Session S12 - Delay and functional differential equations and applications
Talks
Monday, July 12, 11:40 ~ 12:15 UTC-3
A Kupka-Smale Theorem for a Class of Delay-Differential Equations
John Mallet-Paret
Brown University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider generic properties of scalar delay-differential equations of the form $\dot x(t)=f(x(t),x(t-1))$. In particular, we prove that for generic nonlinearities $f$ all equilibria and all periodic solutions are hyperbolic. We note that the corresponding result for equations of the form $\dot x(t)=f(x(t-1))$ remains open. We discuss the significance of the result as it relates to one-parameter families of equations $\dot x(t)=f(x(t),x(t-1),\mu)$ and global continuation of periodic orbits.
Monday, July 12, 12:15 ~ 12:50 UTC-3
Linearized stability and instability for $\dot x(t) = g(x'_t, x_t)$
Bernhard Lani-Wayda
Justus-Liebig-Universität Gießen, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
In joint work in progress with Jaqueline Godoy Mesquita, we try to improve results on stability by linearization, and prove corresponding results in the unstable case.
Joint work with Jaqueline Godoy Mesquita (Universidade de Brasília).
Monday, July 12, 12:50 ~ 13:25 UTC-3
On stability and asymptotics of Mackey-Glass equations with two delays and neutral systems
Elena Braverman
University of Calgary, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
In the first part of our talk, we observe that introduction of two variable delays can change properties of the Mackey-Glass equation $$\dot{x}(t) = r(t) \left[ \frac{a x(h(t))}{1+x^{\nu}(g(t))} - x(t) \right], \quad a>1, ~\nu >0.$$ There may exist non-oscillatory about the positive equilibrium unstable solutions, the effect of possible absolute stability for certain $a$ and $\nu$ disappears. We obtain sufficient conditions for local and global stability of the positive equilibrium and illustrate the stability tests, as well as new effects of two different delays, with examples.
In the second part of the talk, we analyze exponential stability and solution estimates for a delay system $$ \dot{x}(t) - A(t)\dot{x}(g(t))=\sum_{k=1}^m B_k(t)x(h_k(t)) $$ of a neutral type, where $A$ and $B_k$ are $n\times n$ bounded matrix functions, and $g, h_k$ are delayed arguments. Stability tests are applicable to a wide class of linear neutral systems with time-varying coefficients and delays. In addition, explicit exponential estimates for solutions of both homogeneous and non-homogeneous neutral systems are obtained for the first time. These inequalities are not just asymptotic estimates, they are valid on every finite segment and evaluate both short- and long-term behaviour of solutions.
Joint work with Leonid Berezansky (Ben Gurion University of the Negev, Israel).
Monday, July 12, 13:25 ~ 14:00 UTC-3
On persistence of the Nicholson's equation with multiple delays and non-linear harvesting
Melanie Bondorevsky
Universidad de Buenos Aires - IMAS (CONICET), Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
A generalization of the Nicholson's equation is considered. We analyze conditions to guarantee uniform persistence for an $N$-dimensional generalization of the Nicholson's equation with several delays and non-linear harvesting terms. In addition, the existence of $T$-periodic solutions shall be proved and, if the conditions are reversed, we will show that zero is a global attractor.
Joint work with Pablo Amster (Universidad de Buenos Aires - IMAS (CONICET)).
Monday, July 12, 14:15 ~ 14:50 UTC-3
On almost periodic solutions for a model of hematopoiesis with an oscillatory circulation loss rate
Rocio Celeste Balderrama
Universidad de Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
The following nonlinear autonomous delay differential equation was proposed by Mackey and Glass to describe the regulation of hematopoiesis, namely, \begin{equation} \label{MG0} \frac{dP(t)}{dt}=\frac{\lambda \theta^{n}P(t-\tau)}{\theta^n + P^{n}(t-\tau)}-\gamma P(t)\, \hspace{1cm} \hbox{ where } \lambda, \theta, n, \gamma, \tau\in(0,+\infty).\quad (1) \end{equation} Here $P(t)$ is the concentration of cells in the circulating blood and the flux function $f(v)=\frac{\lambda \theta^{n}v }{\theta^n + v^{n}}$ of cells into the blood stream from the stem cell compartment depends on the cell concentration at an earlier time. The delay $\tau$ describes the time between the start of cellular production in the bone marrow and their maturation for release in circulating bloodstream. It is assumed that the cells are lost at a rate proportional to their concentration, namely $\gamma P(t)$.
