SIAM Journal on Applied Dynamical Systems

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

We present a novel and global three-dimensional reduction of a nondimensionalized version of the four-dimensional Hodgkin–Huxley equations [J. Rubin and M. Wechselberger, Biol. Cybernet., 97 (2007), pp. 5–32] that is based on geometric singular perturbation theory. We investigate the dynamics of the resulting three-dimensional system in two parameter regimes in which the flow evolves on three distinct timescales. Specifically, we demonstrate that the system exhibits bifurcations of oscillatory dynamics and complex mixed-mode oscillations, in accordance with the geometric mechanisms introduced in [P. Kaklamanos, N. Popović, and K. U. Kristiansen, Chaos, 32 (2022), 013108], and we classify the various firing patterns in terms of the external applied current. While such patterns have been documented in [S. Doi, S. Nabetani, and S. Kumagai, Biol. Cybernet., 85 (2001), pp. 51–64] for the multiple-timescale Hodgkin–Huxley equations, we elucidate the geometry that underlies the transitions between them, which had not been previously emphasized.

The Kuramoto model provides a prototypical framework for synchronization phenomena in interacting particle systems. Apart from full phase synchrony where all oscillators behave identically, identical Kuramoto oscillators with ring-like nonlocal coupling can exhibit more elaborate patterns such as uniformly twisted states. It was discovered by Wiley, Strogatz, and Girvan in 2006 that the stability of these twisted states depends on the coupling range of each oscillator. In this paper, we analyze twisted states and their bifurcations in the infinite particle limit of ring-like nonlocal coupling. We not only consider traditional pairwise interactions as in the Kuramoto model but also demonstrate the effects of higher-order nonpairwise interactions, which arise naturally in phase reductions. We elucidate how pairwise and nonpairwise interactions affect the stability of the twisted states, compute bifurcating branches, and show that higher-order interactions can stabilize twisted states that are unstable if the coupling is only pairwise.

Motivated by investigating multistationarity in biochemical systems, we address saddle-node bifurcations for chemical reaction networks endowed with general kinetics. At positive equilibria, we identify structural network conditions that guarantee the bifurcation behavior, and we develop a method to identify the proper bifurcation parameters. As a relevant example, we explicitly provide such bifurcation parameters for Michaelis–Menten and Hill kinetics. Examples of applications include reversible feedback cycles, the central carbon metabolism of Escherichia coli, and autocatalytic networks.

Precise conditions are provided for the existence and criticality of Turing bifurcations in a general class of activator-inhibitor reaction-diffusion equations on a one-dimensional infinite domain. The class includes generalized Schnakenberg and Brusselator models, as well as other models with cubic autocatalytic nonlinear terms. Previous numerical work suggests the existence of a bifurcation structure containing localized patterns due to the so-called homoclinic snaking mechanism. This paper provides explicit calculations to justify those results. Two distinct scalings of parameters that lead to tractable normal-form coefficients are considered in the limit that the diffusion ratio \(\delta^2 \to 0\) . First, under a small-parameter scaling, the Turing bifurcation is shown to be always subcritical. Second, a large-parameter scaling reveals the Turing bifurcation to be supercritical, leading, by continuity, to the existence of a codimension-two degenerate bifurcation. The sign of a fifth-order normal form coefficient is also computed, which is shown to have the correct sign for the local birth of homoclinic snaking. For the case of the Brusselator, two such codimension-two points can be found explicitly, as can the leading-order expression for the Maxwell point, in a parameter wedge about which localized patterns emerge. Numerical results are found to be consistent with the theory.

Fluids subject to both thermal and compositional variations can undergo doubly diffusive convection when these properties both affect the fluid density and diffuse at different rates. A variety of patterns can arise from these buoyancy-driven flows, including spatially localized states known as convectons, which consist of convective fluid motion localized within a background of quiescent fluid. We consider these states in a vertical slot with the horizontal temperature and solutal gradients providing competing effects to the fluid density while allowing the existence of a conduction state. In this configuration, convectons have been studied with specific parameter values where the onset of convection is subcritical, and the states have been found to lie on a pair of secondary branches that undergo homoclinic snaking in a parameter regime below the onset of linear instability. In this paper, we show that convectons persist into parameter regimes in which the primary bifurcation is supercritical and there is no bistability, despite coexistence between the stable conduction state and large-amplitude convection. We detail this transition by considering spatial dynamics and observe how the structure of the secondary branches becomes increasingly complex owing to the increased role of inertia at low Prandtl numbers.

