perpendicular $y$-direction. The existence of long wavelength patterns in our model is established analytically using methods from geometric singular perturbation theory, in which a correct parameter scaling is crucial. Subsequently, an Evans function approach yi...

Chimeras in Two-Dimensional Domains: Heterogeneity and the Continuum Limit

Abstract: We consider three different two-dimensional networks of nonlocally coupled heterogeneous phase oscillators These networks were previously studied with identical oscillators, and a number of spatiotemporal patterns were found, mostly as a result of direct numerical simulation Here we take the continuum limit of an infinite number of oscillators and use the Ott--Antonsen ansatz to derive continuum level evolution equations for order parameter--like quantities Most of the patterns previously found in these networks correspond to relative fixed points of these evolution equations, and we show the following results of extensive numerical investigations of these fixed points: their existence and stability, and the bifurcations involved in their loss of stability as parameters are varied Our results answer a number of questions posed by previous authors who studied these networks, and provide a better understanding of these networks' dynamics

A Model for Cell Polarization Without Mass Conservation

Abstract: A system of two Schnakenberg-like reaction-diffusion equations is investigated analytically and numerically. The system has previously been used as a minimal model for concentrations of GTPases inv...

Dynamics of a Time-Delayed Lyme Disease Model with Seasonality

Abstract: In this paper, we propose a time-delayed Lyme disease model incorporating the climate factors. We obtain the existence of a disease-free periodic solution under some additional conditions. Then we introduce the basic reproduction ratio $R_0$ and show that under the same set of conditions, $R_0$ serves as a threshold parameter in determining the global dynamics of the model; that is, the disease-free periodic solution is globally attractive if $R_0 1$. Numerically, we study the Lyme disease transmission in Long Point, Ontario, Canada. Our simulation results indicate that Lyme disease is endemic in this region if no further intervention is taken. We find that Lyme disease will die out in this area if we decrease the recruitment rate of larvae, which implies that we can control the disease by preventing tick eggs from hatching into larvae.

Finite Time Distributions of Stochastically Modeled Chemical Systems with Absolute Concentration Robustness

Abstract: Recent research in both the experimental and mathematical communities has focused on biochemical interaction systems that satisfy an “absolute concentration robustness” (ACR) property. The ACR property was first discovered experimentally when, in a number of different systems, the concentrations of key system components at equilibrium were observed to be robust to the total concentration levels of the system. Follow-up mathematical work focused on deterministic models of biochemical systems and demonstrated how chemical reaction network theory can be utilized to explain this robustness. Later mathematical work focused on the behavior of this same class of reaction networks, though under the assumption that the dynamics were stochastic. Under the stochastic assumption, it was proven that the system will undergo an extinction event with a probability of one so long as the system is conservative, showing starkly different long-time behavior than in the deterministic setting. Here we consider a general class ...

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

Abstract: We study a scalar delay differential equation (DDE) with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state-dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three contain only constant delays. We implemented this expansion and the computation of the normal form coef...

Numerical Computations and Computer Assisted Proofs of Periodic Orbits of the Kuramoto-Sivashinsky-Equation

Abstract: We present numerical results and computer assisted proofs of the existence of periodic orbits for the Kuramoto--Sivashinky equation. These two results are based on writing down the existence of periodic orbits as zeros of functionals. This leads to the use of Newton's algorithm for the numerical computation of the solutions and, with some a posteriori analysis in combination with rigorous interval arithmetic, to the rigorous verification of the existence of solutions. This is a particular case of the methodology developed in [J.-L. Figueras, J.-P. Lessard, M. Gameiro, and R. de la Llave, preprint, arXiv:1605.01086, 2016] for several types of orbits. An independent implementation, covering overlapping but different ground, using different functional setups, appears in [J.-P. Lessard and M. Gameiro, A Posteriori Verification of Invariant Objects of Evolution Equations: Periodic Orbits in the Kuramoto-Sivashinsky PDE, manuscript].

