SCIENTIFIC PROGRAMS AND ACTIVITIES

February 17, 2019

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

Short Thematic Program on Delay Differential Equations
May 2015
Organizing Committee
Odo Diekmann (Utrecht)
Sue Ann Campbell (Waterloo)
Stephen Gourley (Surrey)
Yuliya Kyrychko (Sussex)

Eckehard Schöll (TU Berlin)
Michael Mackey (McGill)
Hans-Otto Walther (Giessen)
Glenn Webb (Vanderbilt)
Jianhong Wu (York)

 

Abstracts

Elena Braverman, University of Calgary
On oscillation and stability of equations with a distributed delay

(joint work with L. Berezansky, S. Zhukovskiy)

In the first part of the talk, we explore nonlinear equations and systems with a delayed positive feedback, where the delay, is, generally, distributed. Such equations are globally asymptotically stable (and intrinsically non-oscillatory), under some natural conditions on the unique positive equilibrium, when the delay is bounded. If there are several equilibrium points, multistability is observed. In the case of the unique positive equilibrium and monotonicity, similar results are obtained for a system of two equations.

In the second part, nonlinear equations with more than one delay included in a nonlinear functions are discussed. Several examples are presented illustrating how replacing the same delay in a nonlinear function with two different delays can completely change

Hermann Brunner, Hong Kong Baptist University and Memorial University of Newfoundland
On the discretization of neutral Volterra functional differential equations with weakly singular kernels and variable delays

The analysis of discretization methods (e. g. collocation or Galerkin-type mathods) for neutral Volterra functional differential equations with variable delay $\tau(t)\ge0$ ,
$$\frac{d}{dt}\left(\int_{-\tau(t)}^0k(s)u(t+s)ds\right)=f(t),\quad t>0;\quad u(t)=\phi(t)\quad(t\le0),
$$ whose kernel has the form $k(s)=|s|^{-\alpha}$ $(0<\alpha<1)$, is only partly understood. I will describe some recent developments and then focus on a number of open problems regarding the above initial value problem and related Volterra-type delay equations.

 

Dirk-Andr\'e Deckert, Ludwig-Maximilians-Universit\"at Munich
Delay equations in electrodynamic

I will give an overview of how delay equations arise naturally in electrodynamics due to the geometry of Minkowski space-time. The corresponding equations of motion form systems of neutral equations involving time-like retarded as well as advanced terms of unbounded delays. To date general global existence and a possible characterization of solutions remain open problems. However, for special situations and toy models, there are several results available. I will review some of these results, outline the involved mathematical techniques and their major obstacles, and report on recent progress of a joint work with D. Dürr and G. Hinrichs concerning the special case of two repelling charges moving along a straight line.


Gregory Derfel, Ben Gurion University of the Negev
On asymptotics of solutions for a class of equations with rescaling

In our talk we present some recent results on asymptotical behaviour of solutions for functional and functional - differential equations with linearly transformed arguments. The class of equations under consideration, includes, generalized pantograph type equations, in particular. We shall discuss different approaches to this problem and describe some applications of equations with rescaling.

 

Odo Diekmann, Utrecht University
Linearizing population models that incorporate variable maturation delay: an example and some speculations

Suppose individuals have a fixed size at birth and start to reproduce upon reaching a larger fixed size. As the individual's growth rate depends on food supply, so does the maturation delay, i.e., the time between being born and producing offspring. The deterministic bookkeeping equations take the form of a coupled system of one renewal equation for the consumer population level birth rate and one delay differential equation for the resource concentration [2]. Assuming that the right hand sides are differentiable, the validity of the principle of linearized stability for such systems is verified in [3]. However, for a large class of reasonable models the right hand side of the renewal equation is NOT differentiable. The aim of the lecture is to give reasons to believe that, nevertheless, the principle holds. One reason is provided by an example [1], another by a not yet completed extension of the theory for linear renewal equations [4].