The goal of this study is to develop a computer-assisted method for proving the existence of periodic solutions for state-dependent delayed perturbations of ordinary differential equations (ODEs). We assume that the unperturbed ODE has an isolated periodic orbit, and derive a set of polynomial inequalities which, if satisfied, imply the existence and uniqueness of a periodic solution in the perturbed system. A general procedure is given, which facilitates the computation of the polynomial coefficients and the optimization of the inputs necessary to satisfy the inequalities on an explicit range of the perturbing parameter. Our approach uses tools from validated numerics and interval arithmetic together with Chebyshev series expansions to obtain the periodic orbit of the ODE as well as the solution of the variational equations. The general procedure is illustrated by proving the existence of some periodic orbits in state-dependent delayed perturbations of the van der Pol equation.

How does a group of agents break indecision when deciding about options with qualities that are hard to distinguish? Biological and artificial multiagent systems, from honeybees and bird flocks to bacteria, robots, and humans, often need to overcome indecision when choosing among options in situations in which the performance or even the survival of the group is at stake. Breaking indecision is also important because in a fully indecisive state, where agents are not biased toward any specific option, the agent group is maximally sensitive and prone to adapt to inputs and changes in its environment. Here, we develop a mathematical theory to study how decisions arise from the breaking of indecision. Our approach is grounded in both equivariant and network bifurcation theory. We model decision from indecision as synchrony-breaking in influence networks in which each node is the value assigned by an agent to an option. First, we show that three universal decision behaviors, namely, deadlock, consensus, and dissensus, are the generic outcomes of synchrony-breaking bifurcations from a fully synchronous state of indecision in influence networks. Second, we show that all deadlock and consensus value patterns and some dissensus value patterns are predicted by the symmetry of the influence networks. Third, we show that there are also many “exotic” dissensus value patterns. These patterns are predicted by network architecture but not by network symmetries through a new synchrony-breaking branching lemma. This is the first example of exotic solutions in an application. Numerical simulations of a novel influence network model illustrate our theoretical results.

The influence of multiplicative stochastic perturbations on the class of asymptotically Hamiltonian systems on the plane is investigated. It is assumed that disturbances do not preserve the equilibrium of the corresponding limiting system, and their intensity decays in time with power-law asymptotics. The paper discusses the long-term asymptotic behavior of solutions and its dependence on the structure and parameters of perturbations. In particular, it is shown that perturbed trajectories can tend to the equilibrium of the limiting system or that new stochastically stable states can arise. The proposed analysis is based on a combination of the averaging method and the construction of stochastic Lyapunov functions. The results obtained are applied to the problem of capture into parametric autoresonance in the presence of noise.

As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension 3, we introduce a \(C^2\) -open set of diffeomorphisms of \(\mathbb R^3\) having two horseshoes with different dimensions of instability. We prove that the unstable set of one horseshoe and the stable set of the other are of Hausdorff dimension nearly 2 whose cross sections are Cantor sets and that the intersection of the unstable and stable sets contains a fractal set of Hausdorff dimension nearly 1. As a corollary we detect \(C^2\) -robust heterodimensional cycles. Our proof employs the theory of normally hyperbolic invariant manifolds and the thicknesses of Cantor sets.

The overexploitation of natural resources by our industrial society questions its long-term sustainability. Recently, a simple nature-society interrelation model, called the HANDY (Human And Nature DYnamics) model, has been proposed by Motesharrei, Rivas, and Kalnay [Ecol. Econ., 101 (2014), pp. 90–102] to address this concern with a special emphasis on the role of the stratification of the society. In this paper we analyze the dynamics of this model, and we explore the influence of two parameters: the nature depletion rate and the inequality factor. We characterize the asymptotic states of the system through a bifurcation analysis, and we derive several quantitative predictions. We show that some collapses are irreversible and, depending on the wealth production factor, bistability regimes can be obtained. In particular, a sustainable equilibrium can coexist with cycles of prosperity and collapse. We discuss the possible policies to prevent undesirable fates.

