Schedule for: 22w5057 - Topics in Multiple Time Scale Dynamics

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Dimension reduction in chemical reaction systems is often based on quasi-steady-state approximations. The mathematical justification of these approximations can be based on the concept of a slow manifold in the framework of geometric singular perturbation theory (GSPT). More recently it was observed, that the dynamics of specific biochemical models with switch-like or oscillatory behavior is organized by several slow manifolds, corresponding to different scaling regimes of the variables. Matching of these different scaling regimes is carried out by the blow-up method which is needed to analyse slow manifolds in situations where normal hyperbolicity breaks down.\\ In this talk, I will survey these developments and highlight some ongoing activities.

A beautiful example of spontaneous pattern formation occurs in certain dryland environments around the globe. Stripes of vegetation alternate with stripes of bare soil, with striking regularity and on a scale readily monitored via satellites. Though the vegetation is a showstopping spectacle, water, which is the limiting resource for these ecosystems, is the unseen player behind the scenes. Water concentrates into the vegetated zones, essentially reinforcing vegetation patterning, via positive feedbacks, and its dynamics play out on the short timescales of the rare storms. In contrast, the vegetation may change very little over decades. I will describe a ``stochastic pulsed precipitation'' model framework that allows us to capture the impacts of variability in storm characteristics, such as storm intensity and duration, as well as seasonality. We identify an intrinsic length scale associated with these storm characteristics that sets the vegetation pattern scale in the model. This work is motivated by the question of how these vulnerable ecosystems might respond to climate change which may lead to increased variability in storm intensity. The work highlighted in this talk was done in collaboration with Punit Gandhi.

The talk will focus on Delayed Hopf Bifurcations in nonlinear reaction-diffusion equations, a phenomenon previously thought to occur mainly in analytic ODEs. In several recent works with Ryan Goh and Theo Vo, the speaker has shown that there can be long delays --past the time of the instantaneous Hopf bifurcation-- before the onset of post-Hopf oscillations occurs. Space-time buffer curves are shown to be important for governing the spatially-dependent time of the delayed onset, as are homogeneous exit time curves. We also present formulas for these curves; quantify how they depend on the main system parameters, including the Hopf frequency, the magnitude of the diffusivity, the source terms, the initial data, and the duration of the approach to the instantaneous Hopf bifurcation; and, show that there is a competition between them. The results presented in the talk are based on joint work with Ryan Goh and Theo Vo.

Nowadays, populations are faced with unprecedented rates of global climate change, habitat fragmentation and destruction causing an accelerating conversion of their living conditions. Critical transitions in ecosystems often called regime shifts lead to sudden shifts in the dominance of species or even to species’ extinction and decline of biodiversity. Many regime shifts are explained as transitions between alternative stable states caused (i) by certain bifurcations when certain parameters or external forcing cross critical thresholds, (ii) by fluctuations or (iii) by extreme events. We address a fourth mechanism which does not require alternative states but instead, the system performs a large excursion away from its usual behaviour when external conditions change too fast. During this excursion, it can embrace dangerously, unexpected states. We demonstrate that predator-prey systems can either exhibit a population collapse or an unexpected large peak in population density if the rate of environmental change crosses a certain critical rate. In reference to this critical rate of change which has to be surpassed, this transition is called rate-induced tipping (R-tipping). Whether a system will track its usual state or will tip with the consequence of either a possible extinction of a species or a large population peak like, e.g., an algal bloom depends crucially on the time scale relations between the ecological timescale and the time scale of environmental change. However, populations have the ability to respond to environmental change due to rapid evolution. We show how such kind of adaptation can prevent rate-induced tipping in predator-prey systems. This mechanism, called evolutionary rescue, introduces a third timescale which needs to be taken into account. Only a large genetic variation within a population would be able to successfully counteract an overcritically fast environmental change. Joint work with Anna Vanselow, Lukas Halekotte, and Sebastian Wieczorek.