[1] O. Diekmann & K. Korvasová Linearization of solution operators for state-dependent delay equations : a simple example Discrete and Continuous Dynamical Systems A , to appear

[2] O. Diekmann, M. Gyllenberg, J.A.J. Metz, S. Nakaoka, A.M. de Roos Daphnia revisited : local stability and bifurcation theory for physiologically structured population models explained by way of an example Journal of Mathematical Biology (2010) 61 : 277-318

[3] O. Diekmann, Ph. Getto, M. Gyllenberg Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars SIAM Journal of Mathematical Analysis 39 (2008) 1023-1069

[4] O.Diekmann & S.M. Verduyn Lunel Jumps allowed : defining semigroups of operators via solutions of Stieltjes renewal equations in preparation


Wayne Enright, University of Toronto
Accurate and reliable approximation of DDEs with variable delays

In recent years, advances in the development of adaptive and reliable numerical methods for initial value problems in ODEs has led to a new generation of numerical methods that can deliver accurate approximate solutions at off-mesh points as well as at the usual underlying discrete mesh associated with the finite interval of interest associated with the problem specification. In this talk, I will describe how this new family of ODE methods has been extended to DDEs and how these methods can reliably and adaptively approximate systems of DDEs that contain multiple state-dependent delays. For these methods, a user need only specify the DDE and a prescribed error tolerance. The method will the generate a piecewise polynomial, defined on the interval of interest, which will satisfy the DDE everywhere to within a small multiple of the tolerance. The "small multiple" will depend only on the problem itself and its associated "conditioning". I will identify the classes of problems that can be solved with these methods as well as characteristics of the problem that can make the solution more difficult to approximate.


Ferenc Hartung, University of Pannonia
On differentiability of solutions with respect to the initial data in differential equations with state-dependent delays

In this talk we consider a class of differential equations with state-dependent delays. We discuss differentiability of the solution with respect to the initial function and the initial time for each fixed time value assuming that the state-dependent time lag function is piecewise monotone increasing. Based on these results, we prove a nonlinear variation of constants formula for differential equations with state-dependent delay.


Qingwen Hu, University of Texas at Dallas
Multi-stand cold metal rolling processes with state-dependent delay

We model multi-stand cold metal rolling processes and investigate chatter vibrations using a system of differential equations which involves state-dependent inter-stand transportation time delay. We show that the model with state-dependent delay can be transformed into a system of equations with both retarded and advanced delays (mixed type delays) and show that the equilibria are unstable if the strip velocity is not controlled. We further show that a delayed feedback control of the strip velocity can stabilize the equilibria under certain conditions. Hopf bifurcation analysis shows that the variation of inter-stand tension can lead to chatter. Numerical simulations are presented to illustrate the results of Hopf bifurcation.


Bernhard Lani-Wayda, Justus-Liebig-Universität Gießen
Symbolic dynamics by the fixed point index for a return map in an equation with state-dependent delay

We describe a general approach to establishing symbolic dynamics, which includes a classical 4-dimensional example by L.P. Shilnikov, but also applies to an equation with state dependent delay which exhibits homoclinic behavior. The method is inspired by the covering relations as used by colleagues from Poland, but technically different. It combines the geometrical and topological description of a return map with homotopies and the Leray-Schauder fixed point index.

Roger Nussbaum, Rutgers University
Analyticity, Non-analyticity and the Krein-Rutman Theorem

It is frequently a subtle problem to find solution(s) of a given functional differential equation or FDE which are defined and bounded on an interval (-(infinity), T]. However, once such a solution has been found, proving that it is infinitely differentiable is usually straightforward. If the FDE in question seems in some formal sense to be analytic, is the solution in question everywhere real analytic? Nowhere real analytic? Given such a solution, can regions of analyticity and non-analyticity coexist? We shall briefly discuss some simple-looking examples where little is known. We shall then apply the Krein-Rutman theorem to a class of compact, positive linear operators L whose nonegative eigenvectors are infinitely differentiable, necessarily display coexistence of analyticity and non-analyticity and satisfy a simple FDE with a variable time lag.