In the real-world phenomena, the environment varies with time. Thus, the following nonautonomous nonlinear delay differential equation with time-varying coefficients and delays, and oscillatory circulation loss rate is a natural extensions of (1)
\begin{equation} \label{MG} x'(t) = \sum_{k=1}^M \lambda_k r_k(t)\frac{x^m(t-\tau_k(t))}{1+x^n(t-\tau_k(t))}-b(t)x(t), \quad (2) \end{equation} where $m\geq 0$, $n, \lambda_k>0$, $b(t),r_k, \tau_k$ are positive almost periodic functions for $k= 1, 2, \ldots, M$ and $b(t)$ is oscillating. The function $ \lambda_k r_k(t)\frac{x^m(t-\tau_k(t))}{1+x^n(t-\tau_k(t))}$ is the flux of cells into the blood stream from the $i-th$ stem cell compartment.
In this work, we establish and prove a fixed point theorem from which some sufficient conditions are deduced on the existence of positive almost periodic solutions for (2). Some particular conditions under the nonlinearity of the equation have been previously considered by authors as fundamental assumption for the study of almost periodic solutions of the model. The aim of this work is to establish results without such assumption. Some examples are given to illustrate our results.
Monday, July 12, 14:50 ~ 15:25 UTC-3
$L^p$-Solutions for Kelvin-Voigt Fluids with Unbounded Memory Terms
Pedro Danizete Damazio
Federal University of Paraná, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work we consider systems of Kevin-Voigt type with distributed memories in the nonlinear and force terms. By setting the problem in the context of the analytic semigroups in Banach spaces, results on existence, uniqueness and regularity of weak, mild and classical solutions are established.
Joint work with Ana Leonor Silvestre (Instituto Superior Técnico - Lisboa, Portugal) and Patrícia Aparecida Manholi (Universidade Tecnológica Federal do Paraná - Curitiba, Brazil).
Monday, July 12, 15:25 ~ 16:00 UTC-3
The multiplicity-induced-dominancy property for scalar delay-differential equations
Guilherme Mazanti
Inria & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
Even in simple situations with time-delays such as that of linear equations with constant coefficients and constant delays, the analysis of the asymptotic behavior of a time-delay system can be a challenging question. From a spectral point of view, this asymptotic behavior can be studied through the spectrum of the system, which is the set of roots of a quasipolynomial, i.e., a function which can be written as a finite sum of products of polynomials and exponentials. Contrarily to the case of polynomials, for which Routh–Hurwitz criterion allows for handy characterizations of the location of their roots in terms of the coefficients of the system, quasipolynomials have infinitely many roots and there is no explicit link in general between their location and the coefficients of the system.
Some recent works have highlighted an interesting property of time-delay systems, called multiplicity-induced-dominancy (MID): for some families of time-delay systems, a spectral value of maximal multiplicity is necessarily the rightmost spectral value in the complex plane, and hence determines the asymptotic behavior of the system. Since then, an important research effort was made to identify families of time-delay systems satisfying the MID property.
After an introductory discussion on the spectral analysis of time-delay systems, this talk will present the MID property and some families of systems for which it is known to hold. We shall present the most common techniques used to prove the MID property, based on the argument principle, factorizations of quasipolynomials in terms of confluent hypergeometric functions, and the analysis of crossing imaginary roots. We will also illustrate its application to the stabilization of control systems and present the main perspectives of this ongoing line of research.
Joint work with Amina Benarab (IPSA & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Catherine Bonnet (Inria & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Islam Boussaada (IPSA & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Yacine Chitour (L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Sébastien Fueyo (Inria & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Silviu-Iulian Niculescu (L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Karim Trabelsi (IPSA, France) and Tomáš Vyhlídal (Czech Technical University in Prague, Czechia).
Tuesday, July 13, 11:05 ~ 11:40 UTC-3
Stochastic age-structured population models derived from the McKendrick-von Foerster equaton
Arkadi Ponossov
Norwegian University of Life Sciences, Norway - This email address is being protected from spambots. You need JavaScript enabled to view it.