We study traveling wave solutions to bistable differential equations on infinite k-ary trees. These graphs generalize the notion of classical square infinite lattices and our results complement those for bistable lattice equations on \(\mathbb{Z}^2_{\times }\). Using comparison principles and explicit lower and upper solutions, we show that wave solutions are pinned for small diffusion parameters. Upon increasing the diffusion, the wave starts to travel with nonzero speed, in a direction that depends on the detuning parameter. However, once the diffusion is sufficiently strong, the wave propagates in a single direction up the tree irrespective of the detuning parameter. In particular, our results imply that changes to the diffusion parameter can lead to a reversal of the propagation direction.

In this paper, we are concerned with the quasi-periodic forced Schrödinger equation subject to Dirichlet boundary condition \(iu_{t}-u_{xx}+\gamma u+g(\beta t)|u|^2u=0,\,\,x\in [0,\pi ],\) where \(\gamma\) is real, and \(g(\beta t)\) is of lower regularity and quasi-periodic in \(t\) with the frequency vector \(\beta =(\beta_1,\ldots,\beta_m)\) . Based on Birkhoff normal form theory and the KAM iterative method for infinite dimensional Hamiltonian systems, we prove the existence of quasi-periodic solutions.

For a mechanical system consisting of a rotator and a pendulum coupled via a small, time-periodic Hamiltonian perturbation, the Arnold diffusion problem asserts the existence of “diffusing orbits” along which the energy of the rotator grows by an amount independent of the size of the coupling parameter, for all sufficiently small values of the coupling parameter. There is a vast literature on establishing Arnold diffusion for such systems. In this work, we consider the case when an additional, dissipative perturbation is added to the rotator-pendulum system with coupling. Therefore, the system obtained is not symplectic but conformally symplectic. We provide explicit conditions on the dissipation parameter, so that the resulting system still exhibits energy growth. The fact that Arnold diffusion may play a role in systems with small dissipation was conjectured by Chirikov. In this work, the coupling is carefully chosen, but the mechanism we present can be adapted to general couplings, and we will deal with the general case in future work.

Multisite phosphorylation is a signaling mechanism well known to give rise to multiple steady states, a property termed multistationarity. When phosphorylation occurs in a sequential and distributive manner, we obtain a family of networks indexed by the number of phosphorylation sites \(n\). This work addresses the problem of understanding the parameter region where this family of networks displays multistationarity, by focusing on the projection of this region onto the set of kinetic parameters. The problem is phrased in the context of real algebraic geometry and reduced to studying whether a polynomial, defined as the determinant of a parametric matrix of size three, attains negative values over the positive orthant. The coefficients of the polynomial are functions of the kinetic parameters. For any \(n\), we provide sufficient conditions for the polynomial to be positive and, hence, preclude multistationarity, and also sufficient conditions for it to attain negative values and, hence, enable multistationarity. These conditions are derived by exploiting the structure of the polynomial and its Newton polytope and employing circuit polynomials. A relevant consequence of our results is that the sets of kinetic parameters that enable or preclude multistationarity are both connected for all \(n\).

The stability analysis of synchronization in time-varying higher-order networked structures (simplicial complexes) is a challenging problem due to the presence of time-varying group interactions. In this context, most of the previous studies have been done either on temporal pairwise networks or on static simplicial complexes. Here, for the first time, we propose a general framework to study the synchronization phenomenon in temporal simplicial complexes. We show that the synchronous state exists as an invariant solution and obtain the necessary condition for it to emerge as a stable state in the fast-switching regime. We prove that the time-averaged simplicial complex plays the role of synchronization indicator whenever the switching among simplicial topologies is adequately fast. We attempt to transform the stability problem into a master stability function form. Unfortunately, for the general circumstances, the dimension reduction of the master stability equation is cumbersome due to the presence of group interactions. However, we overcome this difficulty in two interesting situations based on either the functional forms of the coupling schemes or the connectivity structure of the simplicial complex, and we demonstrate that the necessary condition mimics the form of a master stability function in these cases. We verify our analytical findings by applying them on synthetic and real-world networked systems. We find that the presence of temporality along with the multiway interactions improves the synchronization phenomena as compared to the static higher-ordered or temporal pairwise system. In addition, our results also reveal that with sufficient higher-order coupling and adequately fast rewiring, the temporal simplicial complex achieves synchrony even in a very low connectivity regime.

