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.
