arXiv subjects

Emmanuel Fleurantin

Publications and source records attributed to Emmanuel Fleurantin.

10 recordsLinked to original sources

arXiv2026

Predicting the occurrence of Braess paradox in the synchronization threshold of coupled oscillator systems

We study Braess-type paradoxes in coupled oscillator networks where the addition of an edge to the network leads to an increase in the critical coupling required for the existence of a stable phase-locked synchronized solution. By introducing a continuous edge-weight parameter and constructing an augmented system that allows application of the Implicit Function Theorem, we derive an explicit first-order formula for the sensitivity of the critical coupling, $K'_c(0)$, to small changes in the network structure. This formula reveals that an edge is locally Braess if the ordering of the oscillators being coupled is opposite the ordering of their components in the critical eigenvector at the saddle-node bifurcation, providing a simple criterion for Braess paradox to occur. We then employ this local approximation as a predictor for the occurrence of Braess paradox when the edge is fully incorporated in the network. This prediction is tested numerically in several classes of random networks, demonstrating both high predictive fidelity and the nonlinear limitations of the local approximation. We also examine several explicit examples, highlighting possible motifs by which Braess-type paradoxes occur. Together, these results establish a predictive framework for understanding how incremental changes in network topology can produce counterintuitive synchronization outcomes.

math.DS↗

arXiv2026

Rate-Induced Tipping in a Non-Uniformly Moving Habitat and Determination of the Critical Rate

A habitat that is moving due to environmental change may result in tipping to extinction if the rate at which it moves is too great. We use a scalar reaction-diffusion equation with a non-autonomous reaction term, representing a spatially localized habitat moving from one asymptotic location to another, as a context for studying this phenomenon. The movement is characterized by displacement $d$ and rate parameter $r$. The system admits three steady states in both asymptotic habitat locations: a stable extinction state $u_0^*=0$, an unstable pulse (so-called edge state) $u_1^*(x)>0$, which gives rise to the Allee effect, and a stable pulse (populated base state) $u_2^*(x)>u_1^*(x)$, which corresponds to a thriving population at its carrying capacity. Numerical simulations for a specific model identify a critical displacement $d^*$ and, for $d > d^*$, demonstrate the existence of a \textit{critical rate} $r_c(d)$ at which rate-induced tipping occurs: for $r> r_c$ an initially thriving population becomes extinct due to habitat movement being too rapid. We provide analytical results for two limiting cases. For $r\ll 1$, solutions track the moving base state with error $O(r)$. For $r\gg 1$, solutions converge to the extinction state provided $d$ is sufficiently large. For $d$ too small, no tipping occurs regardless of $r$. Numerical simulations complement and extend these analytical results. At the critical rate $r=r_c(d)$, we identify a pulse-to-pulse heteroclinic connection between the base state at the past asymptotic location and the edge state at the future asymptotic location of the habitat. We also establish the uniqueness of this critical rate and non-degeneracy of the heteroclinic connection as $r$ varies.

math.AP↗

arXiv2026

Dynamics of perturbed elliptical billiard tables

Dynamical billiards consist of a particle on a two-dimensional table, bouncing elastically off a boundary curve. The state of the system is given by two numbers: one describing the location along the curve where the bounce occurs, and another describing the incoming angle of the bounce. Successive bounces define a two-dimensional area preserving map, and iterating this map gives a dynamical system first studied by Birkhoff. One of the simplest smooth table shapes is that of an ellipse, in which case the dynamics of the billiard map is completely integrable. The longstanding Birkhoff conjecture is that elliptical tables are the only smooth convex table for which complete integrability occurs. In this spirit, we present an implicit real analytic method for iterating billiard maps on perturbed elliptical tables. This method allows us to compute local stable and unstable manifolds of periodic orbits using the parameterization method. Globalizing these local manifolds numerically provides insight into the dynamics of the table.

math.DS↗

arXiv2025

On the Spectrum of Schrödinger Operators Interacting at Two Distinct Scales

Schrödinger operators of the form $Δ- W$ on $L^2_{\text{rad}}(\mathbb{R}^3)$, the space of radially symmetric square integrable functions are relevant in a variety of physical contexts. The potential $W$ is taken to be radially symmetric (i.e. $W(x) = W(|x|)$) and to decompose into two components with distinct spatial scales: $W=W_\varepsilon= V_0+V_{1,\varepsilon}$. The second component $V_{1,\varepsilon}(|x|) = \varepsilon^2V_1(\varepsilon |x|)$ represents a scaled potential that becomes increasingly delocalized as $\varepsilon \to 0$. We will assume that both potentials $V_0(r), V_1(r)$ exhibit certain decay properties as $r \to \infty$. We show how the eigenvalue count on the positive real axis is built out of the spectra associated with the two reduced eigenvalue problems on their separate scales. The result is that the total number of eigenvalues of $Δ- W$ is the sum of the number of positive eigenvalues of $Δ- V_0$ and $Δ- V_1$. Our analysis combines dynamical systems techniques with a separation of scales argument, providing a novel framework for studying spectral properties of differential operators where multiple spatial scales interact.