We overview a combination of deterministic and stochastic methods for studying dynamic bifurcations in canonical climate-related models. Our focus is on dominant factors in different scenarios of tipping, that is, where the transition related to the dynamic bifurcation may be advanced or delayed. Previous work has contrasted non-smooth and smooth dynamic bifurcations in the deterministic setting, indicating how noise and “nearly” non-smooth behavior can play a larger role in more realistic tipping models. The presence of high and low frequency forcing must also be considered, resulting in a competition between different important contributions, including stochastic forcing, high and low frequency components, the “non-smoothness” of the underlying bifurcations, bi-stability, and the slow variability of critical physical and environmental process. The analysis points to some fundamental differences in the smooth and non-smooth cases, which lead to a wider variety of tipping mechanisms in non-smooth-like settings.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with a cut-off was popularised by Brunet and Derrida in the 1990s as a model for many-particle systems in which concentrations below a given threshold are not attainable. While travelling wave solutions in cut-off scalar reaction-diffusion equations have since been studied extensively, the impacts of a cut-off on systems of such equations are less well understood. As a first step towards a broader understanding, we consider various coupled two-component reaction-diffusion equations with a cut-off in the reaction kinetics, such as an FKPP-type population model of invasion with dispersive variability due to Cook, a FitzHugh-Nagumo-style model with piecewise linear Tonnelier-Gerstner kinetics and, finally, a more general FKKP-type system with a cut-off in both components that is motivated by models for the spatial spread of hitchhiking traits. Throughout, our focus is on the existence, structure, and stability of travelling fronts, as well as on their dependence on model parameters; in particular, we determine the correction to the front propagation speed that is due to the cut-off. Our analysis is, for the most part, based on a combination of geometric singular perturbation theory and the desingularisation technique known as “blow-up”. Joint work with Zhouqian Miao, Panagiotis Kaklamanos, and Tasso Kaper.

We study a singularly perturbed fast-slow system of two partial differential equations (PDEs) of reaction-diffusion type on a bounded domain. We assume that the reaction terms in the fast variable contain a fold singularity, whereas the slow variable assumes the role of a dynamic bifurcation parameter, thus extending the classical analysis of a fast-slow dynamic fold bifurcation to an infinite-dimensional setting. Our approach combines a spectral Galerkin discretisation with techniques from Geometric Singular Perturbation Theory (GSPT) which are applied to the resulting high-dimensional systems of ordinary differential equations (ODEs). In particular, we show the existence of invariant manifolds away from the fold singularity, while the dynamics in a neighbourhood of the singularity is described by geometric desingularisation, via the blowup technique. Finally, we relate the Galerkin manifolds that are obtained after the discretisation to the invariant manifolds which exist in the phase space of the original system of PDEs.

Phase resetting is a common experimental approach to investigating the behaviour of oscillating neurons. Assuming repeated spiking or bursting, a phase reset amounts to a brief perturbation that causes a shift in the phase of this periodic motion. The observed effects not only depend on the strength of the perturbation, but also on the phase at which it is applied. The relationship between the change in phase after the perturbation and the unperturbed old phase, the so-called phase resetting curve, provides information about the type of neuronal behaviour, although not all effects of the nature of the perturbation are well understood. Mathematically, resetting is closely related to the concept of isochrons of an attracting periodic orbit, which are the submanifolds in its basin of attraction of all points that converge to the periodic orbit with a specific phase. A phase reset maps each isochron in the family of isochrons to another isochron in this family. Recently, we developed a numerical method that computes phase resetting curves in this precise context of mapping one isochron to another. The method is based on the continuation of a multi-segment boundary value problem and can be applied to systems of arbitrary dimension. In this talk, we show how this new approach can be used to study parameter-dependent deformations of phase resetting curves; we give a detailed overview of its properties, and investigate how the resetting behaviour is affected by phase sensitivity in the system. Joint work with Bernd Krauskopf and Peter Langfield.

