Abstract : Time delays appear in several disciplines from engineering and natural sciences. Theory of delay differential equations and infinite dimensional dynamical systems has been extensively developed. Together with the development, applications of time-delay systems have been conducted in several fields including mechanistic engineering and biological sciences. In this mini-symposium, presenting recent progress in the research of time-delay systems from theoretical and application points of view, we aim to promote discussion and collaboration on theoretical and applied research and even deepen our understanding towards the time-delay systems and extend the spectrum of the applications.
Organizer(s) : Kota Ikeda, Tetsuya Ishiwata, Yukihiko Nakata, Junya Nishigchi
[04818] Delay induced self-sustained oscillations in the Nonlinear Noisy Leaky Integrate and Fire model for networks of neurons.
Format : Talk at Waseda University
Author(s) :
Pierre Roux (Mathematical Institute, University of Oxford)
Abstract : The emergence of patterned activity in a neural networks is a key process in human and animal brains. However, since they often arise from the interplay between a large number of cells, these mechanisms are very difficult to encompass whithout the use of simple, consistent and self-contained mathematical models. In this talk, I will present a time-delayed nonlinear partial differential equation, the so-called Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model. In a recent work, my collaborators Kota Ikeda, Delphine Salort, Didier Smets and myself have obtained some new results and insights about the emergence and the shape of periodic self-sustained oscillations in this model. In particular, we have found and studied a simplified version of the problem in the form of a delayed differential equation.
[04601] Global stability of multi-cell reaction systems with arbitrary time delays
Format : Talk at Waseda University
Author(s) :
Hirokazu Komatsu (National Institute of Technology Toyota College)
Abstract : In the present talk, we consider the stability of multi-cell chemical reaction systems with arbitrary time delays for each reaction, in which each intracellular chemical reaction network is weakly reversible and has zero deficiency. By constructing a Lyapunov functional and assuming additional conditions, we can show that any positive solution to the delay differential equation for the system with mass action kinetics globally converges to a positive equilibrium point in the functional state space.
[04324] Time lag monotonicity-breaking in time-delay systems with impulses
Format : Talk at Waseda University
Author(s) :
Kevin Church (CIBC)
Abstract : In this talk, we prove that the solution manifold concept for differential equations with state-dependent delay (DE-SDD) has no "topologically generic" analogue for DE-SDD with impulses. Precisely, the existence of a semiflow is conditional on monotonicity of the time lag. We demonstrate pathologies that occur in the monotonicity-breaking case, which in some instances lead to dynamical behaviour completely different from what is possible in DE-SDD or impulsive differential equations with constant delays.
[03244] Mode Selection Rules for multi-Delay Systems
Format : Talk at Waseda University
Author(s) :
Kin'ya Takahashi (Kyushu Institute of Technology)
Taizo Kobayashi (Kyushu University )
Abstract : We investigate mode selection rules at the first bifurcation for a two-delay system. Selected modes are sensitively changed with the ratio of two delay times, but obey a definite selection rule if the strengths of two delays are fixed. When the strength of the short delay takes negative values, different types of mode selection rules are observed. We explore the underlying mechanism of the change of mode selection rules in the singular perturbation limit.
[04187] Blow-up of solutions to some delay differential equations
Format : Talk at Waseda University
Author(s) :
Tetsuya Ishiwata (Shibaura Institute of Technology)
Yukihiko Nakata (Aoyama Gakuin University)
Abstract : Time lags sometimes play an essential role in the phenomena, and it is also well-known that the delay effects cause instability or oscillation.
In this talk, we consider the effects of time delay for such instabilities from the viewpoint of a finite time blow-up of the solutions and treat some delay differential equations.
We show mathematical results on the blow-up of solutions and give numerical observations.
[03602] Absolute stability and absolute hyperbolicity in systems with time-delays
Format : Online Talk on Zoom
Author(s) :
Serhiy Yanchuk (Potsdam Institute for Climate Impact Research)
Abstract : We present criteria for the absolute stability of DDEs. For a single delay, the absolute stability is shown to be equivalent to asymptotic stability for sufficiently large delays. For multiple delays, the absolute stability is equivalent to asymptotic stability for hierarchically large delays. Additionally, we give necessary and sufficient conditions for a linear DDE to be hyperbolic for all delays. The latter conditions are crucial for determining whether a system can have bifurcations.
[03557] Delay-dependent stability switches in delay differential systems
Format : Talk at Waseda University
Author(s) :
Hideaki Matsunaga (Osaka Metropolitan University)
Abstract : We will summarize some recent results on the stability properties of linear differential systems with delays. Some examples are provided to illustrate the delay-dependent stability switches for a system with delay in the diagonal terms. The proof technique is based on careful analysis of the existence and the transversality of characteristic roots on the imaginary axis. This is a joint work with Yuki Hata.
[05023] Stabilization of periodic orbits with complex characteristic multipliers via DFC
Format : Talk at Waseda University
Author(s) :
Rinko Miyazaki (Shizuoka Univ.)
Dohan Kim (Seoul National University)
Jong Son Shin (Shizuoka University)
Abstract : The delayed feedback control (DFC) proposed by Pyragas (Pyhs. Lett. A, 1992) is a method to stabilize an unstable periodic orbit by using delayed terms. Recently we have succeeded in proving a stabilization regime under certain constraints. In this talk, we will focus on the case where the characteristic multiplier is complex.
[03942] “Mild solutions” for hereditary linear differential systems
Format : Talk at Waseda University
Author(s) :
Junya Nishiguchi (Tohoku University)
Abstract : In this talk, we discuss the variation of constants formula for delay differential equations by introducing the notion of a mild solution, which is a solution under an initial condition having a discontinuous history function. Then the principal fundamental matrix solution is defined as a matrix-valued mild solution, and we obtain the variation of constants formula with this function.
[04145] Linearized instability for neutral functional differential equations with state-dependent delays
Format : Talk at Waseda University
Author(s) :
Jaqueline Godoy Mesquita (Universidade de Brasília)
Bernhard Lani-Wayda (Justus-Liebig University)
Abstract : In this talk, I will present a linearized instability principle for neutral functional differential equations with state-dependent delays. Also, I will discuss open and developing problems in this field. This is a joint work with Professor Bernhard Lani-Wayda from Justus-Liebig University, in Giessen, Germany.
[03765] Morse decomposition of the global attractor for delay differential equations
Format : Talk at Waseda University
Author(s) :
Abel Garab (University of Szeged)
Abstract : We consider unidirectional cyclic systems of delay differential equations of the form
\begin{equation*}
\dot{x}^{i}(t)=g^i(x^i(t),x^{i+1}(t-\tau^{i})), \qquad 0\leq i\leq N,
\end{equation*}
where the indices are understood modulo $N+1$. We show that if the global attractor exists, then it does not contain any superexponential solution (i.e. that converges to $0$ faster than any exponential function). This allows us to construct a Morse decomposition of the global attractor of such equations, which is based on an integer valued Lyapunov function.