The task of modeling and forecasting a dynamical system is one of the oldest problems, and it remains challenging. Broadly, this task has two subtasks: extracting the full dynamical information from a partial observation, and then explicitly learning the dynamics from this information. We present a mathematical framework in which the dynamical information is represented in the form of an embedding. The framework combines the two subtasks using the language of spaces, maps, and commutations. The framework also unifies two of the most common learning paradigms: delay-coordinates and reservoir computing. We use this framework as a platform for two other investigations of the reconstructed system, its dynamical stability and the growth of error under iterations. We show that these questions are deeply tied to more fundamental properties of the underlying system, i.e., the behavior of matrix cocycles over the base dynamics, its nonuniform hyperbolic behavior, and its decay of correlations. Thus, our framework bridges the gap between universally observed behavior of dynamics modeling and the spectral, differential, and ergodic properties intrinsic to the dynamics.

In this work we propose a model that represents the relation between fish ponds, the mosquito population and the transmission of malaria. It has been observed that in the Amazonic region of Acre, in the North of Brazil, fish farming is correlated to the transmission of malaria when carried out in artificial ponds that become breeding sites. Evidence has been found indicating that cleaning the vegetation from the edges of the crop tanks helps to control the size of the mosquito population. We use our model to determine the effective contribution of fish farming practices on malaria transmission dynamics. The model consists of a nonlinear system of ordinary differential equations with jumps at the cleaning time, which act as impulsive controls. We study the asymptotic behavior of the system in function of the intensity and periodicity of the cleaning, and the value of the parameters. In particular, we state sufficient conditions under which the mosquito population is eliminated or persists, and under which the malaria is eliminated or becomes endemic. We prove our conditions by applying results for cooperative systems with concave nonlinearities.

We consider a stochastic partial differential equation close to bifurcation of pitchfork type, where a one-dimensional space changes its stability. For finite-time Lyapunov exponents we characterize regions depending on the distance from bifurcation and the noise strength where finite-time Lyapunov exponents are positive and thus detect bifurcations. One technical tool is the reduction of the essential dynamics of the infinite-dimensional stochastic system to a simple ordinary stochastic differential equation, which is valid close to the bifurcation.

We develop a framework to design optimal entrainment signals that entrain an ensemble of heterogeneous nonlinear oscillators, described by phase models, at desired phases. We explicitly take into account heterogeneity in both oscillation frequency and the type of oscillators characterized by different phase response curves. The central idea is to leverage the Fourier series representation of periodic functions to decode a phase-selective entrainment task into a quadratic program. We demonstrate our approach using a variety of phase models where we entrain the oscillators into distinct phase patterns. Also, we show how the generalizability gained from our formulation enables us to meet a wide range of design objectives and constraints, such as minimum-power, fast entrainment, and charge-balanced controls.

Many biological systems can be modeled as a chemical reaction network with unknown parameters. Data available to identify these parameters are often in the form of a stationary distribution, such as that obtained from measurements of a cell population. In this work, we introduce a framework for analyzing the identifiability of the reaction rate coefficients of chemical reaction networks from stationary distribution data. Working with the linear noise approximation, which is a diffusive approximation to the chemical master equation, we give a computational procedure to certify global identifiability based on Hilbert’s Nullstellensatz. We present a variety of examples that show the applicability of our method to chemical reaction networks of interest in systems and synthetic biology, including discrimination between possible molecular mechanisms for the interaction between biochemical species.

