Odo Diekmann (Utrecht)

Twin semigroups and delay equations

A delay equation is a rule for extending a function of time towards the future on the basis of the (assumed to be) known past. By translation along the extended function one defines a dynamical system. In the standard theory, the fundamental solution does not ‘live’ in the state space. As a consequence, one has to work in order to make the variation-of-constants formula (the main tool for deriving local stability and bifurcation results) operational.

Twin semigroup theory (Diekmann & Verduyn Lunel, 2019) serves to eliminate this anomaly. It builds on ideas of Feller and employs the Pettis integral in the context of a norming dual pair of spaces, as developed by Kunze. It exploits that the rule for extension has finite dimensional range and it allows to cover unbounded perturbations corresponding to neutral equations.

  1. Diekmann, O., & Verduyn Lunel, S. M. (2019). Twin semigroups and delay equations. ArXiv.org. Retrieved from https://arxiv.org/abs/1906.03409

Hans-Otto Walther (Giessen)

Density of short solution segments by variable delay

Simple-looking autonomous delay differential equations

with a real function $f$ and constant time lag $r>0$ can generate complicated (chaotic) solution behaviour, depending on the shape of $f$. The same could be shown for equations where $r$ is replaced with a variable, state-dependent delay $d(x_t)\in[0,r]$, even for the case $f(\xi)=-\alpha\,\xi$ linear, with $\alpha>0$ (Lani-Wayda & Walther, 2016). Here the argument $x_t$ of the delay functional $d$ is the segment, or history, of the solution $x$ between $t-r$ and $t$ defined as the function $x_t:[-r,0]\to\mathbb{R}$ given by $x_t(s)=x(t+s)$.

So the delay alone may be responsible for complicated solution behaviour. In both cases the complicated behaviour which could be established occurs in a thin dust-like invariant subset of the infinite-dimensional Banach space or solution manifold of functions $[-r,0]\to\mathbb{R}$ on which the delay equation defines a semiflow of differentiable solution operators (Lani-Wayda & Walther, 2016) , (Walther, 2003).

The lecture presents results which grew out of an attempt to obtain complicated motion on a larger set with non-empty interior, as certain numerical experiments seem to suggest. In (Walther, 2004) a delay functional $d:Y\to(0,r)$ was constructed on an infinite-dimensional subset $Y$ of the space $C^1([-r,0],\mathbb{R})$, with $r>1$, so that the equation

\begin{align} \tag{1} x’(t)=-\alpha\,x(t-d(x_t)) \end{align}

has a solution whose short segments $x_t|_{[-1,0]}$, $t\ge0$, are dense in the whole space $C^1([-1,0],\mathbb{R})$. This implies a new kind of complicated behaviour of the flowline $[0,\infty)\ni t\mapsto x_t\in C^1_r$.

The set $Y$ in (Walther, 2019) is small in the sense that it has infinite codimension, and it is not smooth like the said solution manifolds of finite codimension. More recent work (Walther, 2019) concerns the construction of a delay functional on an open subset of the space $C^1([-r,0],\mathbb{R})$ so that Eq. (1) defines a nice semiflow on the solution manifold, and has a solution whose short segments are dense in an open subset of the space $C^1([-1,0],\mathbb{R})$.

  1. Lani-Wayda, B., & Walther, H.-O. (2016). A Shilnikov Phenomenon Due to State-Dependent Delay, by Means of the Fixed Point Index. Journal of Dynamics and Differential Equations, 28(3), 627–688. https://doi.org/10.1007/s10884-014-9420-z
  2. Walther, H.-O. (2003). The solution manifold and C^1-smoothness for differential equations with state-dependent delay. J. Differential Equations, 195(1), 46–65. https://doi.org/10.1016/j.jde.2003.07.001
  3. Walther, H.-O. (2004). Smoothness Properties of Semiflows for Differential Equations with State-Dependent Delays. Journal of Mathematical Sciences, 124(4), 5193–5207. https://doi.org/10.1023/B:JOTH.0000047253.23098.12
  4. Walther, H.-O. (2019). A Delay Differential Equation with a Solution Whose Shortened Segments are Dense. Journal of Dynamics and Differential Equations, 31(3), 1495–1523. https://doi.org/10.1007/s10884-018-9655-1
  5. Walther, H.-O. (2019). Solutions with dense short segments from regular delays. Preprint.