math-ph↗

arXiv2025

3D Printing of Invariant Manifolds in Dynamical Systems

Invariant manifolds are one of the key features that organize the dynamics of a differential equation. We introduce a novel approach to visualizing and studying invariant manifolds by using 3D printing technology, combining advanced computational techniques with modern 3D printing processes to transform mathematical abstractions into tangible models. Our work addresses the challenges of translating complex manifolds into printable meshes, showcasing results for the following systems of differential equations: the Lorenz system, the Arneodo-Coullet-Tresser system, and the Langford system. By bridging abstract mathematics and physical reality, this approach promises new tools for research and education in nonlinear dynamics. We conclude with practical guidelines for reproducing and extending our results, emphasizing the potential of 3D-printed manifolds to enhance understanding and exploration in dynamical systems theory.

math.DS↗

arXiv2024

Tipping mechanisms in a carbon cycle model

Rate-induced tipping (R-tipping) occurs when a ramp parameter changes rapidly enough to cause the system to tip between co-existing, attracting states, while noise-induced tipping (N-tipping) occurs when there are random transitions between two attractors of the underlying deterministic system. This work investigates R-tipping and N-tipping events in a carbonate system in the upper ocean, in which the key objective is understanding how the system undergoes tipping away from a stable fixed point in a bistable regime. While R-tipping away from the fixed point is straightforward, N-tipping poses challenges due to a periodic orbit forming the basin boundary for the attracting fixed point of the underlying deterministic system. Furthermore, in the case of N-tipping, we are interested in the case where noise is away from the small noise limit, as it is more appropriate for the application. We compute the most probable escape path (MPEP) for our system, resulting in a firm grasp on the least action path in an asymmetric system of higher scale. Our analysis shows that the carbon cycle model is susceptible to both tipping mechanisms when using the standard formulations.

nlin.CD↗

arXiv2024

Investigating Ocean Circulation Dynamics Through Data Assimilation: A Mathematical Study Using the Stommel Box Model with Rapid Oscillatory Forcings

We investigate ocean circulation changes through the lens of data assimilation using a reduced-order model. Our primary interest lies in the Stommel box model which reveals itself to be one of the most practicable models that has the ability of reproducing the meridional overturning circulation. The Stommel box model has at most two regimes: TH (temperature driven circulation with sinking near the north pole) and SA (salinity driven with sinking near the equator). Currently, the meridional overturning is in the TH regime. Using box-averaged Met Office EN4 ocean temperature and salinity data, our goal is to provide a probability that a future regime change occurs and establish how this probability depends on the uncertainties in initial conditions, parameters and forcings. We will achieve this using data assimilation tools and DAPPER within the Stommel box model with fast oscillatory regimes.

math.DS↗

arXiv2023

A Dynamical Systems Approach for Most Probable Escape Paths over Periodic Boundaries

Analyzing when noisy trajectories, in the two dimensional plane, of a stochastic dynamical system exit the basin of attraction of a fixed point is specifically challenging when a periodic orbit forms the boundary of the basin of attraction. Our contention is that there is a distinguished Most Probable Escape Path (MPEP) crossing the periodic orbit which acts as a guide for noisy escaping paths in the case of small noise slightly away from the limit of vanishing noise. It is well known that, before exiting, noisy trajectories will tend to cycle around the periodic orbit as the noise vanishes, but we observe that the escaping paths are stubbornly resistant to cycling as soon as the noise becomes at all significant. Using a geometric dynamical systems approach, we isolate a subset of the unstable manifold of the fixed point in the Euler-Lagrange system, which we call the River. Using the Maslov index we identify a subset of the River which is comprised of local minimizers. The Onsager-Machlup (OM) functional, which is treated as a perturbation of the Friedlin-Wentzell functional, provides a selection mechanism to pick out a specific MPEP. Much of the paper is focused on the system obtained by reversing the van der Pol Equations in time (so-called IVDP). Through Monte-Carlo simulations, we show that the prediction provided by OM-selected MPEP matches closely the escape hatch chosen by noisy trajectories at a certain level of small noise.

math.DS↗

arXiv2020

Computer Assisted Proofs of Attracting Invariant Tori for ODEs

This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain $C^k$ lower bounds on the regularity of the embedding. In the resonant case we verify the existence of tori which are only $C^0$ and neither star-shaped nor Lipschitz

math.DS↗

arXiv2019

Resonant Tori, Transport Barriers, and Chaos in a Vector Field with a Neimark-Sacker Bifurcation

We make a detailed numerical study of a three dimensional dissipative vector field derived from the normal form for a cusp-Hopf bifurcation. The vector field exhibits a Neimark-Sacker bifurcation giving rise to an attracting invariant torus. Our main goals are to (A) follow the torus via parameter continuation from its appearance to its disappearance, studying its dynamics between these events, and to (B) study the embeddings of the stable/unstable manifolds of the hyperbolic equilibrium solutions over this parameter range, focusing on their role as transport barriers and their participation in global bifurcations. Taken together the results highlight the main features of the global dynamics of the system.

math.DS↗