Overview
This is the 4th meeting in an international conference series on Dynamics of Differential Equations – in Memory of Jack K. Hale. The conference series was established through the joint effort of research groups in dynamics of differential equations from across the Americas, Asia, and Europe, with scientific focus on the advancement and application of dynamical systems as generated by differential equations and their discrete analogs. The series has had successful meetings in Atlanta, USA (2013), Luminy, France (2016), and Hiroshima, Japan (2018).
The 4th meeting will consist of:
● Two Hale Memorial Lectures delivered by Professor John Mallet-Paret;
● One Plenary Session with 10 plenary lectures;
● One Special Session on General Dynamics;
● One Workshop on Dynamics of Infectious Diseases; and
● One Workshop on Dynamics and Evolutionary Equations.
This latter workshop is in honor of Professor Genevieve Raugel, who was a leading expert in the area and also a co-founder of this conference series.
Besides carrying on the general theme of the series, this 4th meeting will give emphasis to the development of emerging areas of dynamics, including stochastic dynamics and the connection of dynamics with data science applications.
Full Schedule
G: Special Session on General Dynamics,
A: Workshop on Dynamics of Infectious Diseases;
R: Workshop on Dynamics & Evolution Equations – in honor of Geneviève Raugel;
Xij: ith (1,2,3,4) day, jth (1,2,3,4) talk in the parallel sessions G, A, and R.
Program
Titles and Abstracts
Hale memorial lectures
John Mallet-Paret, Brown University, USA
LECTURE 1: Analyticity in Delay-Differential Equations
We study classes of delay-differential equations for which the nonlinearities in the equations are analytic functions. One might expect (as with ordinary differential equations) that many solutions, particularly those on the attractor, are analytic functions of time. This is often, but not always, true. We discuss various theorems and examples which illustrate such phenomena.
LECTURE 2: Genericity in Delay-Differential Equations
As noted in the first lecture, solutions of analytic delay-differential equations need not be analytic. Yet numerical simulations of such systems (often with parameters) typically present an analytic picture, with smooth branches of solutions and simple bifurcations. In this spirit we study generic properties of smooth (but not necessarily analytic) delay-differential equations.
Plenary lectures
Title: Full horseshoe for the Galerkin truncations of 2D Navier-Stokes equations with degenerate stochastic forcing
Abstract： In this talk, we will introduce the existence of full horseshoe for the Galerkin truncations of 2D Navier-Stokes equations with degenerate stochastic forcing (Hypoelliptic condition). We will also review weak horseshoe and semi-horseshoe. This is based on joint work with Jianhua Zhang.
Title: Divergence and convergence of Birkhoff Normal Forms.
Abstract: Each real analytic symplectic diffeomorphism admitting a non-resonant fixed point can be formally conjugated to a formal integrable symplectic diffeomorphism, its Birkhoff Normal Form (BNF). I shall discuss two questions. The first one is due to H. Eliasson: are there examples of divergent BNF? The second is the following: is it possible to perturb in the real analytic topology a given real analytic symplectic diffeomorphism so that its BNF converges?
Title: Periodic and connecting orbits for Mackey–Glass type equations
Title: Mathematical models of gene regulation: Biology drives new mathematics
Abstract: Simple bacterial gene regulatory motifs can be viewed from the perspective of dynamical systems theory, and mathematical models of these have existed almost since the statement of the operon concept. In this talk I will review the three basic types of these regulatory mechanisms and the underlying dynamical systems concepts that apply in each case. In the latter part of the talk, I will discuss the exciting mathematical challenges that arise when trying to make honest mathematical models of the underlying biology. These include transcriptional and translational delays, and the fact that these delays maybe state dependent, as well as the interesting and often unsolved problems of characterizing the noise inherent in bacterial dynamics. Surprisingly, the inclusion of state dependent delays can have rather profound effects on the possible dynamical behaviour, and it is unclear if this is reflected in biological behaviour.
Title: On the large-time behavior of bounded solutions of semilinear parabolic equations on R^N
Abstract: We will mainly consider reaction-diffusion equations on the real-line (N=1), adding some results concerning radial solutions in higher space dimensions. Our main focus is the quasiconvergence of bounded solutions, that is, locally uniform convergence of bounded solutions to a set of steady states. While not all bounded solutions have this property in general, we identify a large class of initial data which do give rise to quasiconvergent solutions.
Title: Parabolic points in antiholomorphic and holomorphic dynamics
Title: Breathers and the Implicit Function Theorem
Abstract: Breather solutions are spatially localized, temporally periodic solutions of partial differential equations and lattice systems. They are extremely rare for partial differential equations with constant coefficients but are known to exist in some PDEs with inhomogeneous coefficients. In this talk I’ll describe a new approach to the construction of breathers for special classes of PDEs with spatially periodic coefficients, based on the implicit function theorem.
