Jan Sieber (Exeter)

Characteristic matrices for linear periodic delay differential equations

Abstract

(Szalai, Stépán, & Hogan, 2006) gave a general construction for characteristic matrices for systems of linear delay differential equations with periodic coefficients. Matrices constructed in this way can have a discrete set of poles in the complex plane, which may obstruct their use when determining the stability of the linear system. The talk will present modifications of the original construction such that the poles get pushed into a small neighborhood of the origin of the complex plane (Sieber & Szalai, 2011). Further modifications treat the case of large delays (Sieber, Wolfrum, Lichtner, & Yanchuk, 2013) and the case of large period and delay slightly smaller than the period (Yanchuk, Ruschel, Sieber, & Wolfrum, 2019).

  1. Szalai, R., Stépán, G., & Hogan, S. J. (2006). Continuation of Bifurcations in Periodic Delay-Differential Equations Using Characteristic Matrices. SIAM Journal on Scientific Computing, 28(4), 1301–1317. doi: 10.1137/040618709
  2. Sieber, J., & Szalai, R. (2011). Characteristic Matrices for Linear Periodic Delay Differential Equations. SIAM Journal on Applied Dynamical Systems, 10(1), 129–147. doi: 10.1137/100796455
  3. Sieber, J., Wolfrum, M., Lichtner, M., & Yanchuk, S. (2013). On the stability of periodic orbits in delay equations with large delay. Discrete and Continuous Dynamical Systems, 33(7), 3109–3134.
  4. Yanchuk, S., Ruschel, S., Sieber, J., & Wolfrum, M. (2019). Temporal Dissipative Solitons in Time-Delay Feedback Systems. Phys. Rev. Lett., 123(5), 053901. doi: 10.1103/PhysRevLett.123.053901

Matthias Wolfrum (WIAS Berlin)

Stability properties of temporal dissipative solitons in DDEs

Abstract

Localized states are an universal phenomenon in spatially extended nonlinear systems. Recently, also temporally localized states have gained some attention, mainly driven by applications in various optical systems, where the dynamics of short optical pulses can be described by DDE models. We present a theory for such states, which appear as periodic solutions with a period close to the round trip time of the system. We study such solutions by using the singular limit of large delay. We derive a desingularized equation for the solution profiles, and study the corresponding Floquet spectrum for their stability. To this end, we derive an Evans function and discuss the analogies and differences to the classical theory for localized states in spatially extended systems.

Daniel Franco (UNED Madrid)

Control for delay equations

Abstract

We will consider, in a broad sense, ways to control the asymptotic behaviour of continuous-time delay models. The talk will have two parts. In the first one, we will present a new approach to study the global stability of difference equations and we will show how this approach can be used to extend and complement certain dichotomy results for delay equations (Franco, Guiver, Logemann, & Perán, 2020). In the second part, we will show how a class of forced positive nonlinear delay-differential systems can be seen as feedback control systems. Then, we will provide conditions under which the states of these models are semi-globally persistent and conditions under which the non-zero steady state is stable in a sense developed in control theory (Franco, Guiver, & Logemann, 2021).

  1. Franco, D., Guiver, C., Logemann, H., & Perán, J. (2020). On the global attractor of delay differential equations with unimodal feedback not satisfying the negative Schwarzian derivative condition. Electronic Journal of Qualitative Theory Of Differential Equations, Paper No. 76, 15. doi: 10.14232/ejqtde.2020.1.76
  2. Franco, D., Guiver, C., & Logemann, H. (2021). Persistence and stability for a class of forced positive nonlinear delay-differential systems. Acta Applicandae Mathematicae, 174, Paper No. 1, 42. doi: 10.1007/s10440-021-00414-5

Eugenia Franco (Helsinki)

Renewal equations for measure-valued functions of time describing physiologically structured populations

Abstract

Integral renewal equations (RE) with measure-valued solutions arise naturally when modelling physiologically structured populations. Nevertheless, there are only few available tools to study the long-term behaviour of their measure-valued solutions. In this talk, I will present two assumptions on the kernel characterizing the renewal equation that guarantee asynchronous exponential growth/decline or convergence to a stable distribution for the solution of the renewal equation. These assumptions will be illustrated via two examples: a model of cell growth and fission and a model of waning and boosting of the immunity level against a pathogen. These two models can be formalized also via a partial differential equation (PDE), in the final part of the talk I will sketch the connection between the PDE formulation and the RE formulation.