Continuation and Bifurcation of Grain Boundaries in the Swift--Hohenberg Equation

Abstract: We study grain boundaries between striped phases in the prototypical Swift-Hohenberg equation. We propose an analytical and numerical far- field-core decomposition that allows us to study existence and bifurcations of grain boundaries analytically and numerically using continuation techniques. This decomposition overcomes problems with computing grain boundaries in a large doubly periodic box with phase conditions. Using the spatially conserved quantities of the time-independent Swift-Hohenberg equation, we show that symmetric grain boundaries must select the marginally zig-zag stable stripes. We find that as the angle between the stripes is decreased, the symmetric grain boundary undergoes a parity-breaking pitchfork bifurcation where dislocations at the grain boundary split into disclination pairs. A plethora of asymmetric grain boundaries (with different angles of the far- field stripes either side of the boundary) is found and investigated. The energy of the grain boundaries is then mapped out. We find that when the angle between the stripes is greater than a critical angle, the symmetric grain boundary is energetically preferred while when the angle is less than the critical angle, the grain boundaries where stripes on one side are parallel to the interface are energetically preferred. Finally, we propose a classification of grain boundaries that allows us to predict various non-standard asymmetric grain boundaries.

One-Dimensional Map for the Circadian Modulation of Sleep in a Sleep-Wake Regulatory Network Model for Human Sleep

Abstract: The timing of human sleep is strongly modulated by the 24 h circadian rhythm, and desynchronization of sleep-wake cycles from the circadian rhythm can negatively impact health. To investigate the d...

A Framework for the Numerical Computation and A Posteriori Verification of Invariant Objects of Evolution Equations

Abstract: We develop a theoretical framework for computer-assisted proofs of the existence of invariant objects in semilinear PDEs. The invariant objects considered in this paper are equilibrium points, traveling waves, periodic orbits, and invariant manifolds attached to fixed points or periodic orbits. The core of the study is writing down the invariance condition as a zero of an operator. These operators are in general not continuous, so one needs to smooth them by means of preconditioners before classical fixed point theorems can be applied. We develop in detail all the aspects of how to work with these objects: how to precondition the equations, how to work with the nonlinear terms, which function spaces can be useful, and how to work with them in a computationally rigorous way. In two companion papers, we present two different implementations of the tools developed in this paper to study periodic orbits.

A Predator--2 Prey Fast--Slow Dynamical System for Rapid Predator Evolution

Abstract: We consider adaptive change of diet of a predator population that switches its feeding between two prey populations. We develop a novel 1 fast--3 slow dynamical system to describe the dynamics of the three populations amidst continuous but rapid evolution of the predator's diet choice. The two extremes at which the predator's diet is composed solely of one prey correspond to two branches of the three-branch critical manifold of the fast--slow system. By calculating the points at which there is a fast transition between these two feeding choices (i.e., branches of the critical manifold), we prove that the system has a two-parameter family of periodic orbits for sufficiently large separation of the time scales between the evolutionary and ecological dynamics. Using numerical simulations, we show that these periodic orbits exist, and that their phase difference and oscillation patterns persist, when ecological and evolutionary interactions occur on comparable time scales. Our model also exhibits periodic orb...

Individual Flux Study via Steady-State Poisson--Nernst--Planck Systems: Effects from Boundary Conditions

Abstract: We provide a detailed study for ionic flow through ion channels for the case with three ion species, two positively charged having the same valence and one negatively charged, and with zero permanent charge. The work is based on the general geometric theory developed in [W. Liu, J. Differential Equations, 246 (2009), pp. 428--451] for a quasi-one-dimensional steady-state Poisson--Nernst--Planck model. Our focus is on the effects of boundary conditions on the ionic flow. Beyond the existence of solutions of the model problem, we are able to obtain explicit approximations of individual fluxes and the current-voltage relations, from which effects of boundary conditions on ionic flows are examined in a great detail. Critical potentials are identified and their roles in characterizing these effects are studied. Compared to ionic mixtures with two ion species, a number of new features for mixtures of three ion species arise. Numerical simulations are performed, and numerical results are consistent with our anal...

