Society for Industrial and Applied Mathematics: SIAM Journal on Applied Dynamical Systems: Table of Contents

Unique Reconstruction from Mean-Field Measurements

Narcicegi Kiran — 2026-04-01T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 629-653, June 2026.
Abstract.We address the inverse problem of reconstructing both the structure and the dynamics of a network from mean-field measurements, which are linear combinations of node states. This setting arises in applications where only a few aggregated observations are available, making network inference challenging. We focus on the case when the number of mean-field measurements is smaller than the number of nodes. To tackle this ill-posed recovery problem, we propose a framework that combines localized initial perturbations with sparse optimization techniques. We derive sufficient conditions that guarantee the unique reconstruction of the network’s adjacency matrix from mean-field data and enable recovery of node states and local governing dynamics. Numerical experiments demonstrate the robustness of our approach across a range of sparsity and connectivity regimes. These results provide theoretical and computational foundations for inferring high-dimensional networked systems from low-dimensional observations.

Multihumped Collapsing Solutions in the Nonlinear Schrödinger Problem: Existence, Stability, and Dynamics

S. Jon Chapman — 2026-04-02T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 654-694, June 2026.
Abstract.In the present work we examine multihumped solutions of the nonlinear Schrödinger equation in the blowup regime of the one-dimensional model with power-law nonlinearity, bearing a suitable exponent of [math]. We find that families of such solutions exist for arbitrary pulse numbers, with all of them bifurcating from the critical case of [math]. Remarkably, all of them involve “bifurcations from infinity,” i.e., the pulses come inward from an infinite distance as the exponent [math] increases past the critical point. The position of the pulses is quantified and the stability of the waveforms is also systematically examined in the so-called coexploding frame. Both the equilibrium distance between the pulse peaks and the point spectrum eigenvalues associated with the multihumped configurations are obtained as a function of the blowup rate [math] theoretically, and these findings are supported by detailed numerical computations. Finally, some prototypical dynamical scenarios are explored, and an outlook towards such multihumped solutions in higher dimensions is provided.

Balanced Dynamics in Strongly Coupled Networks

Cristobal Quiñinao — 2026-04-06T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 695-737, June 2026.
Abstract.Many mathematical models of interacting agents assume that individual interactions scale down in proportion to the network size, ensuring that the combined input received from the network does not diverge. In theoretical neuroscience, Sompolinsky and van Vreeswijk proposed in 1996 that, should these scalings be violated (and under appropriate conditions), the system may not diverge but rather approach a balanced state where the inputs to each neuron compensate each other (in neuroscience, where inhibitory currents compensate the excitatory ones). We come back to this observation and formulate here a mathematical conjecture for the occurrence of such behaviors in general stochastic systems of interacting agents. From a mathematical viewpoint, this conjecture can be viewed as a double-limit problem in the space of probability measures, which we discuss in detail, as it provides several possible mathematical avenues for proving this result. We provide some numerical and theoretical explorations of the conjecture in classical models of neuronal networks. Moreover, we provide a complete proof of an asymptotic result consistent with one of the double-limit problems in a one-dimensional model with separable coupling inspired by models of chemically coupled neurons. This proof relies on asymptotic methods, and particularly desingularization techniques used in some PDEs, that we apply here to the mean-field limit of the network as the coupling is made to diverge. From the applications viewpoint, this theory provides an alternative, minimalistic explanation for the widely observed balance of excitation and inhibition in the cerebral cortex not requiring the assumption of the existence of complex regulatory mechanisms.

[math] Extension and Invariant Manifolds for the Compactification of Nonautonomous Systems with Autonomous Limits

Shuang Chen — 2026-04-08T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 738-764, June 2026.
Abstract.We study the compactification of nonautonomous systems with autonomous limits and related dynamics. Although the [math] extension of the compactification was well established, a great number of problems arising in bifurcation and stability analysis require the compactified systems with high-order smoothness. Inspired by this, we give a criterion for the [math] ([math]) extension of the compactification. After compactifying nonautonomous systems, the compactified systems may gain an additional center direction. We prove the existence and uniqueness of center or center-stable manifolds for normally hyperbolic compact invariant manifolds.

Phase Synchronization in Random Geometric Graphs on the Two-Dimensional Sphere

Cecilia De Vita — 2026-04-09T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 765-788, June 2026.
Abstract.The Kuramoto model is a classical nonlinear ODE system designed to study synchronization phenomena. Each equation represents the phase of an oscillator, and the coupling between them is determined by a graph. There is an increasing interest in understanding the relation between the graph topology and the spontaneous synchronization of the oscillators. Abdalla, Bandeira, and Invernizzi [SIAM J. Appl. Dyn. Syst., 23 (2024), pp. 779–790] considered random geometric graphs on the [math]-dimensional sphere and proved that the system synchronizes with high probability as long as the mean number of neighbors and the dimension [math] go to infinity. They posed the question about the behavior when [math] is small. In this paper, we prove that synchronization holds for random geometric graphs on the two-dimensional sphere, with high probability as the number of nodes goes to infinity, as long as the initial conditions converge to a smooth function. We conjecture a similar behavior for more general simply connected closed Riemannian manifolds, but we expect global synchronization to fail if the manifold is not simply connected, as was shown in De Vita, Bonder, and Groisman [SIAM J. Appl. Dyn. Syst., 24 (2025), pp. 1–15] and suggested in Cirelli et al. [SIAM J. Appl. Math., 85 (2025), pp. 1719–1748].