Neural field equations (NFEs) have become an important tool in mathematical neuroscience to model dynamical behavior in the brain on macroscopic scales. They can be thought of as spatially continuous extensions of large-scale neuronal networks, and can therefore be interpreted as infinite-dimensional dynamical systems. A main difficulty in the analysis of NFEs is posed by their nonlocal nature, which is typically represented in such models by the convolution of a synaptic kernel with a nonlinear transformation of the neuronal activity variable. NFEs often feature slowly-varying adaptation variables, which are highly relevant to the regulation of neuronal dynamics. For example, such variables can represent short-term synaptic plasticity. A rigorous fast-slow analysis of these models has so far not been explored. More generally, mathematical research on infinite-dimensional fast-slow systems of differential equations appears scarce in comparison to the vast literature that is available for ODE models. I will present preliminary results on a functional analytic approach to the study of trajectories in the slow manifold of fast-slow NFEs. This approach relies on analyzing the behavior of bounded continuous solutions of the equation of perturbed motion at such trajectories. Such an approach has the advantage that it is closely related to methods based on exponential dichotomies for infinite-dimensional and non-autonomous dynamical systems. Therefore it could provide an integrated theoretical framework for multiple timescale dynamics in NFEs.

Several basic mechanisms of chaotic dynamics in phenomenological and biologically plausible models of individual neurons are discussed. We show that chaos occurs at transitions between generic activity types in neurons such as tonic spiking, bursting, and quiescence, where the system can also become bi-stable. Bifurcations underlying these transitions give rise to period-doubling cascades, various homoclinic and saddle phenomena, torus breakdown, and chaotic mixed-mode oscillations in diverse model of individual neurons.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Stochastic resonance can occur when a multi-stable system is subject to both periodic and random perturbations. For suitable parameter values, the system can respond to the perturbations in a way that is close to periodic. This phenomenon was initially proposed as an explanation for glacial cycles in the Earth's climate. While its role in that context remains controversial, stochastic resonance has since been observed in many physical and biological systems. This talk will focus on stochastic resonance in parabolic SPDEs, such as the Allen-Cahn equation, when they are driven by a periodic perturbation and by space-time white noise. We will discuss both the case of one spatial dimension, in which the equation is well-posed, and the case of two spatial dimensions, in which a renormalisation procedure is required. \\ This talk is based on joint works with Barbara Gentz and Rita Nader.\\

Anomalous diffusion and Lévy flights, which are characterized by the occurrence of random discrete jumps of all scales, have been observed in a plethora of natural and engineered systems, ranging from the motion of molecules to climate signals. Mathematicians have recently unveiled mechanisms to generate anomalous diffusion, both stochastically and deterministically. However, there exists to the best of our knowledge no explicit example of a spatially extended system which exhibits anomalous diffusion without being explicitly driven by Levy noise. We provide the first explicit example of a stochastic partial differential equation which albeit only driven by normal Gaussian noise supports anomalously diffusive propagating front solutions. This is an entirely emergent phenomenon without explicitly built-in mechanisms for anomalous diffusion. This is joint work with Chunxi Jiao.

Detecting bifurcation points for stochastic partial differential equations is a subtle task, because even for finite-dimensional stochastic systems the question of how to describe a bifurcation is not fully answered. There are several concepts of bifurcations, which can lead to different results. For instance, using order-preserving random dynamical systems, the famous result by Crauel and Flandoli indicates that additive noise destroys a pitchfork bifurcation. However, we show that even in the presence of additive noise a phenomenological bifurcation still occurs. This can be explained by a different qualitative behavior of the equilibrium before and after the bifurcation and it can be quantified by finite-time Lyapunov exponents. This talk is based on a joint work with Alex Blumenthal and Maximilian Engel and on an ongoing work with Dirk Blömker.

Systems with multiple time scales appear in a wide variety of applications. Yet, their mathematical analysis is challenging already in the context of ODEs, where many decades were needed to develop a more comprehensive theory based upon invariant manifolds, desingularization, variational equations, and many other techniques. Yet, for PDEs the progress has been extremely slow due to many obstacles in generalizing several ODE methods. In my talk, I shall report on several recent advances for fast-slow PDEs, namely the extension of slow manifold theory for unbounded operators driving the slow variables, and the design of blow-up methods for PDEs to tackle normal hyperbolicity.