Title: Dynamical aspects of chemotaxis models with logistic source on the whole space
Abstract: Chemotaxis models are used to describe the movements of mobile species or living organisms in response to certain chemicals in their environments. The current talk is concerned with various dynamical aspects of chemotaxis models with logistic source on the whole space. I will first discuss the global existence of positive classical solutions of such chemotaxis models, which is fundamental for the study of various dynamical aspects. Next, I will discuss the asymptotic behavior of globally defined positive solutions with strictly positive initial functions. I will then consider the spatial spreading dynamics of positive solutions with compactly supported or front-like initial functions.
Title: Modeling Delay in Population Growth Differential and Difference Equation Models
Abstract: Depending upon the underlying assumptions and reason for including delay in a model of population growth, different strategies for deriving the models will be discussed. Discrete delay differential equations, integro-differential equations and difference equations with both discrete and distrbuted delay will be considered. In all cases the delay is included in the growth of the population consistent with the decline of the population predicted by the model so that those contributing to the growth survive the delay period. In each case there is an important critical delay threshold that depends on the length of the delay and other parameters in the model. If the length of the time delay exceeds this threshold, the models predict that the population will go extinct for any non-negative initial conditions. Below this threshold, there is at least one positive equilibrium. The number of positive equilibria, their stability, and how their magnitude depends on the length of the delay will be discussed.
Title: Assorted problems of delay differential equations modeling impact of individual behavioral adaptation on disease transmission in the population
Abstract: I will introduce a few classes of delay differential equations/renewal equations arising from viral infection spread vector-borne disease spread in the respective populations. These equations have some particular components for individual behaviors, leading to renewal and delay equations governing the behavioural adaptation dynamics. I will talk about some preliminary results on the qualitative properties (multiple waves and multi-stability) of the disease spread and behavioural adaptation coupled syst
Special Session on General Dynamics
Workshop on Dynamics of Infectious Diseases
Workshop on Dynamics and Evolutionary Equations
– in honor of Geneviève Raugel (1951 – 2019)
Organizer: Romain Joly and Yingfei Yi
As a part of the 4th International Conference on Dynamics of Differential Equations – in memory of Jack K. Hale, this special workshop is in honor of Professor Geneviève Raugel.
Professor Raugel obtained her Ph.D degree from Uniersité Rennes in 1978. Her career had been associated with CNRS, first as a researcher (1978-1994) then a research director (1994-2019). In particular, she worked at the Orsay Math Lab of CNRS affiliated to the University of Paris-Sud since 1989. Her early research interest lied in numerical analysis concerning finite element discretization of PDEs and related bifurcation problems. Since the mid of 1980s, she started working on infinite dimensional dynamical systems especially on the dynamics of evolution equations including reaction-diffusion equations, damped wave equations, and Navier-Stokes equations. Widely regarded as a top researcher and a world-leading expert in these areas, she was especially recognized by her fundamental contributions to the subjects of global attractor and disturbances, thin domain problems, and infinite dimensional Morse-Smale structures.
Professor Raugel has made many distinguished services to the international mathematical community. In particular, she was a co-founder of this conference series, co-chaired its executive committee, and organized the 2nd conference of the series in Luminy, France in 2016.
Title: Universal dynamics of invasion fronts
Abstract: Front propagation into unstable states plays an important role in organizing structure formation in many spatially extended systems. When a trivial background state is pointwise unstable, localized perturbations typically grow and spread with a selected speed, leaving behind a selected state in their wake. A fundamental question of great interest is to predict the propagation speed and the state selected in the wake. The marginal stability conjecture postulates that speeds can be universally predicted via a marginal spectral stability criterion. In this talk, we will present background on the marginal stability conjecture and present some ideas of our recent conceptual proof of the conjecture in a model-independent framework focusing on systems of parabolic equations.
Title: Semi-discrete optimal transport problems: finding the network structure
by exploiting star shapedeness and use of Newton’s method
Abstract: In this talk, we propose a novel implementation of Newton’s method for solving semi-discrete optimal transport problems for cost functions which is a positive combination of $p$-norms, $1<p<\infty$. It is well understood that the problem reduces to finding a subdivision of a bounded region in Laguerre cells, and we prove that the Laguerre cells are star-shaped with respect to the target points. By exploiting the geometry of the Laguerre cells, we obtain an efficient and reliable implementation of Newton’s method to find the sought network structure. We give some implementation details and numerical results. Joint work with Daniyar Omarov, Georgia Tech.
Title: A global bifurcation diagram for a one-parameter family of nonautonomous scalar ODE’s driven by a minimal flow.