Bistability of Traveling Waves and Wave-Pinning States in a Mass-Conserved Reaction-Diffusion System: From Bifurcations to Implications for Actin Waves

Jack M. Hughes — 2026-04-09T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 789-835, June 2026.
Abstract.Eukaryotic cells demonstrate a wide variety of dynamic patterns of filamentous actin (F-actin) and its regulators. Some of these patterns play important roles in cell functions, such as distinct motility modes, which motivate this study. We devise a polynomial mass-conserved reaction-diffusion model for active and inactive Rho-GTPase and F-actin in the cell cortex. The mass-conserved Rho-GTPase system promotes F-actin, which feeds back to inactivate the former. We study the model on a one-dimensional periodic domain (edge of thin sheet-like cell) using bifurcation theory in the framework of spatial dynamics, complemented with numerical simulations. Among several discussed bifurcations, the analysis centers on the study of the codimension-2 long wavelength and finite wavenumber Hopf instability, in which we describe a rich structure of steady wave-pinning states (a.k.a. mesas, obeying the Maxwell construction), propagating coherent solutions (fronts and excitable pulses), and traveling and standing waves, all distinguished by mass conservation regimes and classified by domain sizes. Specifically, we highlight the unexpected conditions for bistability between steady wave-pinning and traveling wave states on moderate domain sizes, i.e., unfolding through domain length. These results uncover and exemplify possible mechanisms of coexistence, robustness, and transitions between distinct cellular motility modes, including directed migration, turning, and ruffling. More broadly, the results indicate that nongradient reaction-diffusion models comprising mass conservation have distinct pattern formation mechanisms that motivate further investigations, such as the unfolding of codimension-3 instabilities and T-points.

Data-Driven Identification of Attractors Using Machine Learning

Marcio Gameiro — 2026-04-13T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 836-860, June 2026.
Abstract.In this paper we explore challenges in developing a topological framework in which machine learning can be used to robustly characterize global dynamics. Specifically, we focus on learning a useful discretization of the phase space of a flow on a compact, hyperrectangle in [math] from a neural network trained on labeled orbit data. A characterization of the structure of the global dynamics is obtained from approximations of attracting neighborhoods provided by the phase space discretization. The perspective that motivates this work is based on Conley’s topological approach to dynamics, which provides a means to evaluate the efficacy and efficiency of our approach.

Numerical Periodic Normalization at Codim 1 Bifurcations of Limit Cycles in DDEs

Maikel M. Bosschaert — 2026-04-17T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 861-901, June 2026.
Abstract.Recent work by B. Lentjes, L. Spek, M. M. Bosschaert, and Yu. A. Kuznetsov [J. Dynam. Differential Equations, 37 (2023), pp. 815–858; and J. Differential Equations, 423 (2025), pp. 631–694] on periodic center manifolds and normal forms for bifurcations of limit cycles in delay differential equations (DDEs) motivates the derivation of explicit computational formulas for the critical normal form coefficients of all codimension one bifurcations of limit cycles. In this paper, we derive such formulas via an application of the periodic normalization method in combination with the functional analytic perturbation framework for dual semigroups (sun-star calculus). The explicit formulas allow us to distinguish between nondegenerate, sub- and supercritical bifurcations. To efficiently apply these formulas, we introduce the characteristic operator as this enables us to use robust numerical boundary-value algorithms based on orthogonal collocation. Although our theoretical results are proven in a more general setting, the software implementation and examples focus on DDEs with discrete delays. The actual implementation is described in detail and its effectiveness is demonstrated on various models.

The Role of Mutants in the Spatiotemporal Progression of Inflammatory Bowel Disease: Three Classes of Permanent Form Traveling Waves

Blaine van Rensburg — 2026-04-20T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 902-939, June 2026.
Abstract.Despite its high prevalence and impact on the lives of those affected, a complete understanding of the cause of inflammatory bowel disease (IBD) is lacking. In this paper, we investigate a novel mechanism which proposes that mutant epithelial cells are significant to the progression of IBD since they promote inflammation and are resistant to death. We develop a simple model encapsulating the propagation of mutant epithelial cells and immune cells which results from interactions with the intestinal barrier and bacteria. Motivated by the slow growth of mutant epithelial cells, and relatively slow response rate of the adaptive immune system, we use singular perturbation theory via the method of matched asymptotic expansions to determine the one-dimensional slow invariant manifold that characterizes the leading order dynamics at all times beyond a passive initial adjustment phase. The dynamics on this manifold are controlled by a bifurcation parameter [math] which depends on the ratio of growth to decay rates of all components except mutants and determines three distinct classes of permanent-form traveling waves that describe the propagation of mutant epithelial and immune cells. These are obtained from scalar reaction-diffusion equations with the reaction being (i) a bistable nonlinearity with a cut-off, (ii) a cubic Fisher nonlinearity, and (iii) a Kolmogorov–Petrovskii–Piskunov or Fisher nonlinearity. Our results suggest that mutant epithelial cells are critical to the progression of IBD. However, their effect on the speed of progression is subdominant.