We formulate an effective numerical scheme that can readily, and accurately, calculate the dynamics of weakly interacting multi-pulse solutions of the quintic complex Ginzburg–Landau equation (QCGLE) in one space dimension. The scheme is based on a global center-manifold reduction where one considers the solution of the QCGLE as the composition of individual pulses plus a remainder function, which is orthogonal to the adjoint eigenfunctions of the linearized operator about a single pulse. This center-manifold projection overcomes the difficulties of other, more orthodox, numerical schemes, by yielding a fast-slow system describing “slow” ordinary differential equations for the locations and phases of the individual pulses, and a “fast” partial differential equation for the remainder function. With small parameter \(\epsilon =e^{-\lambda_r d_0}\) where \(\lambda_r\) is a constant and \(d_0\gt 0\) is the minimal pulse separation distance, we write the fast-slow system in terms of first-order and second-order correction terms only, a formulation which is solved more efficiently than the full system. This fast-slow system is integrated numerically using adaptive time-stepping. Results are presented here for two- and three-pulse interactions. For the two-pulse problem, cells of periodic behavior, separated by an infinite set of heteroclinic orbits, are shown to “split” under perturbation creating complex spiral behavior. For the case of three-pulse interaction a range of dynamics, including chaotic pulse interaction, is found. While results are presented for pulse interaction in the QCGLE, the numerical scheme can also be applied to a wider class of parabolic PDEs.

A positively invariant ball and a global attractor for the porous media lattice system are obtained. By the implicit Euler scheme, the continuous-time model is discretized as a discrete-time porous media lattice system, which generates a discrete semigroup on the positively invariant ball for sufficiently small step sizes. It is shown that the discrete semigroup has a numerical attractor, which converges upper semicontinuously to the global attractor as the step size tends to zero. A new subject on enlarged attractors for lattice systems is investigated, which includes unique existence and convergence of attractors as well as solutions on the larger initial space. The second-order Taylor expansion and discretization error on the enlarged regular space are established to prove the upper semicontinuity of the enlarged numerical attractors with respect to the step size, while an upper bound of the enlarged attractors is provided to establish the lower semicontinuity in special cases.

This work studies front formation in the Allen–Cahn equation with a parameter heterogeneity which slowly varies in space. In particular, we consider a heterogeneity which mediates the local stability of the zero state and subsequent pitchfork bifurcation to a nontrivial state. For slowly varying ramps which are either rigidly propagating in time or stationary, we rigorously establish existence and stability of positive, monotone fronts and give leading order expansions for their interface location. For nonzero ramp speeds, and sufficiently small ramp slopes, the front location is determined by the local transition between convective and absolute instability of the base state and leads to an \(\mathcal{O}(1)\) delay beyond the instantaneous pitchfork location before the system jumps to a nontrivial state. The slow ramp induces a further delay of the interface controlled by a slow passage through a fold of strong- and weak-stable eigenspaces of the associated linearization. We introduce projective coordinates to desingularize the dynamics near the trivial state and track relevant invariant manifolds all the way to the fold point. We then use geometric singular perturbation theory and blow-up techniques to locate the desired intersection of invariant manifolds. For stationary ramps, the front is governed by the slow passage through the instantaneous pitchfork bifurcation with inner expansion given by the unique Hastings–McLeod connecting solution of Painlevé’s second equation. We once again use geometric singular perturbation theory and blow-up techniques to track invariant manifolds into a neighborhood of the nonhyperbolic point where the ramp passes through zero and to locate intersections.

This manuscript is aimed at addressing several long-standing limitations of dynamic mode decomposition in the application of Koopman analysis. Principal among these limitations are the convergence of associated dynamic mode decomposition (DMD) algorithms and the existence of Koopman modes. To address these limitations, two major modifications are made, where Koopman operators are removed from the analysis in light of Liouville operators (known as Koopman generators in special cases), and these operators are shown to be compact for certain pairs of Hilbert spaces selected separately as the domain and range of the operator. While eigenfunctions are discarded in the general analysis, a viable reconstruction algorithm is still demonstrated, and the sacrifice of eigenfunctions realizes the theoretical goals of DMD analysis that have yet to be achieved in other contexts. However, in the case where the domain is embedded in the range, an eigenfunction approach is still achievable, where a more typical DMD routine is established, but that leverages a finite rank representation that converges in norm. The manuscript concludes with the description of two DMD algorithms that converge when a dense collection of occupation kernels, arising from the data, are leveraged in the analysis.