A widespread method of producing a stochastic population model consists in appending an additive stochastic noise to the model’s deterministic counterpart. It is shown in the talk that this straightforward construction may result in a wrong equation, at least in the case of age-structured population models derived from the McKendrick-von Foerster equaton (MFE), the biological “conservation law”. This well-elaborated transformation technique produces many popular deterministic differential equations with time-delay such as Nicholson’s blowflies model, the recruitment-delayed model etc.
If one takes MFE with a stochastically perturbed mortality rate as a starting point and use a similar transformation technique, then, as it is demonstrated in the talk, one obtains stochastic versions of the above population models, which are very different from those one can construct by simply appending an additive stochasticity to deterministic equations. In particular, it is shown that stochastic Nicholson’s blowflies model should contain both additive and multiplicative stochastic noises.
Joint work with Lev Idels (Vancouver Island University, Canada) and Ramazan Kadiev (Dagestan State University, Russian Federation).
Tuesday, July 13, 11:40 ~ 12:15 UTC-3
On solution manifolds
Hans-Otto Walther
Justus-Liebig-Universitaet Giessen, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
Differential equations with state-dependent delays which generalize the scalar example \[ x'(t)=g(x(t),x(t-d(x_t))) \] where $g:\mathbb{R}^2\to\mathbb{R}$ and $d:C([-r,0],\mathbb{R})\to[0,r]$ are continuously differentiable, and with $x_t:[-r,0]\to\mathbb{R}$ given by $x_t(s)=x(t+s)$, define semiflows of differentiable solution operators on an associated submanifold of the state space $C^1=C^1([-r,0],\mathbb{R}^n)$. When written in the general form \[ x'(t)=f(x_t) \] with a map $f:C^1\supset U\to\mathbb{R}^n$ then the associated manifold is \[ X_f=\{\phi\in U:\phi'(0)=f(\phi)\}. \] We obtain results on the nature of $X_f$.
1. If all delays in the system are bounded away from zero then a projection $C^1\to C^1$ onto the subspace \[ H=\{\phi\in C^1:\phi'(0)=0\}=X_0 \] defines a diffeomorphism of $X_f$ onto an open subset of $H$. In other words, $X_f$ is a graph over $H$.
2. There exist $g$ and $d$ with $d(\phi)>0$ everywhere and $\inf\,d=0$ so that the manifold $X_f$ associated with the scalar example above does not admit any graph representation.
3. If all delays in the system are strictly positive (but not necessarily bounded away from zero) then $X_f$ has an ''almost graph representation`` which implies that it is diffeomorphic to an open subset of $H$.
Tuesday, July 13, 12:15 ~ 12:50 UTC-3
Laguerre-domain modeling and identification of linear discrete-time delay systems
Rosane USHIROBIRA
Inria, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we propose a closed-form Laguerre-domain representation of discrete linear time-invariant systems with constant input time delay. This representation is shown to be useful in system identification setups involving square-summable signals, usually appearing in biomedical applications, and where experimental protocols do not allow for the persistent excitation of the system dynamics. We illustrate the utility of our proposed system representation on a problem of drug kinetics estimation from clinical data.
Joint work with Viktor Bro (Uppsala University, Sweden) and Alexander Medvedev (Uppsala University, Sweden).
Tuesday, July 13, 12:50 ~ 13:25 UTC-3
Random recurrent neural networks with delays
Xiaoying Han
Auburn University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we will introduce an infinite lattice model of recurrent neural networks with random connection strengths among nodes, and various types of temporal delays. The existence and structure of random pullback attractors will be discussed.
Joint work with Peter Kloeden( Tuebingen University, Germany), Yejuan Wang (Lanzhou University, China) and Meiyu Sui (Hebei Normal University, China).
Tuesday, July 13, 13:40 ~ 14:15 UTC-3
A delay differential equation with an impulsive self-support condition
Márcia Federson
Universidade de São Paulo, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We investigate an autonomous impulsive system proposed as a model of drugs absorption and consisting of a linear differential delay equation and an impulsive self-support condition. A representation of the general solution in terms of the fundamental solution is obtained. We also show that the impulsive self-support condition generates periodic auto-oscillations given by fixed points of a first return map.
Joint work with Istvan Gyori (University of Pannonia, Hungary), Jaqueline G. Mesquita (Universidade de Brasília, Brazil) and Plácido Táboas (Universidade de São Paulo, Brazil).
Tuesday, July 13, 14:15 ~ 14:50 UTC-3