August 21, 2019


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)

KONSTANTIN BLYUSS, University of Sussex
Dynamics of coupled oscillators with distributed-delay coupling

In this talk I will discuss different aspects of the dynamics of systems of oscillators with
distributed-delay coupling. Using the example of coupled Stuart-Landau oscillators with
different choices of delay distributions, we derive the conditions for amplitude death depending
on coupling parameters, as well as the average frequency and frequency detuning.
I will demonstrate the emergence of different branches of phase-locked solutions,
and discuss their stability for different delay distributions. The talk will conclude with
the discussion of open problems for systems of coupled oscillators with distributed-delay


Otti D'Huys,1 Nicholas D. Haynes,1 Johannes Lohmann 1,2 and Daniel J. Gauthier1
Extreme transients in autonomous Boolean networks

1 Department of Physics, Duke University, Durham, North Carolina, USA
2 Institut fuer Theoretische Physik und Kontrolle, Technische Universitaet Berlin, Germany

Autonomous Boolean networks are known to display complex dynamics, originating from the
absence of an external clock, internal time delays and the non-ideal behavior of the logic
gates. We study experimentally such networks on a field-programmable gate array (FPGA). In
particular, we show that networks consisting of only a few logic elements can produce
transients that can last up to a billion times of the typical timescale of the dynamics. In ring
networks, we find oscillatory transients and characterize their duration in terms of coupling
delays, asymmetries and noise. In a network involving two delay loops, we observe, beside
chaotic transients, a multitude of periodic patterns.

INGO FISCHER, Campus Universitat de les Illes Balears
Delays in Physical Systems: Nuisance, Challenges and Opportunities
Time delays in feedback or coupling occur in a variety of physical systems, ranging from
high-speed machining to photonic systems. Such delays can create dynamical instabilities,
that have been a nuisance in many applications, but delays can also be employed
to stabilize and control dynamical systems. Physical systems, and in particular optical
systems, have proven excellent to study the influence of delay experimentally and theoretically
under well-controlled conditions [1]. In fact, such studies have boosted the interest
in delay systems. Moreover, the gained insights and the good control over such systems
have in recent years been inspiring applications, specifically using delayinduced properties.
In this presentation, we discuss this development and provide recent examples of
fundamental aspects of delay systems, as well as applications of delay-dynamical systems.
In particular we will present a photonic systems, allowing to study the influence of statedependent
delay [2] and an application, showing how delay systems can be employed for
neuro-inspired information processing.

[1] M.C. Soriano, J. Garca-Ojalvo, C.R. Mirasso, I. Fischer, Rev. Mod. Phys. 85,
421470 (2013).

[2] Jade Martnez-Llins, Xavier Porte, Miguel C. Soriano, Pere Colet, and Ingo Fischer,
accepted for publication in Nature Communications (2015).

PHILIPP HOVEL, Technische Universit¨at Berlin
Control of cluster synchronization in delay-coupled oscillators by network

In my presentation, I will discuss an adaptive control scheme for the control of in-phase
and cluster synchronization in delay-coupled networks of Stuart-Landau oscillators. This
paradigmatic normal form arises naturally in an expansion of systems close to a Hopf bifurcation.
Based on the considered, automated control scheme, the speed-gradient method,
the topology of a network adjusts itself in a self-organized manner such that the target
state is realized. I will demonstrate that the emerging topology of the network is modulated
by the coupling delay. If the delay time is a multiple of the system’s eigenperiod, the
coupling within a cluster and to neighboring clusters is on average positive (excitatory),
while the coupling to clusters with a phase lag close to p is negative (inhibitory). For
delay times equal to odd multiples of half of the eigenperiod, the opposite holds: Nodes
within one cluster and of neighboring clusters are coupled by inhibitory links, while the
coupling to clusters distant in phase state is excitatory. In addition, the control scheme
is able to construct networks such that they exhibit not only a given cluster state but
also oscillate with a prescribed frequency. Finally, I will illustrate the effectiveness of the
speed-gradient method in cases, where only part of the network is accessible.