Large Deviations of a Network of Neurons with Dynamic Sparse Random Connections

James MacLaurin — 2026-04-23T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 940-963, June 2026.
Abstract.In this work, we determine the limiting dynamics of a system of interacting particles indexed by a lattice [math]. The connections are random, sparse, and unscaled, so that the system converges in the large size limit due to the probability of a connection between any two particles decreasing as the system size increases. The particles are also subject to noise (such as independent Brownian Motions). The method of proof is to assume a process-level (or Level 3) large deviation principle (LDP) for the double-layer empirical measure for the noise and connections and then apply a series of transformations to this to obtain an LDP for the process-level empirical measure of our system. Although it is not explicitly necessary, we expect that most applications of this work should involve an assumption of stationarity of the probability law for the noise and connections given translations of the lattice, so that the system converges to an ergodic probability law in the large size limit. This work synthesizes the theory of large-size limits of interacting particles with that of random graphs and matrices. It is therefore relevant to neuroscience and social networks theory in particular.

Global Stability of Wright-Type Equations with Negative Schwarzian

Mauro Díaz — 2026-04-23T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 964-997, June 2026.
Abstract. Simplicity of the 37/24-global stability criterion announced by E. M. Wright in 1955 and rigorously proved by B. Bánhelyi et al. in 2014 for the delayed logistic equation raised the question of its possible extension for other population models. In our study, we answer this question by extending the 37/24-stability condition for the Wright-type equations with decreasing smooth nonlinearity [math] which has a negative Schwarzian and satisfies the standard negative feedback and boundedness assumptions. The proof contains the construction and careful analysis of qualitative properties of certain bounding relations. To validate our conclusions, these relations are evaluated at finite sets of points; for this purpose, we systematically use interval analysis.

Bifurcations of Riemann Ellipsoids

Fahimeh Mokhtari — 2026-04-27T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 998-1041, June 2026.
Abstract.We give an account of the various changes in the stability character of three types of Riemann ellipsoids by establishing the occurrence of different quasi-periodic Hamiltonian bifurcations. Suitable symplectic changes of coordinates, that is, linear and nonlinear normal form transformations are performed, leading to the characterization of the bifurcations responsible for the stability changes. Specifically we find three types of bifurcations, namely, Hamiltonian pitchfork, saddle-center, and Hamiltonian–Hopf in the four-degree-of-freedom Hamiltonian system resulting after reducing out the symmetries of the problem. The approach is mainly analytical up to a point where nondegeneracy conditions have to be checked numerically.

Hamiltonian Triplet Interactions: Areal and Perimetric Forces

James D. Meiss — 2026-04-30T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 1042-1069, June 2026.
Abstract.Gravitational and electromagnetic interactions are Hamiltonian systems with forces between pairs of particles. We propose an alternative: Hamiltonian dynamics with triplet interactions between point particles. Our system has a potential energy that depends on the shape of the triangle for each triplet. Similar multibody forces occur in many physical systems, such as polarizable molecules, nucleon interactions, and colloids, but typically are combined with more conventional two-body forces. We focus on potentials that depend only on the triangle perimeter or on its area. The resulting forces point toward a center of the triangle, either the incenter or the orthocenter, respectively. For the planar case, the resulting system has six degrees of freedom but can be reduced to three since it conserves the total momentum and angular momentum. The dynamics often exhibits chaotic motion, but there are a number of special solutions, such as equilateral and isosceles triangles, and perturbations of these can lie on invariant tori. Numerical investigations of several examples show families of such regular trajectories as well as examples of chaotic dynamics.

Bifurcation Analysis for Gene Regulatory Networks Embedding a Toggle-Switch Model: Application to X Chromosome Inactivation

Frédérique Clément — 2026-05-06T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 1070-1105, June 2026.
Abstract.We analyze models of minimal gene regulatory networks (GRNs) introduced in the context of X chromosome inactivation (XCI), an epigenetic process occurring in placental female mammals. A core module in these GRNs is a 2D toggle-switch model resulting from a mutual inhibition. We first perform a bifurcation analysis of the toggle-switch model, where the main difficulty arises from the unknown coordinates of the fixed points. In the symmetric case, we prove the occurrence of a pitchfork bifurcation and compute explicitly the bifurcation lines. In the asymmetric case, we design a constructive numerical approach, providing one simultaneously with the bifurcation lines and fixed-point coordinates, and illustrate the occurrence of a saddle-node bifurcation using numerical continuation tools. Bistability in the toggle-switch accounts for the possible choice between an activated ([math]) and inactivated ([math]) state in the dynamics of a single X chromosome. We then proceed to a thorough analysis of 4D and 6D GRN models representing the joint dynamics of the X chromosome pair to (i) ensure the stability of the mono-inactivated ([math]) state and then (ii) exclude the stability of the bi-inactivated ([math]) and bi-activated ([math]) states. Studying the 6D GRN involves the analysis of toggle-switch models coupled through a state-dependent parameter. Combining proper changes of variables and reparameterization, we manage to study both the transient and asymptotic behavior from a 2D phase plane analysis. We further restrict the parameter space to meet quantitative specifications on the relative gene expression level between the [math] and [math] states.