Many authors have demonstrated the utility of the geometric blow-up method as a tool for studying dynamic bifurcations in finite-dimensional slow-fast systems. We focus on the development and application of the geometric blow-up method for PDEs, applied in particular to the scalar Swift-Hohenberg equation with a slow parameter drift on an unbounded domain. In order to understand the dynamics near the dynamic Turing bifurcation which is responsible for the onset of patterned states, we show that the classical multiple scales approximation from modulation theory can be reformulated as a (geometric) blow-up transformation. This leads to an approximating set of non-autonomous Ginzburg-Landau equations, which can be analysed in the blown-up space. Analysing these equations and quantifying the magnitude of the approximation allows for a rigorous description of the solutions in a rich class of weighted Sobolev spaces. We also prove the existence of delayed stability loss phenomena.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider the FitzHugh--Nagumo PDE in the so-called oscillatory regime in which one observes spatially oscillatory patterns that invade an unstable steady state. The resulting pattern is selected from a family of periodic traveling wave train solutions by an invasion front in the layer problem. Using geometric singular perturbation techniques, we construct pushed and pulled pattern-forming fronts as heteroclinic orbits between the unstable steady state and a periodic orbit representing the wave train in the wake. In the case of pushed fronts, the associated wave train necessarily passes near a pair of nonhyperbolic fold points on the critical manifold. We also briefly discuss implications for the stability of the wave trains and the challenges introduced by the fold points in the spectral stability problem. This is joint work with Montie Avery, Björn de Rijk, and Arnd Scheel.

An issue in the stability analysis of periodic traveling waves in spatially extended systems is that the linearization about the wave possesses continuous spectrum, which is parameterized by the Floquet-Bloch variable and touches the origin due to translational invariance. Thus, the fine structure of the spectrum close to the origin is of importance, but often delicate to determine. In this talk we show how scale separation in singularly perturbed systems can be used to reduce complexity. We apply our techniques to study the spectral stability of periodic waves in the FitzHugh-Nagumo system, which are selected by invasion fronts. A challenge is that those waves are typically `nonhyperbolic' in the sense that large and small spatial eigenvalues interact at so-called fold points. This is joint work with Montie Avery, Paul Carter and Arnd Scheel.

In ecosystems that cannot sustain dense uniform vegetation, patterns of concentrated vegetation emerge to leverage benefits of dense growth and economize overall resources. As environmental conditions change, such patterns adapt by varying the spacing between and number of vegetated regions. In a prototypical model for pattern formation, the complex Ginzburg-Landau equation, we study the dynamics of wavenumbers in spatially periodic solutions. As parameters slowly pass through an Eckhaus instability, patterns with higher wavenumber become unstable. We predict the resulting drop in wavenumber and the time delay of this transition based on spatio-temporal resonances. Joint work with Anna Asch, Montie Avery, and Anthony Cortez.

Following pattern formation away from onset is still a challenge as no general theory is available. Here, we consider a singularly perturbed three-component reaction-diffusion system and show how the singular perturbed structure can be used to study pattern formation away from onset. We show analytically how a near-equilibrium periodic pattern emerges through a Hamiltonian-Hopf bifurcation and, upon continuing in a system parameter, evolves to various far-from equilibrium periodic patterns that can be described rigorously by geometrical singular perturbation techniques.\\ This is joint work with Christopher Brown, Gianne Derks, and David J.B.~Lloyd from the University of Surrey in the UK.

In 2018, Carter and Sandstede made use of geometric singular perturbation theory and blow-up analysis to determine the mechanism behind parametric pulse replication in the Fitzhugh-Nagumo equation: the presence of a canard point in the associated traveling-wave ODE leads to the existence of a one-parameter family of solutions the authors termed a homoclinic banana. As one travels along the banana, a transition occurs from single-pulse solutions to double pulses, with an intermediate phase of single pulses with oscillatory tails. As is typical with canard points, this transition takes place in an exponentially thin region of parameter space. Subsequently, in 2021 Carter et al. numerically observed the phenomenon of temporal pulse replication: starting with an initial condition close to a one-pulse on the banana causes the associated solution to mimic the parametric transition dynamically, despite the parameters being fixed. The authors speculated that the parametric transition may generate a nearby invariant manifold which guides the temporal transition. In this talk, we discuss the accompanying stability results for the parametric transition, why those results preclude a simple resolution of temporal pulse replication, and the numerical explorations carried out to address these challenges.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Air is a compressible medium. Yet, experience shows that sound waves play a negligible role in the vast majority of meteorologically relevant atmospheric processes. Nevertheless, the family of sound-proof flow models, which correspond to the incompressible or zero-Mach number approximations in engineering fluid mechanics, has met with severe scepticism from a large fraction of the meteorological community since they were first introduced many decades ago. In this lecture, I will elucidate reasons for this scepticism, explain that a thorough analysis of nearly sound-free atmospheric flows involves a non-standard asymptotic three-scale problem, discuss formal estimates of the range of validity of available sound-proof models, and describe ongoing research aiming at an associated rigorous proof.