The extinction of species is a major problem of concern with a large literature. We investigate a differential equations model for population interactions with the goal of determining when several species (i.e., coordinates of a bounded solution) must die out or “go extinct” and must do so exponentially fast. Typically each coordinate represents the population density of a different species. For our main tool, we create what we call “die-out” Lyapunov functions. A given system may have several or many such functions, each of which is a function of a different set of coordinates. That die-out function implies that one of the species in its subset must die out exponentially fast—for almost every choice of coefficients of the system. We create a “team” of die-out functions that work together to show that \(k\) species must die, where \(k\) is determined separately. Second, we present a “trophic” condition for generalized Lotka–Volterra systems that guarantees that there is a trapping region that is globally attracting. That implies that all solutions are bounded.

Reaction-diffusion models with nonlocal constraints naturally arise as limiting cases of coupled bulk-surface models of intracellular signalling. In this paper, a minimal, mass-conserving model of cell-polarization on a curved membrane is analyzed in the limit of slow surface diffusion. Using the tools of formal asymptotics and calculus of variations, we study the characteristic wave-pinning behavior of this system on three dynamical timescales. On the short timescale, generation of an interface separating high- and low-concentration domains is established under suitable conditions. Intermediate timescale dynamics are shown to lead to a uniform growth or shrinking of these domains to sizes that are fixed by global parameters. Finally, the long timescale dynamics reduce to area-preserving geodesic curvature flow that may lead to multi-interface steady state solutions. These results provide a foundation for studying cell polarization and related phenomena in biologically relevant geometries.

Dynamic mode decomposition (DMD) has recently become a popular tool for the nonintrusive analysis of dynamical systems. Exploiting proper orthogonal decomposition (POD) as a dimensionality reduction technique, DMD is able to approximate a dynamical system as a sum of spatial bases evolving linearly in time, thus enabling a better understanding of the physical phenomena and forecasting of future time instants. In this work we propose an extension of DMD to parameterized dynamical systems, focusing on the future forecasting of the output of interest in a parametric context. Initially all the snapshots—for different parameters and different time instants—are projected to a reduced space; then DMD, or one of its variants, is employed to approximate reduced snapshots for future time instants. Exploiting the low dimension of the reduced space, we then combine the predicted reduced snapshots using regression techniques, thus enabling the possibility of approximating any untested parametric configuration in the future. This paper depicts in detail the algorithmic core of this method; we also present and discuss three test cases for our algorithm: a simple dynamical system with a linear parameter dependency, a heat problem with nonlinear parameter dependency, and a fluid dynamics problem with nonlinear parameter dependency.

For reaction networks with zero-one stoichiometric coefficients (or simply zero-one networks), we prove that if a network admits a Hopf bifurcation, then the rank of the stoichiometric matrix is at least four. As a corollary, we show that if a zero-one network admits a Hopf bifurcation, then it contains at least four species and five reactions. As applications, we show that there exist rank-four subnetworks, which have the capacity for Hopf bifurcations/oscillations, in two biologically significant networks: the mitogen-activated-protein-kinase cascade and the extracellular signal-regulated kinase network. We provide a computational tool for computing all four-species, five-reaction, zero-one networks that have the capacity for Hopf bifurcations.

The rotating shallow water equations with f-plane approximation and nonlinear bottom drag are a prototypical model for midlatitude geophysical flow that experience energy loss through simple topography. Motivated by numerical schemes for large-scale geophysical flow, we consider this model on the whole space with horizontal kinetic energy backscatter source terms built from negative viscosity and stabilizing hyperviscosity with constant parameters. We study its interplay with linear and nonsmooth quadratic bottom drag through the existence of coherent flows. Our results highlight that backscatter can have undesired amplification and selection effects, generating obstacles to energy distribution. We find that decreasing linear bottom drag destabilizes the trivial flow and generates nonlinear flows that can be associated with geostrophic equilibria (GE) and inertia-gravity waves (IGWs). The IGWs are periodic travelling waves, while the GE are stationary and can be studied by a plane wave reduction. We show that for isotropic backscatter both bifurcate simultaneously and supercritically, while for anisotropic backscatter the primary bifurcations are GE. In all cases, the presence of nonsmooth quadratic bottom drag implies unusual scaling laws. For the rigorous bifurcation analysis, by Lyapunov–Schmidt reduction, care has to be taken due to this lack of smoothness and since the hyperviscous terms yield a lack of spectral gap at large wave numbers. For purely smooth bottom drag, we identify a class of explicit such flows that behave linearly within the nonlinear equations: amplitudes can be steady and arbitrary, or grow exponentially and unboundedly. We illustrate the results by numerical computations and present extended branches in parameter space.

