arXiv subjects

Theodore Vo

Publications and source records attributed to Theodore Vo.

15 recordsLinked to original sources

arXiv2026

Singular Turing bifurcations and spatial canard solutions in nonlinear reaction-diffusion systems

We study a general class of nonlinear reaction-diffusion equations that model pattern-forming systems. The class includes the Gierer-Meinhardt PDE, Brusselator model, and van der Pol PDE, as well as the Gray-Scott, Klausmeier, Lengyel-Epstein, Schnakenberg PDEs and others of activator-inhibitor type. In the limit in which the activator diffusivity is much smaller than that of the inhibitor, these PDEs exhibit singular Turing bifurcations, which have only recently begun to receive attention. We analytically establish that the spatially-periodic solutions that emerge in both the sub-critical and super-critical cases of singular Turing bifurcations are spatially-periodic canard solutions. These canard patterns are new types of spatially-periodic solutions that --just beyond the Turing point-- have fast-slow structure in space, rather than the classical sinusoidal profile. In addition, their amplitude grows more rapidly than the classical square root growth for Turing patterns. Indeed, even for parameter values that differ by one part in a hundred from the Turing point, they can have $\mathcal{O}(1)$ amplitude. We also establish the existence of general spatially-dependent canard solutions with fast-slow structure. Our analysis focuses on the spatial ODEs that govern the time-independent solutions. We show that these ODEs have a reversible folded saddle-node singularity of type II asymptotically close to the singular Turing point, that reversible folded saddles occur for parameters away from it, and that the true and faux canards of these folded singularities are the mechanisms responsible for creating the spatial canard solutions in the general class of PDEs.

math.DS↗

arXiv2026

A symmetric mechanism for symmetry-breaking in oscillator networks with strong nonlinear coupling

In this article, we describe and analyse a novel mechanism for symmetry-breaking in minimal symmetrically coupled identical slow/fast oscillator networks with strong nonlinear mutually inhibitory coupling. We show that the symmetry-breaking, surprisingly, originates from the canard dynamics of a folded node that lies on the axis of symmetry. By applying geometric singular perturbation theory and the blow-up technique to a normal form, we determine the geometric mechanisms by which the {\em symmetric folded node} induces symmetry-breaking. More specifically, we show that (i) the fold curve of the coupled system is orthogonal to the axis of symmetry at the symmetric folded node; (ii) there is only one primary maximal canard (either strong or weak, depending on parameters), which always lies on the axis of symmetry and is the axis of rotation for the twisting of solutions; and (iii) the number of rotations is the key local diagnostic feature that breaks the symmetry. Our work is closely related to that of Kristiansen and Pedersen [SIAM J. Appl. Dyn. Syst., {\bf 22} (2023)] on symmetrically coupled FitzHugh-Nagumo oscillators with strong linear inhibitory gap junctional coupling, however, we consider nonlinear coupling and we identify and study multiple sub-types of their `cusped singularities'. We demonstrate our theoretical results by applying them to a model of the eukaryotic cell cycle in which the symmetric folded node plays a key role in rhythmogenesis. More specifically, we study periodic and quasi-periodic symmetry-breaking mixed-mode oscillatory attractors of the cell cycle model. We show that the local twisting induced by the symmetric folded node is the local mechanism that both breaks the symmetry and generates the small-amplitude oscillations in the mixed-mode dynamics.

math.DS↗

arXiv2025

A coordinate-independent Pontryagin-Rodygin theorem for slow-fast averaging

The slow drift along a manifold of periodic orbits is a key mathematical structure underlying bursting dynamics in many scientific applications. While classical averaging theory, as formalised by the Pontryagin-Rodygin theorem, provides a leading-order approximation for this slow drift, the connection to the underlying geometry described by Geometric Singular Perturbation Theory (GSPT)--also known as Fenichel theory--is often not explicit, particularly at higher orders. This paper makes that connection rigorous and constructive using the parametrisation method. We provide a detailed, self-contained exposition of this functional analytic technique, showing how it synthesizes the geometric insight of invariant manifold theory with a systematic, perturbative algorithm. By treating the manifold's embedding and the reduced flow as coupled unknowns, the method provides a constructive proof of an averaged system that is guaranteed to be geometrically consistent with the persistence of the normally hyperbolic manifold to any order. We translate the abstract theory into a concrete computational procedure using Floquet theory, spectral analysis, and the Fredholm alternative, yielding a practical guide for computing high-accuracy, higher-order averaged models, and we demonstrate its implementation, both analytically and numerically, through specific examples.

math.DS↗

arXiv2025

Stable and unstable spatially-periodic canards created in singular subcritical Turing bifurcations in the Brusselator system