Abstract: The description of nonautonomous bifurcation patterns is of growing interest in the scientific community, both for its theoretical interest and for its possible applications to the analysis of mathematical models. In the talk, in the wake of [1] and using results and methods of [2], conditions on the coefficients of the one-parameter family $x'(t)=\varepsilon(a(t)+b(t)\,x (t)) + c(t)\,x^2 (t) -x^3 (t)$ as $\varepsilon$ varies are established to describe the global diagram of the motion. Such conditions include the recurrency of the coefficients. Joint work with Carmen Nunez.
[1] R. Fabbri, R. Johnson, F. Mantellini, “A nonautonomous saddle-node bifurcation pattern”, Stochastics and Dynamics, Vol. 4, No. 4 (2004) 335-350.
[2] J. Duenas, C. Nunez, R. Obaya, “Bifurcation theory of attractors and minimal sets in d-concave nonautonomous scalar ordinary differential equations”, Journal of Differential Equations, 361 (2023) 138-182.
Title: Computing the Global Dynamics of Parameterized Systems of ODEs
Abstract: We present a combinatorial topological method to compute the dynamics of a parameterized family of ODEs. A discretization of the state space of the systems is used to construct a combinatorial representation from which recurrent versus non-recurrent dynamics is extracted. Algebraic topology is then used to validate and characterize the dynamics of the system. We will discuss the combinatorial description and the algebraic topological computations and will present applications to systems of ODEs arising from gene regulatory networks.
Title: Combinatorial structure of continuous dynamics in gene regulatory networks
Abstract: We first describe the mathematical foundation of DSGRN (Dynamic Signatures Generated by Regulatory Networks), an approach that provides a combinatorial description of global dynamics of a network over its entire parameter space. Finite description allows comparison of parameterized dynamics between hundreds of networks to discard networks that are not compatible with experimental data. We also describe a close connection of DSGRN to Boolean network models that allows us to view DSGRN as a platform for bifurcation theory of Boolean maps. We discuss several applications of this methodology to systems biology.
Title: Stochastic Speed and Shape Corrections for Travelling Patterns in Random Media
Abstract: We describe a framework to uncover the shape and speed corrections caused by adding noise-terms to PDEs that support travelling wave solutions. We consider both parabolic and dispersive systems, and noise terms that can be Lipschitz or non-Lipschitz. By using perturbative techniques, we avoid the need to use the comparison principle. In some cases, it is no longer sufficient to use the standard linear operators associated to the deterministic travelling waves, which uncovers a surprising connection with singular perturbation theory. Joint work with C. Hamster and R. Westdorp.
Title: Linearized instability for differential equations with dependence on the past derivative
Abstract: We provide a criterion for instability of equilibria of equations in the form $\dot x(t) = g(x_t’, x_t)$, which includes neutral delay equations with state-dependent delay. The criterion is based on a lower bound $\Delta >0$ for the delay in the neutral terms, on regularity assumptions of the functions in the equation, and on spectral assumptions for an approximating semigroup. Estimates in the $C^1$-norm, a manifold containing the state space $X_2$ of the equation, another manifold contained in $X_2$, and an invariant cone method are used for the proof. A mechanical example illustrates possible applications. (Joint work with Jaqueline Godoy Mesquita).
Title: Disturbing the Big Bang
Abstract: We will give an introductory talk on the dynamics of the Big Bang singularity and perturbations thereof. This topic has attracted a great deal of attention of both mathematicians and physicists since the heuristic approach of Belinski–Khalatnikov–Lifshitz (known as BKL conjecture/picture) and the Mixmaster attractor construction of Misner. We will see how a specific perturbation of the Big Bang singularity will unravel well-known and brand-new dynamical features. Moreover, we will see how such perturbations yield good (or bad) approximating schemes of the usual Einstein’s general theory of relativity. These results were fruit of collaborations with K.E. Church (U Montreal), V.H. Daniel (Columbia U), J. Hell (FU Berlin), O. Hénot (McGill U), J.P. Lessard (McGill U), H. Sprink (FU Berlin) and C. Uggla (Karlstad U).
Title: Permanent charge effects on ionic flows
Abstract: Permanent charge (mathematical model for protein structure) is the most important structure of an ion channel and is the major controller for properties of ionic flows through the ion channel. Another important factor for ionic flow specifics is the boundary conditions. In this talk, we will report our studies toward an understanding of effects of interplays between permanent charge and boundary conditions on ionic flow via a quasi-one-dimensional Poisson-Nernst-Planck (PNP) model.
Permanent charges considered here are restricted to a special case of piecewise constant with one non-zero portion. For ionic mixtures with one cation (positively charged) species and one anion (negatively charged) species, a fairly rich behavior of permanent charge effects has been revealed from rigorous analyses based on a geometric framework for PNP and from numerical simulations guided by the analytical results. For ionic mixtures with three ion species: two cations and one anion, richer behavior is expected and our preliminary analysis and case studies identify, in concrete manner, a number of these, including some surprising ones.