This paper investigates the stability and instability of a generalized reaction-diffusion-advection system, which describes a cross-talk between antigens and immune cells via chemokines in the immune system. Based on the spectral analysis, energy estimates, and bootstrap techniques, we first consider a linearized problem around the positive constant equilibrium and show how the reaction term and chemotactic strength affect the stability and instability of the linearized problem. Then we go a further step to study the stability and instability for the nonlinear system based on the linearized results. Moreover, the numerical simulations are carried out to support our analytically theoretical results. Our results not only extend the previous ones of [S. Lee et al., J. Math. Biol., 75 (2017), pp. 1101–1131] but also involve some new conclusions.

We describe a class of vector fields exhibiting abundant switching near a heteroclinic network: for every neighborhood of the network and every infinite admissible path, the set of initial conditions within the neighborhood that follows the path has positive Lebesgue measure. The proof relies on the existence of “large” strange attractors in the terminology of Broer, Simó, and Tatjer [Nonlinearity, 11 (1998), pp. 667–770] near a heteroclinic tangle unfolding an attracting network with a two-dimensional heteroclinic connection. For our class of vector fields, any small nonempty open ball of initial conditions realizes infinite switching. We illustrate the theory with a specific one-parameter family of differential equations, for which we are able to characterize its global dynamics for almost all parameters.

We leverage data-driven model discovery methods to determine governing equations for the emergent behavior of heterogeneous networked dynamical systems. Specifically, we consider networks of coupled nonlinear oscillators whose collective behavior approaches a limit cycle. Stable limit cycles are of interest in many biological applications, as they model self-sustained oscillations (e.g. heartbeats, chemical oscillations, neurons firing, circadian rhythm). For systems that display relaxation oscillations, our method automatically detects boundary (time) layer structures in the dynamics, fitting inner and outer solutions and matching them in a data-driven manner. We demonstrate the method on well-studied systems: the Rayleigh oscillator and the van der Pol oscillator. We then apply the mathematical framework to networks of semisynchronized Kuramoto, Rayleigh, Rössler, and FitzHugh–Nagumo oscillators, as well as heterogeneous combinations thereof, showing that the discovery of coarse-grained variables and dynamics can be accomplished with the proposed architecture.

Traditional models of opinion dynamics, in which the nodes of a network change their opinions based on their interactions with neighboring nodes, consider how opinions evolve either on time-independent networks or on temporal networks with edges that follow Poisson statistics. Most such models are Markovian. However, in many real-life networks, interactions between individuals (and hence the edges of a network) follow non-Poisson processes and thus yield dynamics with memory-dependent effects. In this paper, we model opinion dynamics in which the entities of a temporal network interact and change their opinions via random social interactions. When the edges have non-Poisson interevent statistics, the corresponding opinion models have non-Markovian dynamics. We derive a family of opinion models that are induced by arbitrary waiting-time distributions (WTDs), and we illustrate a variety of induced opinion models from common WTDs (including Dirac delta distributions, exponential distributions, and heavy-tailed distributions). We analyze the convergence to consensus of these models and prove that homogeneous memory-dependent models of opinion dynamics in our framework always converge to the same steady state regardless of the WTD. We also conduct a numerical investigation of the effects of waiting-time distributions on both transient dynamics and steady states. We observe that models that are induced by heavy-tailed WTDs converge more slowly to a steady state than models that are induced by WTDs with light tails (or with compact support) and that entities with longer waiting times exert more influence on the mean opinion at steady state.