In this article, we study the Brusselator partial differential equation (PDE) in the limit in which the diffusivity of the activator is much smaller than that of the inhibitor. The PDE robustly exhibits a subcritical Turing bifurcation that, in this limit, is labeled as a singular Turing bifurcation. We show that families of spatially-periodic canard solutions emerge from this subcritical singular Turing bifurcation. Then, right after they emerge, the solutions lose their purely sinusoidal structure ($e^{ik_Tx}$, where $k_T$ is the critical wavenumber at the Turing bifurcation) and gain a distinct multi-scale spatial structure. They consist of segments along which the components vary gradually in space, interspersed with short intervals on which the activator component exhibits pulses and steep gradients. The branches of these spatially-periodic canards undergo a saddle-node bifurcation. Some of the large-amplitude patterns on the upper branches are attractors of the PDE, and unstable patterns with small pulses that exist below the folds appear to guide the evolution of data to the attractors. We also analyze the spatial ordinary differential equations (ODEs) that govern time-independent solutions. We show that the spatial ODE system has a folded singularity known as a reversible folded saddle-node of type II (RFSN-II) point that coincides with the Turing bifurcation in the singular limit. We demonstrate that, for parameter values close to the Turing bifurcation, the true and faux canards of the RFSN-II point are responsible for generating the spatially-periodic canards, and for parameter values away from the Turing value there is a reversible folded saddle point whose true and faux canards generate the spatially-periodic canard solutions. Overall, we identify the RFSN-II and RFS folded singularities and their canards as new selection mechanisms for subcritical bifurcations.

math.DS↗

arXiv2024

Les Canards de Turing

In this article, we study a system of reaction-diffusion equations in which the diffusivities are widely separated. We report on the discovery of families of spatially periodic canard solutions that emerge from {\em singular Turing bifurcations}. The emergence of these spatially periodic canards asymptotically close to the Turing bifurcations, which are reversible 1:1 resonant Hopf bifurcations in the spatial ODE system, is an analog in spatial dynamics of the emergence of limit cycle canards in the canard explosions that occur asymptotically close to Hopf bifurcations in time-dependent ODEs. In the full PDE system, we show that for most parameter values under study the Turing bifurcation is sub-critical, and we present the results of some direct numerical simulations showing that several of the different types of spatial canard patterns are attractors in the prototypical PDE. To support the numerical discoveries, we use geometric desingularization and geometric singular perturbation theory to demonstrate the existence of these families of spatially periodic canards. Crucially, in the singular limit, we study a novel class of {\em reversible folded singularities}. In particular, there are two reversible folded saddle-node bifurcations of type II (RFSN-II), each occurring asymptotically close to a Turing bifurcation. We derive analytical formulas for these singularities and show that their canards play key roles in the observed families of spatially periodic canard solutions. Then, for an interval of values of the bifurcation parameter further below the Turing bifurcation and RFSN-II point, the spatial ODE also has spatially periodic canard patterns, however these are created by a reversible folded saddle (instead of the RFSN-II). It also turns out that there is an interesting scale invariance, so that some components of some spatial canards exhibit nearly self-similar dynamics.

math.DS↗

arXiv2022

Fronts in the wake of a parameter ramp: slow passage through pitchfork and fold bifurcations

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 non-trivial 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 non-zero 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 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 de-singularize 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 Painlev\'{e}'s second equation. We once again use geometric singular perturbation theory and blow-up to track invariant manifolds into a neighborhood of the non-hyperbolic point where the ramp passes through zero and to locate intersections.

math.DS↗

arXiv2021

A new class of chimeras in locally coupled oscillators with small-amplitude, high-frequency asynchrony and large-amplitude, low-frequency synchrony

Chimeras are surprising yet important states in which domains of decoherent (asynchronous) and coherent (synchronous) oscillations co-exist. In this article, we report on the discovery of a new class of chimeras, called {\it mixed-amplitude chimera states}, in which the structures, amplitudes, and frequencies of the oscillations differ substantially in the decoherent and coherent regions. These mixed-amplitude chimeras exhibit domains of decoherent small-amplitude oscillations (phase waves) coexisting with domains of stable and coherent large-amplitude or mixed-mode oscillations. They are observed in a prototypical bistable partial differential equation with spatially homogeneous kinetics and purely local, isotropic diffusion. New bifurcations are identified in which the mixed-amplitude chimeras emerge from, or are annihilated in, common large-amplitude solutions. Also, key singularities, folded nodes and folded saddles, arising commonly in multi-scale, bistable systems play important roles, and these have not previously been studied in systems with chimeras. The discovery of these mixed-amplitude chimeras is an important advance for understanding some processes in neuroscience, pattern formation, and physics which involve both small-amplitude and large-amplitude oscillations. It may also be of use for understanding some aspects of EEG recordings from animals that exhibit unihemispheric slow-wave sleep.