  1. Franco, E., Gyllenberg, M., & Diekmann, O. (2021). One dimensional reduction of a renewal equation for a measure-valued function of time describing population dynamics. Acta Applicandae Mathematicae, 175, Paper No. 12, 67. doi: 10.1007/s10440-021-00440-3
  2. Franco, E., Diekmann, O., & Gyllenberg, G. (2022). Modelling physiologically structured populations: renewal equations and partial differential equations. https://arxiv.org/abs/2201.05323

Frits Veerman (Leiden)

Noninvasive control of singular patterns

Abstract

Noninvasive (Pyragas) control aims to control the stability of special solutions such as periodic orbits, in general n-dimensional dynamical systems by adding a control term that vanishes on the ’target’ solution - hence the term ’noninvasive’ - but has a nontrivial (local) structure in the neighbourhood of the target solution, thereby influencing its stability properties. Hence, a suitable choice of the control term can stabilize the target solution. This approach can also be applied to infinite-dimensional dynamical systems, such as reaction-diffusion systems, where stationary patterns are obvious candidates for ’special solutions’ to be stabilized. For a wide range of patterns in (scalar) reaction-diffusion equations, Isabelle Schneider (FU Berlin) has shown that a well-chosen combination of spatio-temporal delay can stabilize said patterns. In our most recent work, we show that the techniques and ideas developed in this scalar setting can be extended to a class of singularly perturbed reaction diffusion systems, where scale separation plays a key role in the (constructive) existence and stability analysis of so-called singular patterns. Incorporating different strategies (proportional, time-developed) in the existing Evans function framework, our preliminary results show that single homoclinic singular pulses can always be stabilized using proportional control, while the efficacy of time-developed feedback depends on the structure of the pulse spectrum (joint work with Isabelle Schneider, FU Berlin).

Mark van den Bosch (Leiden)

Delay differential equations driven by Lévy processes of finite intensity: on the existence of invariant measures for Mackey-Glass type equations

Abstract

We consider a class of stochastic delay differential equations driven by Lévy processes, where jumps have mean zero, do not exceed a certain height, and happen finitely often in any compact time interval. The class of delay equations contains the original Mackey-Glass equation and Nicholson’s blowflies equation. We prove the global existence of strong solutions and investigate when they are bounded in probability. In here, the equation is allowed to be non-autonomous. Both the drift and noise term must be locally Lipschitz and may depend on the past. Subsequently, for autonomous systems, we look into the existence of stationary solutions via the Krylov-Bogoliubov method.

Bram Lentjes (Utrecht)

Center manifolds and periodic normal forms for bifurcations of limit cycles in DDEs

Abstract

Iooss proved in (Iooss, 1988) the existence of a periodic smooth finite dimensional center manifold near a non-hyperbolic periodic orbit in finite dimensional ODEs and provided in addition critical normal forms to describe the dynamics on this center manifold. Using these normal forms, Kuznetsov et al. (Kuznetsov, Govaerts, Doedel, & Dhooge, 2005) derived via a periodic normalization procedure, explicit formulas to compute the critical normal form coefficients for all codim 1 bifurcations of limit cycles. These formulas are useful to detect codim 2 points and allow us to distinguish between sub- and supercritical bifurcations. The first aim of this talk is to generalize the center manifold theorem from Iooss, via the functional analytic framework of sun-star calculus, towards the infinite dimensional setting of (classical) DDEs. The second aim of this talk is to lift the periodic normalization procedure towards the DDE-setting. In the final part of the talk, I will illustrate via discrete DDEs how these normal form coefficients could be implemented in a continuation software.

  1. Iooss, G. (1988). Global characterization of the normal form for a vector field near a closed orbit. Journal of Differential Equations, 76(1), 47–76. doi: 10.1016/0022-0396(88)90063-0
  2. Kuznetsov, Y. A., Govaerts, W., Doedel, E. J., & Dhooge, A. (2005). Numerical Periodic Normalization for Codim 1 Bifurcations of Limit Cycles. SIAM Journal on Numerical Analysis, 43(4), 1407–1435. doi: 10.1137/040611306