A Novel Route to Oscillations via Noncentral SNICeroclinic Bifurcation: Unfolding the Separatrix Loop between a Saddle-node and a Saddle

Kateryna Nechyporenko — 2026-05-08T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 1106-1131, June 2026.
Abstract.In this paper, we investigate saddle-node to saddle separatrix-loops that we term SNICeroclinic bifurcations. They are generic codimension-two bifurcations involving a heteroclinic loop between one nonhyperbolic and one hyperbolic saddle. A particular codimension-three case is the noncentral SNICeroclinic bifurcation. We unfold this bifurcation in the minimal dimension (planar) case where the nonhyperbolic point is assumed to undergo a saddle-node bifurcation. Applying the method of Poincaré return maps, we present a minimal set of perturbations that captures all qualitatively distinct behaviors near a noncentral SNICeroclinic loop. Specifically, we study how variation of the three unfolding parameters leads to transitions from heteroclinic and homoclinic loops; saddle-node on an invariant circle (SNIC); and periodic orbits, as well as equilibria. We show that although the bifurcation has been largely unexplored in applications, it can act as an organizing center for transitions between various types of saddle-node and saddle separatrix loops. It is also a generic route to oscillations that are both born and destroyed via global bifurcations compared to the commonly observed scenarios involving local (Hopf) and, in some cases, global (homoclinic or SNIC) bifurcations.

Existence of Invariant Probability Measures for Stochastic Differential Equations with Finite Time Delay

M. van den Bosch — 2026-06-02T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 1132-1182, June 2026.
Abstract.We provide sufficient conditions for the existence of invariant probability measures for generic stochastic differential equations with finite time delay. This is achieved by means of the Krylov–Bogoliubov method. Furthermore, we focus on stochastic delay equations whose deterministic coefficient satisfies a one-sided bound, which enables us to show that boundedness in probability of a solution [math] entails boundedness in probability of its solution segment [math]. This implies that for a large set of systems, we can infer that an invariant measure exists if only there is at least one solution that is bounded in probability. Applications include, but are not limited to, the stochastic Mackey–Glass equations [M. van den Bosch, O. W. van Gaans, and S. M. Verduyn Lunel, SIAM J. Appl. Dyn. Syst., 25 (2026)] and the stochastic Wright’s equation [M. van den Bosch, O. W. van Gaans, and S. M. Verduyn Lunel, Stochastic Wright’s Equation: Existence of Invariant Measures, preprint, arXiv:2605.09805, 2026]. The noise driving the dynamical system is allowed to be an integrable Lévy process.

Analytic Extended Dynamic Mode Decomposition

Alexandre Mauroy — 2026-06-02T07:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 2, Page 1183-1206, June 2026.
Abstract.We develop a novel extended dynamic mode decomposition (EDMD)-type algorithm that captures the spectrum of the Koopman operator defined on a reproducing kernel Hilbert space of analytic functions. This method, which we call analytic EDMD, relies on an orthogonal projection on polynomial subspaces, which is equivalent to a data-driven Taylor approximation. In the case of dynamics with a hyperbolic equilibrium, analytic EDMD demonstrates excellent performance to capture the lattice-structured Koopman spectrum based on the eigenvalues of the linearized system at the equilibrium. Moreover, it yields the Taylor approximation of associated principal eigenfunctions. Since the method preserves the triangular structure of the operator, it does not suffer from spectral pollution, and, moreover, arbitrary accuracy on the spectrum can be reached with a fixed finite dimension of the approximation and with a (possibly nonuniform) sampling over an arbitrary set of nonzero measure. The performance of analytic EDMD is illustrated with numerical examples and is assessed through a comparative study with related methods. Finally, the method is complemented with theoretical results, proving strong convergence of the eigenfunctions and providing error bounds on the spectrum estimation.

Bounded-Confidence Models of Multidimensional Opinions with Topic-Weighted Discordance

Grace Jingying Li — 2026-01-02T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 1-41, March 2026.
Abstract.People’s opinions on a wide range of topics often evolve over time through their interactions with others. Models of opinion dynamics primarily focus on one-dimensional opinions, which represent opinions on one topic. However, opinions on various topics are rarely isolated; instead, they can be interdependent and correlated. In a bounded-confidence model (BCM) of opinion dynamics, agents are receptive to each other only if their opinions are sufficiently similar. We extend classical agent-based BCMs—namely, the Hegselmann–Krause BCM, which has synchronous interactions, and the Deffuant–Weisbuch BCM, which has asynchronous interactions—to a multidimensional setting, in which the opinions are multidimensional vectors representing opinions of different topics and opinions on different topics are interdependent. To measure opinion differences between agents, we introduce topic-weighted discordance functions that account for opinion differences in all topics. We define regions of receptiveness for our models, and we use them to characterize the steady-state opinion clusters and provide an analytical approach to compute these regions. In addition, we numerically simulate our models on various networks with initial opinions drawn from a variety of distributions. When initial opinions are correlated across different topics, our topic-weighted BCMs yield significantly different results in both transient and steady states compared to baseline models, where the dynamics of each opinion topic are independent.