In a manner analogous to the partitioning by the spectral density of the variance of a stochastic process among frequency components, the bispectral density partitions the third statistical moment into contributions from interacting frequency pairs. This quantity allows the asymmetry of fluctuations of a stochastic process to be related to timescale interactions. Various quantities of geophysical interest (e.g. ocean surface wave power density, wind speed) can be related to the square of processes which well approximated as Gaussian. In this talk, I show how the bispectral density of a squared Gaussian process $x_{t}$ can be expressed as a convolution-type integral involving the spectral density of $x_{t}$. This integral can be evaluated analytically for several classical stochastic processes (e.g. Ornstein-Uhlenbeck process, damped oscillator). The relevance of these results to the physical characterization of non-Gaussian variability in atmosphere/ocean fields will be discussed.

A general, data-informed and theory-guided variational approach based on analytic parameterizations of unresolved scales/variables is presented to address the closure problem of stochastic systems. It relies on the Optimal Parameterizing Manifold (OPM) framework introduced in (Chekroun et al, J. Stat. Phys. 179, 2020) which allows, for deterministic turbulent systems away from the instability onset, to derive useful analytic formulas for such parameterizations. These are obtained as homotopic deformations of parameterizations near criticality such as e.g. arising in center manifold reduction, and whose homotopy parameters are optimized away from criticality using data from the full model. Contrarily to other nonlinear-parameterization approaches such as those based on invariant/inertial or slow manifolds, the superiority of the OPM approach lies in its ability to get rid of constraining spectral gap or timescale separation conditions. In this work, this program is extended to stochastic partial differential equations (SPDEs) driven by additive noise, either white or of jump type. Analytic formulas of stochastic OPMs are derived. These parameterizations are optimized using a single solution path and are shown to represent efficiently the interactions between the noise and nonlinear terms in a given reduced state space, for the other solution paths. Path-dependent coefficients depending on the noise history are shown to play a key role in these parameterizations especially when the noise is acting along the "orthogonal direction” of the reduced state space. Applications to stochastic transitions in SPDEs will be presented. This talk is based on a joint work with Honghu Liu and James C. McWilliams.

Please join us in TCPL foyer for a group photo. Virtual participants- please turn your cameras on.

The identification of persistent forecastable structures in complicated or high-dimensional dynamics is vital for a robust prediction (or manipulation) of such systems in a potentially sparse-data setting. Such structures can be intimately related to so-called collective variables known for instance from statistical physics. We have recently developed a first data-driven technique to find provably good collective variables in molecular systems. Here we will show that these generalize to other applications as well, such as fluid dynamics and social dynamics.

In this talk, we consider the long-term evolution of particle distributions in a time-inhomogeneous vector field with diffusion. The time dependence is governed by an ergodic driving system and particle distributions evolve in time through the Perron-Frobenius semigroup of the Fokker-Planck equation. We consider and connect two points of view: a) The Perron-Frobenius semigroup as a linear cocycle over the ergodic driving system and its Lyapunov-spectrum; b) The infinitesimal generator of the Perron-Frobenius semigroup on the augmented phase space and its spectrum. Joint work with Maximilian Engel, Gary Froyland, Peter Koltai.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Consider a finite-size system of coupled harmonic oscillators, and assume that the oscillators are commensurate and the coupling structure is circulant. We will present an averaging result for stochastic differential equations which will allow us to show that weak multiplicative-noise coupling can amplify some of the systems’ eigenmodes and, hence, lead to asymptotic eigenmode synchronization.\\ \textit{Reference:} PhD thesis of Christian Wiesel (formerly University of Bielefeld)

Many complex processes exhibit transient dynamics - intriguing or important dynamical behaviors over a relatively long but finite time period. A fundamental issue is to understand transient dynamics of different mechanisms. In this talk, we focus on a class of randomly perturbed processes arising in chemical reactions and population dynamics where species only persist over finite time periods and go to extinction in the long run. To capture such transient persistent dynamics, we use quasi-stationary distributions (QSDs) and study their noise-vanishing asymptotic. Special attention will be paid to essential differences between models with and without environmental noises. The talk ends up with some discussions.