nlin.PS↗

arXiv2020

Delayed Hopf bifurcation and space-time buffer curves in the Complex Ginzburg-Landau equation

In this article, the phenomenon of delayed Hopf bifurcations (DHB) in reaction-diffusion PDEs is analyzed in the cubic Complex Ginzburg-Landau equation with a slowly-varying parameter. We use the classical asymptotic methods of stationary phase and steepest descents to show that solutions which approach the attracting quasi-steady state (QSS) before the Hopf bifurcation remain near that state for long times after the Hopf bifurcation and the QSS has become repelling. In the complex time plane, the phase function of the linear PDE has a saddle point, and the Stokes and anti-Stokes lines are central to the asymptotics. The nonlinear terms are treated by applying an iterative method to the mild form of the PDE given by perturbations about the linear particular solution. This tracks the closeness of solutions near the attracting and repelling QSS. Next, we show that beyond a key Stokes line through the saddle there is a space-time buffer curve along which the particular solution of the linear PDE ceases to be exponentially small, causing the solution of the nonlinear PDE to diverge from the repelling QSS and exhibit large-amplitude oscillations. The homogeneous solution also stops being exponentially small in a spatially dependent manner, as determined also by the initial time. We find four different cases of DHB, depending on the competition between the homogeneous and particular solutions, and we quantify how these depend on system parameters. Examples are presented for each case, with uni-modal, spatially-periodic, smooth step, and algebraically-growing source terms. Also, rich spatio-temporal dynamics are observed in the post-DHB oscillations. Finally, it is shown that large-amplitude source terms can be designed so that solutions spend substantially longer times near the repelling QSS, and hence region-specific control over the delayed onset of oscillations can be achieved.

math.DS↗

arXiv2020

Unsteady dynamics of a classical particle-wave entity

A droplet bouncing on the surface of a vertically vibrating liquid bath can walk horizontally, guided by the waves it generates on each impact. This results in a self-propelled classical particle-wave entity. By using a one-dimensional theoretical pilot-wave model with a generalized wave form, we investigate the dynamics of this particle-wave entity. We employ different spatial wave forms to understand the role played by both wave oscillations and spatial wave decay in the walking dynamics. We observe steady walking motion as well as unsteady motions such as oscillating walking, self-trapped oscillations and irregular walking. We explore the dynamical and statistical aspects of irregular walking and show an equivalence between the droplet dynamics and the Lorenz system, as well as making connections with the Langevin equation and deterministic diffusion.

physics.flu-dyn↗

arXiv2019

Why Pacing Frequency Affects the Production of Early Afterdepolarizations in Cardiomyocytes: An Explanation Revealed by Slow/Fast Analysis of a Minimal Model

Early afterdepolarizations (EADs) are pathological voltage oscillations in cardiomyocytes that have been observed in response to a number of pharmacological agents and disease conditions. EADs are small voltage fluctuations that occur during the plateau of an action potential. Although a single-cell behavior, EADs can lead to tissue-level arrhythmias, including ventricular tachycardia. Much is currently known about the biophysical mechanisms (i.e., the roles of ion channels and intracellular calcium stores) for EADs, due partially to the development and analysis of mathematical models. This includes the application of slow/fast analysis, which takes advantage of timescale separation inherent in the system to simplify its analysis. We take this further, using a minimal 3D model to demonstrate that the phase-2 EADs are canards that are formed in the neighborhood of a folded node singularity. This knowledge allows us to determine the number of EADs that can be produced for a given parameter set without performing computer simulations, and provides guidance on parameter changes that can facilitate or inhibit EAD production. With this approach, we demonstrate why periodic stimulation, as would occur in an intact heart, preferentially facilitates EAD production when applied at low frequencies,. We also explain the origin of complex alternan dynamics that can occur with intermediate-frequency stimulation, in which varying numbers of EADs are produced with each stimulation. These revelations fall out naturally from an understanding of folded node singularities, but are hard or impossible to glean from a knowledge of the biophysical mechanism for EADs alone. Therefore, an understanding of the canard mechanism is a useful complement to an understanding of the biophysical mechanism that has been developed over years of experimental and computational investigations.

q-bio.TO↗

arXiv2018

Delayed bifurcation phenomena in reaction-diffusion equations: persistence of canards and slow passage through Hopf bifurcations

In the context of a spatially extended model for the electrical activity in a pituitary lactotroph cell line, we establish that two delayed bifurcation phenomena from ODEs ---folded node canards and slow passage through Hopf bifurcations--- persist in the presence of diffusion. For canards, the single cell (ODE) model exhibits canard-induced bursting. Numerical simulations of the PDE reveal rich spatio-temporal canard dynamics, and the transitions between different bursts are mediated by spatio-temporal maximal canards. The ODE model also exhibits delayed loss of stability due to slow passage through Hopf bifurcations. Numerical simulations of the PDE reveal that this delayed stability loss persists in the presence of diffusion. To quantify and predict the delayed loss of stability, we show that the Complex Ginzburg-Landau equation exhibits the same property, and derive a formula for the space-time boundary that acts as a buffer curve beyond which the delayed onset of oscillations must occur.