TITLE: Critical transitions in nonautonomous ordinary equations with concave derivative
ABSTRACT: A function with finite asymptotic limits gives rise to a transition equation between a “past system” and a “future system”. We analyze this situation in the cases of nonautonomous coercive scalar ODEs with concave derivative with respect to the state variable, assuming the existence of three hyperbolic solutions for the limit systems, in which case the upper and lower ones are attractive. The different global dynamical possibilities are described in terms of the internal dynamics of the pullback attractor: cases of tracking of the two hyperbolic attractive solutions or lack of it (tipping) arise. This analysis allows us to present cases of rate-induced critical transitions, as well as cases of phase-induced and size-induced tipping. Our conclusions are applied in models of mathematical biology and population dynamics: we describe rate-induced tracking phenomena causing extinction of a native species or invasion of a non-native one. This is a joint work with Jesús Dueñas and Rafael Obaya.
Title: Analysis and modeling of the modular Burgers equation
Abstract: I will review the state of art in analysis and modeling of the Burgers’ equation with a modular advection term. The modular Burgers’ equation admits a traveling viscous shock with a single interface, which is stable against smooth and exponentially localized perturbations. In contrast, it can be shown with the help of energy estimates and numerical simulations that the evolution of shock waves with multiple interfaces leads to finite-time coalescence of two consecutive interfaces. A precise scaling law of the finite-time extinction is supported by the interface equations and by numerical simulations
Title: Selection criteria for invasion fronts
Abstract: I will overview recent efforts that strive to characterize both the speed of invasion fronts and the selected states in the wake of invasion, from a conceptual, somewhat model independent point of view. The first part of the talk will be concerned with pulled and pushed fronts and the marginal stability conjecture, focusing on analytical and computational tools for determining wavespeeds. In the second part, I will present some intriguing examples of invasion mechanisms that are based on higher spatio-temporal resonances. This is joint work with Montie Avery and Matt Holzer.
Title: Projection Techniques and Reduced Order Modeling for Data Assimilation
Abstract: Data assimilation (DA) techniques combine physical model forecasts with data or observations to predict future model states typically with some measure of uncertainty. In Bayesian DA this is done by expressing the posterior as proportional to the likelihood times the prior. Two common classes of techniques are Kalman Filter (KF) based techniques that parameterize the uncertainty in the posterior distribution as a Gaussian and Particle Filter (PF) based techniques that express posterior distributions in terms of weighted particles (physical model states) but generally suffer from curse of dimensionality issues. Motivated by computational dynamical systems techniques we develop techniques initially based on the use of reduced order physical models but more recently emphasizing the use of reduced order observational or data models. We illustrate the effectiveness of the techniques developed on both fast and slow systems, in particular versions of the Lorenz ’96 model and discretized 2D shallow water equation.
Title: On solution manifolds
Title: Response solutions in degenerate, quasi-periodically forced nonlinear oscillators
Abstract. For a quasi-periodically forced differential equation, response solutions are quasi-periodic ones whose frequency vector coincides with that of the forcing function and they are known to play a fundamental role in the harmonic and synchronizing behaviors of quasi-periodically forced oscillators. These solutions are well-understood in quasi-periodically perturbed nonlinear oscillators either in the presence of large damping or in the non-degenerate cases with small or free damping. This talk will present some recent results on the existence of responsive solutions in degenerate, quasi-periodically forced nonlinear oscillators with small or free damping.
Title: Modelling the impact of human precaution on disease dynamics and its evolution
Abstract: In the past three years, we have witnessed or seen or felt that the human (host) behaviour/precaution during an epidemic/pandemic can play a significant role in controlling the disease dynamics. In this talk, we discuss, by mathematical models, the impact of human precaution level on disease dynamics as well as the evolution of the precaution level.
Organizing Committee
- Luca Dieci – Georgia Institute of Technology
- Hans-Otto Walther – Universitaet Giessen
- Huaiping Zhu – York University (Chair)
Scientific Committee
- Jose Maria Arrieta – Complutense University of Madrid
- Alexandre Carvalho – University of São Paulo
- Giorgio Fusco – Università degli studi dell’Aquila
- Romain Joly – Université Grenoble-Alpes
- Carlos Rocha – University of Lisbon
- Kunimochi Sakamoto – Hiroshima University
- Sjoerd Verduyn Lunel – Utrecht University
- Huaiping Zhu – York University
- Yingfei Yi – University of Alberta (Chair)
Canadian local coordinating Committee
- Jacques Bélair – Université de Montréal
- Sue Ann Campbell – University of Waterloo
- Michael Li – University of Alberta
- Yingfei Yi – University of Alberta
- Xiaoqiang Zhao – Memorial University
- Huaiping Zhu – York University (Chair)