Realizations Through Weakly Reversible Networks and the Globally Attracting Locus

Abhishek Deshpande — 2026-01-02T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 42-76, March 2026.
Abstract.We investigate the possibility that for any given reaction rate vector [math] associated with a network [math], there exists another network [math] with a corresponding reaction rate vector that reproduces the mass-action dynamics generated by [math]. Our focus is on a particular class of networks for [math], where the corresponding network [math] is weakly reversible. We show that strongly endotactic two-dimensional networks with a two-dimensional stoichiometric subspace, as well as certain endotactic networks under additional conditions, exhibit this property. In particular, we show that being endotactic is necessary for the dynamics of a network to be included in the dynamics of a weakly reversible network. Additionally, we establish a strong connection between this family of networks and the locus in the space of rate constants in which the corresponding dynamics admits globally stable steady states.

Beyond Boolean Networks: New Tools for the Steady State Analysis of Multivalued Networks

Juliana García Galofre — 2026-01-02T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 77-107, March 2026.
Abstract.Boolean networks can be viewed as functions on the set of binary strings of a given length, described via logical rules. They were introduced as dynamic models into biology, in particular as logical models of intracellular regulatory networks involving genes, proteins, and metabolites. Since genes can have several modes of action depending on their expression levels, binary variables are often not sufficiently rich, requiring the use of multivalued networks instead. In this paper, we explore the multivalued generalization of Boolean networks by writing the standard Boolean operations in terms of the operations from multivalued logic. We recall the basic theory of this mathematical framework and give a novel algorithm for computing the fixed points that in many cases has essentially the same complexity as in the binary case. Our approach provides a biologically intuitive representation of the network. Furthermore, it uses tools to compute lattice points in rational polytopes, tapping a rich area of algebraic combinatorics as a source for combinatorial algorithms for network analysis. An implementation of the algorithm is provided.

Computing a Connection Matrix and Persistence Efficiently from a Morse Decomposition

Tamal K. Dey — 2026-01-02T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 108-130, March 2026.
Abstract.Morse decompositions partition the flows in a vector field into equivalent structures. Given such a decomposition, one can define a further summary of its flow structure by what is called a connection matrix. These matrices, a generalization of Morse boundary operators from classical Morse theory, capture the connections made by the flows among the critical structures—such as attractors, repellers, and orbits—in a vector field. Recently, in the context of combinatorial dynamics, an efficient persistence-like algorithm to compute connection matrices has been proposed in Dey, Lipiński, Mrozek, and Slechta [SIAM J. Appl. Dyn. Syst., 23 (2024), pp. 81–97]. We show that, actually, the classical persistence algorithm with exhaustive reduction retrieves connection matrices, both simplifying the algorithm of Dey et al. and bringing the theory of persistence closer to combinatorial dynamical systems. We supplement this main result with an observation: the concept of persistence as defined for scalar fields naturally adapts to Morse decompositions whose Morse sets are filtered with a Lyapunov function. We conclude by presenting preliminary experimental results.

Computer Assisted Discovery of Integrability via SILO: Sparse Identification of Lax Operators

Jimmie Adriazola — 2026-01-02T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 131-159, March 2026.
Abstract.We formulate the discovery of Lax integrability of Hamiltonian dynamical systems as a symbolic regression problem, which, loosely speaking, seeks to maximize the compatibility between a pair of Lax operators and the known Hamiltonian of the dynamical system. Our approach is first tested on the simple harmonic oscillator. We then move on to the Henon–Heiles system, i.e., a two-degree-of-freedom system of nonlinear oscillators. The integrability of the Henon–Heiles system is critically dependent on a set of three parameters within its Hamiltonian, a fact that we leverage to assess the robustness of our approach in detecting the integrability of this system with respect to the parameter dependence of the Hamiltonian. We then adapt our method to canonical examples of Hamiltonian partial differential equations, including the Korteweg–de Vries and cubic nonlinear Schrödinger equations, again testing robustness against nonintegrable perturbations of their respective Hamiltonians. In all examples, our approach reliably confirms or denies the integrability of the equations of interest. Moreover, by appropriately adjusting the loss function and applying thresholded [math] regularization to enforce sparsity in the operator weights, we successfully recover accurate forms of the Lax pairs despite wide initial hypotheses on the operators. Some of the relevant Lax pairs, notably for the Henon–Heiles system and the Korteweg–de Vries equation, are distinct from the ones that are typically reported in the literature. The Lax pairs that our methodology discovers warrant further mathematical and computational investigation, and we discuss extensively the opportunities for further improvement of SILO as a viable tool for interpretable exploration of integrable Hamiltonian dynamical systems.