math.DS↗

arXiv2017

Modeling the Dynamics of Glacial Cycles

This article is concerned with the dynamics of glacial cycles observed in the geological record of the Pleistocene Epoch. It focuses on a conceptual model proposed by Maasch and Saltzman [J. Geophys. Res.,95, D2 (1990), pp. 1955-1963], which is based on physical arguments and emphasizes the role of atmospheric CO2 in the generation and persistence of periodic orbits (limit cycles). The model consists of three ordinary differential equations with four parameters for the anomalies of the total global ice mass, the atmospheric CO2 concentration, and the volume of the North Atlantic Deep Water (NADW). In this article, it is shown that a simplified two-dimensional symmetric version displays many of the essential features of the full model, including equilibrium states, limit cycles, their basic bifurcations, and a Bogdanov-Takens point that serves as an organizing center for the local and global dynamics. Also, symmetry breaking splits the Bogdanov-Takens point into two, with different local dynamics in their neighborhoods.

math.DS↗

arXiv2017

Dynamical systems analysis of the Maasch-Saltzman model for glacial cycles

This article is concerned with the internal dynamics of a conceptual model proposed by Maasch and Saltzman [J. Geophys. Res., 95, D2 (1990) 1955-1963] to explain central features of the glacial cycles observed in the climate record of the Pleistocene Epoch. It is shown that, in most parameter regimes, the long-term system dynamics occur on certain intrinsic two-dimensional invariant manifolds in the three-dimensional state space. These invariant manifolds are slow manifolds when the characteristic time scales for the total global ice mass and the volume of North Atlantic Deep Water are well- separated, and they are center manifolds when the characteristic time scales for the total global ice mass and the volume of North Atlantic Deep Water are comparable. In both cases, the reduced dynamics on these manifolds are governed by Bogdanov-Takens singularities, and the bifurcation curves associated to these singularities organize the parameter regions in which the model exhibits glacial cycles. This work was submitted March 30, 2017.

math.DS↗

arXiv2016

On the Existence of and Relationship between Canards and Torus Canards in Forced Slow/Fast Systems

Canards are special solutions of slow/fast systems which are ubiquitous in neuroscience and electrical engineering. Two distinct classes of canard solutions have been identified and carefully studied: folded singularity canards and torus canards. Recently, an explicit and analytic relationship between these seemingly unrelated families of solutions was established in the classical forced van der Pol equation (Burke et al., J. Nonlinear Sci. 26:405--451, 2015). In this article, we generalize the results of Burke et al. (2015) to the broader class of time-periodically forced planar slow/fast systems, which includes the forced van der Pol and the forced FitzHugh-Nagumo equations. We analytically determine the parameter values in this class of systems for which the two types of canard solutions exist, and show that the branches of primary canards of folded singularities continue into those of the torus canards as the forcing frequency is increased. We illustrate our results in the paradigm problem of the forced FitzHugh-Nagumo system.

math.DS↗

arXiv2016

Generic Torus Canards

Torus canards are solutions of slow/fast systems that alternate between attracting and repelling manifolds of limit cycles of the fast subsystem. A relatively new dynamic phenomenon, torus canards have been found in neural applications to mediate the transition from spiking to bursting via amplitude-modulated spiking. In $\mathbb{R}^3$, torus canards are degenerate: they require one-parameter families of 2-fast/1-slow systems in order to be observed and even then, they only occur on exponentially thin parameter intervals. The addition of a second slow variable unfolds the torus canard phenomenon, making them generic and robust. That is, torus canards in slow/fast systems with (at least) two slow variables occur on open parameter sets. So far, generic torus canards have only been studied numerically, and their behaviour has been inferred based on averaging and canard theory. This approach, however, has not been rigorously justified since the averaging method breaks down near a fold of periodics, which is exactly where torus canards originate. In this work, we combine techniques from Floquet theory, averaging theory, and geometric singular perturbation theory to show that the average of a torus canard is a folded singularity canard. In so doing, we devise an analytic scheme for the identification and topological classification of torus canards in $\mathbb{R}^4$. We demonstrate the predictive power of our results in a model for intracellular calcium dynamics, where we explain the mechanisms underlying a novel class of elliptic bursting rhythms, called amplitude-modulated bursting, by constructing the torus canard analogues of mixed-mode oscillations. We also make explicit the connection between our results here with prior studies of torus canards and torus canard explosion in $\mathbb{R}^3$, and discuss how our methods can be extended to slow/fast systems of arbitrary (finite) dimension.

math.DS↗