Spectral identification of networks using sparse measurements

Abstract: We propose a new method to recover global information about a network of interconnected dynamical systems based on observations made at a small number (possibly one) of its nodes. In contrast to classical identification of full graph topology, we focus on the identification of the spectral graph-theoretic properties of the network, a framework that we call spectral network identification. The main theoretical results connect the spectral properties of the network to the spectral properties of the dynamics, which are well-defined in the context of the so-called Koopman operator and can be extracted from data through the dynamic mode decomposition algorithm. These results are obtained for networks of diffusively-coupled units that admit a stable equilibrium state. For large networks, a statistical approach is considered, which focuses on spectral moments of the network and is well-suited to the case of heterogeneous populations. Our framework provides efficient numerical methods to infer global information ...

Mixed-Mode Oscillations and Twin Canard Orbits in an Autocatalytic Chemical Reaction

Abstract: A mixed-mode oscillation (MMO) is a complex waveform with a pattern of alternating small-amplitude oscillations (SAOs) and large-amplitude oscillations (LAOs). MMOs have been observed experimentally in many physical and biological applications, but most notably in chemical reactions. We are interested in MMOs of an autocatalytic chemical reaction that can be modeled by a system of three ordinary differential equations with one fast and two slow variables. This difference in time scales provides a mechanism for generating small and large oscillations. Provided the time-scale ratio $\varepsilon$ is sufficiently small, geometric singular perturbation theory predicts the existence of two-dimensional locally invariant manifolds called slow manifolds. Slow manifolds and their intersections, which occur along so-called canard orbits, give great insight into the mechanisms for generating SAOs. The mechanisms for LAOs are less well understood and involve analysis of the global dynamics. We study the autocatalytic ...

Feedback Control for Systems with Uncertain Parameters Using Online-Adaptive Reduced Models

Abstract: We consider control and stabilization for large-scale dynamical systems with uncertain, time-varying parameters. The time-critical task of controlling a dynamical system poses major challenges: using large-scale models is prohibitive, and accurately inferring parameters can be expensive, too. We address both problems by proposing an offline-online strategy for controlling systems with time-varying parameters. During the offline phase, we use a high-fidelity model to compute a library of optimal feedback controller gains over a sampled set of parameter values. Then, during the online phase, in which the uncertain parameter changes over time, we learn a reduced-order model from system data. The learned reduced-order model is employed within an optimization routine to update the feedback control throughout the online phase. Since the system data naturally reflects the uncertain parameter, the data-driven updating of the controller gains is achieved without an explicit parameter estimation step. We consider t...

The Stability and Slow Dynamics of Localized Spot Patterns for the 3-D Schnakenberg Reaction-Diffusion Model

Abstract: On a bounded three-dimensional domain $\Omega$, a hybrid asymptotic-numerical method is employed to analyze the existence, linear stability, and slow dynamics of localized quasi-equilibrium multispot patterns of the Schnakenberg activator-inhibitor model with bulk feed-rate $A$ in the singularly perturbed limit of small diffusivity $\varepsilon^2$ of the activator component. By approximating each spot as a Coulomb singularity, a nonlinear system of equations is formulated for the strength of each spot. To leading order in $\varepsilon$, two types of solutions are identified: symmetric patterns for which all strengths are identical, and asymmetric patterns for which each strength takes on one of two distinct values. The $\mathcal{O}(\varepsilon)$ correction to the strengths is found to depend on the spatial configuration of the spots through a certain Neumann Green's matrix $\mathcal{G}$. When $\mathbf{e} = (1,\dots,1)^T$ is not an eigenvector of $\mathcal{G}$, a detailed numerical and (in the case of two ...

Folded Saddles and Faux Canards

Abstract: We study the two parameter family of faux canards associated with the folded saddle singularity within the folded singularity normal form. Recently, rotational behavior of folded saddle faux canards has been reported by Vo and Wechselberger in [SIAM J. Math. Anal., 47 (2015), pp. 3235--3283] where they studied examples of systems close to a folded saddle-node type I limit. This is a surprising observation and merits a closer look at faux canards which have been somewhat neglected in the literature. We address this gap in canard knowledge and provide a comprehensive analysis of folded saddle faux canards, both numerically and analytically. We show that for certain values of $\mu$---the eigenvalue ratio of the associated folded singularity within the reduced flow---faux canards may possess rotations about the primary faux canard, and that the stable and unstable fast manifolds (i.e., the nonlinear stable and unstable fast fiber bundles) of the primary faux canard form the boundaries of sets of solutions wit...