In this talk, I will present our recent result about how to use a numerical coupling method to classify dynamics at different time scales. I will first discuss how to use numerical coupling technique to estimate the speed of convergence of a stochastic differential equation towards its steady state. Then I will show how to connect the property of deterministic dynamics and coupling time distributions by running the coupled process with different magnitudes of noise. Applications of studying loss surfaces of deep neural networks will be demonstrated.

In this talk we present some qualitative behaviour in finite-time windows in a stochastic Brusselator, where one of the parameters in the classical model is assumed to be random. In particular, we focus our attention on the so called Finite-Time Lyapunov Exponents (FTLE), which are an indicator of exponential expansion/contraction of nearby orbits in short time scales. On the other hand, we make use of the Covariant Lyapunov Vectors (CLV) in order to analyse the fast-slow structure when the parameters of the system change. Specifically, we study the relation between their alignment within the fast regime. Joint work with Maximilian Engel.

Michaelis-Menten kinetic is one of the most used when modelling enzyme-, or more generally catalytic-, reactions. In the case of homogeneous medium, i.e. the (bio-)chemical concentrations depend solely on time, both formal and rigorous derivations of MM from mass action kinetic have been studied extensively and thoroughly in the last decades. For heterogeneous medium, the modelling should take into account the diffusion of substances, which leads to a system of partial differential equations. In this case, interestingly, only formal derivation of MM from mass action kinetic has been investigated. In this talk, we present, up to our knowledge, the first rigorous derivation of MM in the presence of diffusion. The proof utilises an improved duality technique and a modified energy method. This is based on a joint work with Bao-Ngoc Tran (University of Graz).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we investigate the stationary profiles of a convection-diffusion model for unidirectional pedestrian flows in domains with a single entrance and exit. The inflow and outflow conditions at both the entrance and exit as well as the shape of the domain have a strong influence on the structure of stationary profiles, in particular on the formation of boundary layers. We are able to relate the location and shape of these layers to the inflow and outflow conditions as well as the shape of the domain using geometric singular perturbation theory, both in the case of closing channels and for more intricate geometries such as bottlenecks. Furthermore, we confirm and interpret our analytical results by means of computational experiments in connection with real-life applications.

Laplacian dynamics has been extensively used as a paradigmatic model for discrete linear diffusion processes. Nonlinear extensions of Laplacian dynamics display a richer behaviour that, for example, could model more complex diffusive phenomena on networked systems. We present a study of absolute Laplacian flows (ALFs) under small perturbations. The algebraic properties of such systems lead naturally to singular perturbations, acting as a drift in the (nonlinear) diffusion process. The main goal is to describe the near-consensus behaviour; in order to accomplish that we employ techniques from geometric singular perturbation theory, equivariant dynamical systems theory, and algebraic graph theory.

We employ a Lin’s method set-up to compute a surface of heteroclinic connections between two saddle slow manifolds in the four-dimensional Olsen model for peroxidase-oxidase reaction. As will be shown, this surface organises the return mechanism of mixed-mode oscillations that also involve a slow passage through a Hopf bifurcation. Joint work with Elle Musoke and Hinke M.~Osinga.

The Brusselator is one of the oldest systems studied in spatial dynamics, first conceived as a result of Turing's landmark work on the formation of stripe patterns in the 1960's. Despite the decades and myriad studies since then, it remains a system of interest due to its ability to display a zoo of different complex behaviors. In this work we look at a newly discovered behavior, snaking of contact defects. Numerical studies by Tzou et al. (2013), found that an interaction between two distinct types of stable oscillations allows for the production of contact defects: a temporally constant core region sitting in a temporally oscillating background. Their results suggest that these patterns form through a process called snaking. In this work, we look to extend their numerical studies with the aim of producing a proof for the existence and stability of these patterns and the snaking bifurcation that produces them.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.