Therese Mur (Santiago de Chile and Giessen)

The impact of variable delay on the stability of a periodic orbit (poster)

This poster presents results from the author’s thesis, which is a case study of the impact of variable delay on the stability of a periodic orbit. If in a simple delay differential equation the originally constant delay is made variable then attraction to a periodic orbit is weakened, there are bifurcations in the Floquet spectrum, and for sufficiently large variability of the delay the periodic orbit becomes unstable, and a period-doubling bifurcation occurs.

Thomas Erneux (Brussels)

Hopf bifurcation for large delays, isolas of periodic solutions, multi-rhythmicity

For oscillators subject to a delayed feedback, the Hopf bifurcation is singular in the limit of large delays (Sieber, Wolfrum, Lichtner, & Yanchuk, 2013; Yanchuk & Giacomelli, 2015). In this limit, the Hopf slow time amplitude equation is a delay differential equation by itself with surprising properties. We review this particular problem in the light of recent experiments (Kovalev et al., 2019).

Even if there are no Hopf bifurcations from a basic steady state, isolated branches of periodic solutions may coexist with a stable steady state. We illustrate this specific phenomenon with the time-delayed FitzHugh-Nagumo equations, a minimal model in neurobiology (Erneux, Weicker, Bauer, & Hövel, 2016; Weicker, Keuninckx, Friart, Danckaert, & Erneux, 2016).

Multi-rhythmicity is the property that an oscillator subject to a delayed feedback may admit several coexisting branches of periodic solutions with distinct periods. We review this phenomenon for two different problems in optics by comparing experiments and simulations of rate equation models (Friart, Verschaffelt, Danckaert, & Erneux, 2014; Weicker, Erneux, Rosin, & Gauthier, 2015; Weicker, Keuninckx, Friart, Danckaert, & Erneux, 2016).

  1. Sieber, J., Wolfrum, M., Lichtner, M., & Yanchuk, S. (2013). On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems - A, 33(1078-0947_2013_7_3109), 3109. https://doi.org/10.3934/dcds.2013.33.3109
  2. Yanchuk, S., & Giacomelli, G. (2015). Dynamical systems with multiple long-delayed feedbacks: Multiscale analysis and spatiotemporal equivalence. Phys. Rev. E, 92(4), 042903. https://doi.org/10.1103/PhysRevE.92.042903
  3. Kovalev, A. V., Islam, M. S., Locquet, A., Citrin, D. S., Viktorov, E. A., & Erneux, T. (2019). Resonances between fundamental frequencies for lasers with large delayed feedbacks. Phys. Rev. E, 99(6), 062219. https://doi.org/10.1103/PhysRevE.99.062219
  4. Erneux, T., Weicker, L., Bauer, L., & Hövel, P. (2016). Short-time-delay limit of the self-coupled FitzHugh-Nagumo system. Phys. Rev. E, 93(2), 022208. https://doi.org/10.1103/PhysRevE.93.022208
  5. Weicker, L., Keuninckx, L., Friart, G., Danckaert, J., & Erneux, T. (2016). Multirhythmicity for a time-delayed FitzHugh-Nagumo system with threshold nonlinearity. Control of Self-Organizing Nonlinear Systems, 337–354.
  6. Friart, G., Verschaffelt, G., Danckaert, J., & Erneux, T. (2014). All-optical controlled switching between time-periodic square waves in diode lasers with delayed feedback. Opt. Lett., 39(21), 6098–6101. https://doi.org/10.1364/OL.39.006098
  7. Weicker, L., Erneux, T., Rosin, D. P., & Gauthier, D. J. (2015). Multirhythmicity in an optoelectronic oscillator with large delay. Phys. Rev. E, 91(1), 012910. https://doi.org/10.1103/PhysRevE.91.012910

Jan Sieber (Exeter)

Spectrum of linear time-periodic delay equations in the limit of large delay

The talk will discuss the asymptotics of the spectrum for periodic orbits for two cases in the limit of large delay: first, the case of bounded period and, second, the case of period close to the delay. The latter case is similar to traveling pulses in partial differential equations. The talk will also discuss practical issues that occur when trying to compute eigenvalues numerically.

Babette de Wolff (Berlin)

Pseudospectral approximation for Hopf bifurcation of delay equations

Pseudospectral approximation for delay equations was introduced in 2005 by Breda et al. as a tool to approximate eigenvalues of delay equations by eigenvalues of ordinary differential equations. The pseudospectral approximation technique has been proposed as a method for numerical bifurcation analysis, because of the specific structure of the approximating ODEs.

In this talk, we will discuss how the pseudospectral approximation can be used as a numerical tool for the Hopf bifurcation and we analytically study convergence of the Lyapunov coefficient.