How Time and Pollster History Affect U.S. Election Forecasts under a Compartmental Modeling Approach

Ryan Branstetter — 2026-01-02T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 160-195, March 2026.
Abstract.In the months leading up to political elections in the United States, forecasts are widespread and take on multiple forms, including projections of what party will win the popular vote, state ratings, and predictions of vote margins at the state level. It can be challenging to evaluate how accuracy changes in the lead-up to Election Day or to put probabilistic forecasts into historical context. Moreover, forecasts differ between analysts, highlighting the many choices in the forecasting process. With this as motivation, here we take a more comprehensive view and begin to unpack some of the choices involved in election forecasting. Building on a prior compartmental model of election dynamics, we present the forecasts of this model across months, years, and types of race. By gathering together monthly forecasts of presidential, senatorial, and gubernatorial races from 2004–2022, we provide a larger-scale perspective and discuss how treating polling data in different ways affects forecast accuracy. We conclude with our 2024 election forecasts [45] (upcoming at the time of writing), and a postscript following the 2024 elections.

Bridging the Gap Between Koopmanism and Response Theory: Using Natural Variability to Predict Forced Response

Niccolò Zagli — 2026-01-06T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 196-229, March 2026.
Abstract.The fluctuation-dissipation theorem is a cornerstone result in statistical mechanics that can be used to translate the statistics of the free natural variability of a system into information on its forced response to perturbations. By combining this viewpoint on response theory with the key ingredients of Koopmanism, it is possible to deconstruct virtually any response operator into a sum of terms, each associated with a specific mode of natural variability of the system. This dramatically improves the interpretability of the resulting response formulas. We show here on three simple yet mathematically meaningful examples how to use the extended dynamic mode decomposition algorithm on an individual trajectory of the system to compute with high accuracy correlation functions as well as Green functions associated with acting forcings. This demonstrates the great potential of using Koopman analysis for the key problem of evaluating and testing the sensitivity of a complex system.

Emergence of Supercritical Hopf Bifurcations in Coupled Oscillator Ensembles and How Desynchronization Can Facilitate Energy-Optimal Phase Resetting

Dan Wilson — 2026-01-06T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 230-259, March 2026.
Abstract.Phase resetting, involving shifting the timing of oscillations by a prespecified amount, is a fundamental control problem arising in the investigation of oscillatory dynamical systems. This work investigates the relationships between phase resetting of the collective rhythm and desynchronization in populations of coupled oscillators in the context of an optimal control problem. Conditions on the emergence of a supercritical Hopf bifurcation are considered for a general model of coupled oscillators, and the Hopf normal form is used to simplify the analysis. Two solution archetypes for phase resetting emerge: weak resetting, which does not influence interoscillator synchrony, and strong resetting, whereby the network is first desynchronized before resynchronizing with the correct phase. Analytical expressions for the energy expenditure associated with these solutions yield general conditions for which desynchronization is expected during the course of phase resetting. In particular, strong phase resetting is more efficient than weak resetting when the following conditions are met: 1) the relaxation rate of the collective rhythm to its limit cycle is sufficiently slow relative to its natural frequency, and 2) the required phase shifts are sufficiently large.

Random Attraction in TASEP with Time-varying Hopping Rates

Lars Grüne — 2026-01-08T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 260-278, March 2026.
Abstract.The totally asymmetric simple exclusion principle (TASEP) is a fundamental model in nonequilibrium statistical mechanics. It describes the stochastic unidirectional movement of particles along a 1D chain of ordered sites. We consider the continuous-time version of TASEP with a finite number of sites and with time-varying hopping rates between the sites. We show how to formulate this model as a nonautonomous random dynamical system (NRDS) with a finite state-space. We provide conditions guaranteeing that random pullback and forward attractors of such an NRDS exist and consist of singletons. In the context of the nonautonomous TASEP, these conditions imply almost sure synchronization of the individual random paths. This implies in particular that perturbations that change the state of the particles along the chain are filtered out in the long run. We demonstrate that the required conditions are tight by providing examples where these conditions do not hold and consequently the forward attractor does not exist or the pullback attractor is not a singleton. The results in this paper generalize our earlier results for autonomous TASEP in [] and contain these as a special case.