DANIEL GAUTHIER, Duke University
Reservoir computing using autonomous time-delay Boolean networks

Daniel J. Gauthier, Nicholas D. Haynes, Otti D'Huys, David P. Rosin, Duke University, Durham, North Carolina, USA 27708
Miguel C. Soriano and Ingo Fischer, Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain

I will discuss our efforts to develop reservoir computers using autonomous Boolean logic elements with time-delay feedback. Reservoir computers are based on recurrent networks with randomly connected nodes. Time-series data is typically fed into the reservoir using random connections and only the weights of the output layer are adjusted to optimize the performance of the system for a particular task, such as time-series forecasting and signal classification. In one set of experiments, we use a single Boolean logic element with two time-delay links to realize a reservoir computer that classifies up to three bits of a 400-MHz clock rate input binary bit stream. The system generates a chaotic transient with a consistency window that ranges from 30 to 300 ns. When the reservoir is optimized with respect to the time delays of the links, we find that the input waveform can be classified with low error for up to 70 ns even though the inputs are provided to the reservoir for only 7.5 ns. Finally, I will discuss preliminary research on using a reservoir computer to classify objects in high-speed (400 Mframes/s) video imagery. In particular, we use a 100-node, randomly-connected autonomous time-delay Boolean network to classify a rotating object in a 10x10 pixel image. This experiment highlights the power of a reservoir computer for undertaking complex tasks that cannot be achieved using other approaches, such as Deep Learning networks, and highlights the use of massively parallel logic chips for realizing reservoir computers.

SABINE H. L. KLAPP, Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin
Delay-differential equations for driven soft matter systems

In this talk I will present concepts for and examples of time-delayed feedback control
for the dynamics of driven soft matter systems, particularly colloidal suspensions. Colloidal
particles are ideal, theoretically and experimentally accessible, ”model” systems
due to their large size and the tunabilty of their interactions. In the first part, I will discuss
spatially confined colloidal suspensions under shear flow, analyzed by many-particle
(Brownian Dynamics) computer simulations. By varying the externally applied shear rate
(”open-loop” control) these colloidal films display a sequence of states characterized by
pinning, shear-induced melting, laning, and moving crystalline order with synchronized
oscillations of the particles. We then show how this rich dynamics changes under feedback
control targeting the shear stress. Within the many-particle simulations, this can be
realized by supplementing the original equations of motion for N interacting particles by
a suitable relaxation equation involving a relaxation (”delay”) time scale tc. It turns out
that the stress feedback control significantly influences the dynamics. Moreover, the time
tc (relative to relevant intrinsic time scales of the many-body system) plays a key role for
the stability of certain states. In the second part, I will present examples of feedback control
for one-dimensional colloidal systems, based on Fokker-Planck equations with time
delay. We show that the delayed feedback control can lead to a variety of phenomena
such as reversal of current, the enhancement of transport in ratchet systems, and the
suppression of density fluctuations.

BERND KRAUSKOPF, The University of Auckland
Pulsed lasers with delay

We consider a semiconductor laser with a saturable absorber whose output is fed back
to itself via an optical loop. This setup has been realised recently with a VSCEL with
saturable absorber. The motivating idea is that this system may produce laser pulses
with low jitter and specified repetition times in response to an initial pulse. A bifurcation
analysis of the Yamada rate equation model with a delay term reveals effects of the optical
feedback on the pulsing properties of the laser. These theoretical results will be put into
the context of some initial experimental measurements.

RACHEL KUSKE, University of British Columbia
New pattern dynamics in stochastic PDEs with Pyragas control

Recent computational studies demonstrate complex dynamics for PDE’s with Pyragas
control. On the one hand the control of patterns via delayed feedback is an attractive
means for pattern resilience, but on the other hand there is the potential for additional
complex behaviour. The interaction of stochastic effects and Pyragas control generates
new pattern dynamics that do not occur when these features appear in isolation. We
give a new analysis for stochastic PDE’s with delay, capturing novel spatio-temporal
pattern mechanisms in the Swift-Hohenberg equation (SHE) with Pyragas control and
noise that are not observed for the standard SHE. We demonstrate the connection between
traveling waves that appear via coherence resonance-type phenomena and compare these
to multimode patterns generated by delays and noise in other nonlinear settings.