A key theorem provides necessary and sufficient conditions under which, for L in our class of operators, the spectral radius r(L) of L satisfies r(L)>0.


Robert Szczelina, Jagiellonian University, Cracow
Rigorous integration of delay differential equations and applications

In this talk we want to present method for rigorous integration of delay differential equations (DDEs) that may be used to conduct computer-assisted proofs of dynamical phenomena occurring in DDEs.

By a computer-assisted proof we mean a computer program which rigorously checks assumptions of abstract theorems about existence of some dynamical property. In recent years there were many important applications of computer-assisted proofs in discrete maps, ordinary differential equations (ODEs) and (dissipative) partial differential equations (PDEs). see for example [1,2,3] and references therein. By the rigorous integrationwe understand a computer program which produces strict bopunds for the solution of the system - in this case the bounds are a finite set of computer-representable numbers $\bar{B}$ which describe a subset $B$ of some functional space with a property that a real solution $x(t)$ of a given DDE system belongs to $B$. Ny dybamical propereties we understand the exitence of stationaray points, periodic orbits, homo- or heteroclinic connections and chaos.

In the talk we will discuss briefly the theory and topological tools that link finite-dimensional rigorous representations to the real solutions of DDEs and we will present exemplary application to prove existence of a periodic solution in a highly nonlinear, non-monotonous scalar DDE with constant delay (smooth and non-monotonous variant of the systems considered in [4]). We will also discuss possible applications in the case of DDEs with variable delay.

References

[1] D. Wilczak and P. Zgliczy\'nski. Computer-assisted proof of the existence of homoclinic tangency
for the H\'enon map and for the forced damped pendulum. SIAM J. Appl. Dyn. Syst., 8 (4):1632-1663, 2009.

[2] R. Szczelina and P. Zgliczy\'nski. A homoclinic orbot in a planar singular ODE - a computer-assisted proof.
SIAM J. Appl. Dyn. Syst., 12 (3): 1541-1565, 2013.

[3] P. Zgliczy\'nski. Rigorous numerics for dissipative partial differential equations II. Periodic orbit for the Kuramoto-Sivashinsky PDE; a computer-assisted proof. Found. Comput. Math., 4 (2): 157-185, April 2004.

[4] T. Krisztin and G. Vas. Large-amplitude periodic solutions for differential equations wioth delayed monotone positive feedback. J. Dyn. Differential Equations 23 (2011), 727-790.


Bruce van Brunt, Massey University of New Zealand
Higher order asymptotics of solutions to an initial boundary value problem involving a functional argument

In this talk we consider an initial boundary value problem that involves a partial differential equation containing a functional term. The pde comes from a simple cell growth model, which is a special case of a more general fragmentation type equation. The leading order asymptotics in time are well-known, but it is possible to solve the problem directly and thereby obtain the higher order asymptotics. We sketch the idea behind the solution technique and make a connection between the higher order terms and a certain eigenvalue problem.



Hans-Otto Walther, Giessen University
Semiflows for delay differential equations on Fr\'echet manifolds

We construct a semiflow of continuously differentiable solution operators for functional differential equations
$x'(t)=f(x_t)$ with $f$ defined on an open subset of the Fr\'echet space $C^1=C^1((-\infty,0],\mathbb{R}^n)$. This semiflow lives on a submanifold $X\subset C^1$ of finite codimension. The hypotheses are that the functional $f$ is continuously differentiable (in the Michel-Bastiani sense) and the derivatives have a mild extension property. The result applies to autonomous delay differential equations in the general case, with delay state-invariant or state-dependent, bounded or unbounded.

[1] H. O. Walther, {\it The semiflow of a delay differential equation on its solution manifold in $C^1((-\infty,0],\mathbb{R}^n)$.}
Preprint, 29 pp, 2015.