High-order Approximations of the Canard Explosion in a Delayed van der Pol System

Shu Zhang — 2026-01-09T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 279-303, March 2026.
Abstract.In the present work, we investigate the canard explosion occurring in a neurodynamics model–the van der Pol system with a time-delayed feedback. We assume that the feedback gain is small (on the same order as the time scale difference) such that the delay differential equation (DDE) can be reduced to a planar ordinary differential equation (ODE) on a non-local center manifold. We then expand the ODE flow on this center manifold in the small parameter and apply a nonlinear time transformation to obtain high-order approximations of the critical value and the critical manifold simultaneously. A significant advantage of the present method is that the tedious computations of the center manifold reduction with normal form usually involved in solving delay differential equations are avoided. We prove that each perturbation order of the critical manifold can be expressed as a polynomial of a spatial variable. This greatly simplifies the computations and makes high-order computations possible. We also compare the proposed approach with the existing small-delay expansion of which DDEs are reduced to ODEs by assuming a small delay value. While the latter cannot predict the critical manifold due to the discontinuity, our approach provides accurate and continuous approximations. More significantly, the analytical results obtained by the nonlinear time transformation method go far beyond the existing first-order results, showing an excellent agreement with the numerical simulations even for large delay values. The procedure developed in this work is efficient but simple. Because our method predicts both the critical value and the critical manifold accurately, it has great potential for practical applications in dynamical systems with time-delayed coupling and small coupling coefficients, such as controlling the occurrence of the neuronal spiking and its amplitude, in the future.

Identifiability of Directed-Cycle and Catenary Linear Compartmental Models

Saber Ahmed — 2026-01-28T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 304-350, March 2026.
Abstract.A parameter of a mathematical model is structurally identifiable if it can be determined from noiseless experimental data. Here, we examine the identifiability properties of two important classes of linear compartmental models: directed-cycle models and catenary models (models for which the underlying graph is a directed cycle or a bidirected path, respectively). Our main result is a complete characterization of the directed-cycle models for which every parameter is (generically locally) identifiable. Additionally, for catenary models, we give a formula for their input-output equations. Such equations are used to analyze identifiability, so we expect our formula to support future analyses into the identifiability of catenary models. Our proofs rely on prior results on input-output equations, and we also use techniques from linear algebra and graph theory.

Bifurcations of the Hénon Map with Additive Bounded Noise

Jeroen S.W. Lamb — 2026-01-28T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 351-374, March 2026.
Abstract.We numerically study bifurcations of attractors of the Hénon map with additive bounded noise with spherical reach. The bifurcations are analyzed using a finite-dimensional boundary map. We distinguish between two types of bifurcations: topological bifurcations and boundary bifurcations. Topological bifurcations describe discontinuous changes of attractors, and boundary bifurcations occur when singularities of an attractor’s boundary are created or destroyed. We identify correspondences between topological and boundary bifurcations of attractors and local and global bifurcations of the boundary map.

An Extension of the Chemostat Model with Linear Coupling Between Species

Tewfik Sari — 2026-02-04T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 375-417, March 2026.
Abstract.We investigate the classical model of competition of two populations in the chemostat when a linear coupling between the populations is taken into account and the removal rates of the populations are distinct from the dilution rate and their yield coefficients are also distinct. This model extends a model of wall growth and a model of lateral gene transfer, previously studied in the literature. We show the existence and uniqueness of the coexistence equilibrium at which the populations coexist, provided that the input concentration of the chemostat exceeds a critical value, or, equivalently the dilution rate does not exceed a critical value that can be computed explicitly. In contrast with the particular cases of this model, previously studied in the literature, the positive equilibrium can be unstable with the appearance of Hopf bifurcations and sustainable oscillations. We construct the operating diagram of the system, which is the two-parameter bifurcation diagram with respect to the operating parameters, that are the dilution rate of the chemostat and its input nutrient concentration. This study reveals a rich variety of dynamical behaviors, including the emergence and disappearance of stable and unstable limit cycles through Hopf bifurcations and limit point of cycles bifurcations. Furthermore, codimension-two bifurcations such as cusp and generalized Hopf points are identified, highlighting complex transitions in the system’s dynamics.

[math]-Independent Boolean Networks

Julio Aracena — 2026-02-06T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 418-438, March 2026.
Abstract.This paper proposes a new parameter for studying Boolean networks: the independence number. We establish that a Boolean network is [math]-independent if, for any set of [math] variables and any combination of binary values assigned to them, there exists at least one fixed point in the network that takes those values at the given set of [math] indices. In this context, we define the independence number of a network as the maximum value of [math] such that the network is [math]-independent. This definition is closely related to widely studied combinatorial designs, such as “[math]-strength covering arrays,” also known as Boolean sets with all [math]-projections being surjective. Our motivation arises from understanding the relationship between a network’s interaction graph and its fixed points, which deepens the classical paradigm of research in this direction by incorporating a particular structure on the set of fixed points, beyond merely observing their quantity. Specifically, among the results of this paper, we highlight a condition on the in-degree of the interaction graph for a network to be [math]-independent, we show that all unate networks are at most [math]-independent, and we construct [math]-independent networks for all possible [math] in the case of monotone networks with a complete interaction graph.

Helfrich Cylinders—Instabilities, Bifurcations, and Amplitude Equations

Alexander Meiners — 2026-02-19T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 439-484, March 2026.
Abstract.Combining local bifurcation analysis with numerical continuation and bifurcation methods we study bifurcations from cylindrical vesicles described by the Helfrich equation with volume and area constraints, with a prescribed periodicity along the cylindrical axis. The bifurcating solutions are in two main classes, axisymmetric (pearling), and nonaxisymmetric (coiling, buckling, and wrinkling), and depending on the spontaneous curvature and the prescribed periodicity along the cylinder axis we obtain different stabilities of the bifurcating branches, and different secondary bifurcations.