Uniform approximation of solutions by elimination of intermediate species in deterministic reaction networks

Abstract: Chemical reactions often proceed through the formation and the consumption of intermediate species. An example is the creation and subsequent degradation of the substrate-enzyme complexes in an enzymatic reaction. In this paper we provide a setting, based on ordinary differential equations, in which the presence of intermediate species has little effect on the overall dynamics of a biological system. The result provides a method to perform model reduction by elimination of intermediate species. We study the problem in a multiscale setting, where the species abundances as well as the reaction rates scale to different orders of magnitudes. The different time and concentration scales are parameterized by a single parameter $N$. We show that a solution to the original reaction system is uniformly approximated on compact time intervals to a solution of a reduced reaction system without intermediates and to a solution of a certain limiting reaction systems, which does not depend on $N$. Known approximation tech...

Equilibria for an Aggregation Model with Two Species

Abstract: We consider an aggregation model for two interacting species. The coupling between the species is via their velocities, which incorporate self- and cross-interactions. Our main interest is categorizing the possible steady states of the considered model in two dimensions. Notably, we identify their regions of existence and stability in the parameter space. For assessing the stability we use a combination of variational tools (based on the gradient flow formulation of the model and the associated energy) and linear stability analysis (perturbing the boundaries of the species' supports). We rely on numerical investigations for those steady states that are not analytically tractable. Finally we perform a two-scale expansion to characterize the steady state in the limit of asymptotically weak cross-interactions.

High-Order Parameterization of Stable/Unstable Manifolds for Long Periodic Orbits of Maps

Abstract: This paper develops seminumerical methods for computing high-order polynomial approximations of stable/unstable manifolds attached to long periodic orbits in discrete time dynamical systems Our approach extends a standard multiple shooting scheme for periodic orbits, allowing us to compute invariant manifolds for periodic orbits without considering compositions of the map This leads to a system of conjugacy equations characterizing the complete collection of chart maps, with one chart attaching a local stable/unstable manifold to each point along the periodic orbit We develop a formal series solution for the system of conjugacy equations and show that the coefficients of the series are determined by recursively solving certain linear systems of equations We derive the recursive equations for a number of example problems in dimensions two and three, with both polynomial and transcendental nonlinearities, and present some numerical results which illustrate the utility of the method We also highlight so

Spatially Localized Self-Assembly Driven by Electrically Charged Phase Separation

Abstract: Self-assembly driven by phase separation coupled to Coulombic interactions is fundamental to a wide range of applications, examples of which include soft matter lithography via di-block copolymers, membrane design using polyelectrolytes, and renewable energy applications based on complex nano-materials, such as ionic liquids. The most common mean field framework for these problems is the nonlocal Cahn--Hilliard, such as the Ohta--Kawasaki model. Unlike the common investigations of spatially extended patterns, the focus here is on the emergence of spatially localized states in both the classical and the extended Ohta--Kawasaki model. The latter also accounts for (i) asymmetries in long-range Coulomb interactions that are manifested by differences in the dielectric response, and (ii) asymmetric short-range interactions that correspond to differences in the chemical potential between two materials or phases. It is shown that in one space dimension there is a multiplicity of coexisting localized solutions, wh...

Memory Matters in Synchronization of Stochastically Coupled Maps

Abstract: Synchronization of stochastically coupled chaotic oscillators is a topic of intensive research for its ubiquitous application across natural and technological systems. Several breakthroughs have been made over the last decade in understanding the underpinnings of stochastic synchronization. Yet, most of the literature has focused on memoryless switching, where the coupling between the oscillators intermittently changes independently of the switching history. Here, we analytically investigate the synchronization of two one-dimensional coupled nonlinear maps under Markovian switching. We linearize the system in the vicinity of the synchronous solution and examine the mean square asymptotic stability of the error dynamics. By leveraging state-of-the-art techniques in jump linear systems, fundamentals of ergodic theory, and perturbation analysis, we elucidate the potential of Markovian switching in manipulating the stability of synchronization. We focus on chaotic tent maps, for which we compute exact, closed...