Stabilization of Propagating Bubbles Using Control-Based Continuation

João Vitor Nogueira Fontana — 2026-02-20T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 485-517, March 2026.
Abstract.We explore the potential of feedback control to stabilize and detect unstable steady propagation modes for confined, deformable bubbles moving through a fluid-filled Hele-Shaw channel. We use a depth-averaged model as a numerical experiment to develop and test the capabilities of control-based continuation (CBC) in this system, with our work being the first use of CBC in free-surface fluid dynamics. We construct a suitable stabilizing feedback gain, with control actuation delivered via fluid injection at the sides of the channel and determined from real-time observations of the bubble shape as viewed from above. We show how this feedback gain can be used to stabilize, detect, and observe unstable states, without prior knowledge of their behavior. As the steady states in fact involve steady bubble propagation along the channel length, we investigate both the idealized case of comoving actuators, as well as the more plausible scenario when actuation is delivered through an array of equally-spaced injection points.

A Theoretical Approach for Gait Generation of the CPG Controller by Using the Time-Delay Coupled Oscillators

Xiao Yu — 2026-02-24T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 518-553, March 2026.
Abstract.In this paper, a new theoretical approach of gait generation is presented for the legged robots by using delay-coupling oscillators. First, a general theory of the central pattern generator (CPG) is constructed through the time delay coupled oscillators with a unidirectional ring network structure. Based on the symmetric Hopf bifurcation, the parameter condition for rhythm generation is obtained. The parameter group is divided into different regions by Hopf bifurcation curves, where the periodic solutions exhibit distinct spatiotemporal patterns in each region. Furthermore, to investigate the impact of these spatiotemporal patterns on the gait generation of multilegged robots, the proposed theoretical approach is applied to simulate various gaits for hexapod and biped robots using six FitzHugh–Nagumo (FHN) oscillators and two Stuart–Landau (SL) oscillators, respectively. The findings reveal that the rhythm signals generated by the CPG controller can prompt the robot to exhibit multiple stable gaits. The CPG controllers, composed of four and eight Van der Pol (VDP) oscillators, are designed to control the hip and knee joint movements of quadruped robots, which achieves the coordinated movement of a robot’s leg joints. To this end, a large number of numerical simulations are provided to validate the correctness of the proposed theoretical approach.

Filtered Finite State Projection Method for the Analysis and Estimation of Stochastic Biochemical Reaction Networks

Elena D’Ambrosio — 2026-03-05T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 554-587, March 2026.
Abstract.Time-lapse microscopy has become increasingly prevalent in biological experimentation, as it provides single-cell trajectories that unveil valuable insights into underlying networks and their stochastic dynamics. However, the limited availability of fluorescent reporters typically constrains tracking to only a few network species. Addressing this challenge, the dynamic estimation of hidden state components becomes crucial, for which stochastic filtering presents a robust mathematical framework. Yet, the complexity of biological networks often renders direct solutions to the filtering equation intractable due to high dimensionality and nonlinear interactions. In this study, we establish and rigorously prove the well-posedness of the filtering equation for the time evolution of the conditional distribution of hidden species. Focusing on continuous-time, noise-free observations within a continuous-time discrete state-space Markov chain model, we develop the filtered finite state projection (FFSP) method. This computational approach offers an approximated solution by truncating the hidden species’ state space, accompanied by computable error bounds. We illustrate the effectiveness of FFSP through diverse numerical examples, comparing it with established filtering techniques such as the Kalman filter, extended Kalman filter, and particle filter. Finally, we show an application of our methodology with real time-lapse microscopy data. This work not only advances the application of stochastic filtering to biological systems but also contributes towards more accurate implementation of biomolecular feedback controllers.

Analysis of Dynamics near Heteroclinic Networks in [math] with a Projected Map

David C. Groothuizen Dijkema — 2026-03-05T08:00:00Z

SIAM Journal on Applied Dynamical Systems, Volume 25, Issue 1, Page 588-628, March 2026.
Abstract.The usual method to analyze the stability of heteroclinic cycles and networks is with transition matrices that are derived from return maps. In this paper, we introduce an extension to this methodology, the projected map, which we define by identifying trajectories that have, in a certain sense, qualitatively the same dynamics. The projected map is a discrete, piecewise-smooth map of one dimension fewer than the order of the transition matrix. We use these maps to describe the dynamics of trajectories near three heteroclinic networks in [math] with four equilibria. We find in all three cases that the onset of trajectories that switch between cycles of the network is caused by either a fold bifurcation or a border-collision bifurcation, where fixed points of the map no longer exist in the corresponding function’s domain of definition. We are able to show that a given initial condition near any of these three networks is asymptotic to only one subcycle and cannot switch between subcycles multiple times, resolving a 30-year-old claim by Brannath. We are also able to generalize certain results to all quasi-simple networks, proving that a border-collision bifurcation of the projected map, corresponding to a condition on the eigenvectors of certain transition matrices, causes a cycle to lose stability.