Yuliya Kyrychko, University of Sussex
Dynamics of neural and genetic networks with discrete and distributed delays

In this talk I will present an artificial Hopfield-type neural network model, where one subsystem receives a
delayed input from another subsystem. This model includes a combination of both discrete and distributed delay, where
distributed time delays represent the neural feedback between the two subsystems, and the discrete
delays describe the neural interactions within each of the two subsystems. Stability properties
are investigated for different commonly used distribution kernels, and the results are compared to
the corresponding results on stability analysis for networks with no distributed delays. It is shown
how boundaries of stability region of the trivial equilibrium point can be obtained
analytically for the cases of delta, uniform and gamma distributions. In the second part of the talk,
I will show the effects of transcriptional and translational time delays on the dynamics
of gene regulatory networks, that are known to be fundamental for many life processes.
Conditions for stability and Hopf bifurcation of the positive equilibrium are established in terms
of the overall time delay and other system parameters. Numerical simulations are performed to
support analytical conclusions and to illustrate the behaviour of the model in different dynamical regimes.

LAURENT LARGER, University of Franche-Comte
Delay dynamics explored through signal and information photonic processing

Photonic is a modern experimental science which is particularly well matching many specific
requirements for observing complex dynamics involving delays: the speed of light
combined to broadband, long, and highly transparent optical fibers allow for the generation
of highly controllable delays from 100s of ps to 100s of s, acting equally on any
temporal fluctuations of the light parameters (intensity, or phase, or even wavelength)
ranging from 10s of ps to the DC. The development of high performance optoelectronic
devices for optical telecommunications has also enabled a wide set of tools to manipulate
various kinds of dynamics which can be designed experimentally so that the dynamics can
be effectively ruled by well identified delay differential equations. In this contribution, we
propose to illustrate this rich potential of photonic systems arranged in an optoelectronic
delayed feedback loop, emphasizing on the cross-fertilization between application driven
research, and fundamental investigation of dynamical complexity exhibited by delay dynamics.
More specifically, we will report on how space-time analogy of delay dynamics
has been recently used on the one hand to demonstrate and design novel brain-inspired
photonic processors, and on the other hand to explore chimera states in delay systems.

XINZHI LIU, University of Waterloo
Optical delays for improving the dynamic behavior of passively mode-locked lasers

There has been a growing interest in hybrid dynamical systems in recent years. Such
systems often undergo vector field switching and/or state jumps due to sudden changes in
model characteristics. Hybrid dynamical systems arise from a wide variety of applications
such as switching circuits in power electronics, mechanical systems subject to impacts,
multimedia switching communication networks, orbital transfer of spacecraft. This talk
will discuss some of the recent results on existence, stability and control of hybrid dynamical
systems with time delays.

ANDRE LONGTIN, University of Ottawa
Delay-induced linearization and paradoxical oscillations in feedforward nets

This talk will first consider the description of delay-differential equations in the presence of sinusoidal or random forcing.
In both cases we present a scheme to describe the reduced dynamics of the system on the center manifold. Theoretical results
are in good agreement with numerics, but discrepancies emerge away from the bifurcation. We also present work on the stabilization
of a bifurcation by a delay. Finally we present a novel mechanism for oscillation generation in networks. It relies on delayed correlations
arising from direct stimulation and indirect feedforward inhibition due to the primary stimulation. The analysis further reveals how delay
can be a source of linearization in neural nets.

KATHY LUDGE, Freie Universit¨at Berlin,
Optical delays for improving the dynamic behavior of passively mode-locked lasers

Integrated multisection semiconductor light sources are promising candidates for on-chip
optical data communication. Among these passively mode-locked lasers are of particular
interest. Consisting of an absorbing and an amplifying section, these devices are able
to produce fast and regular pulse trains that are, for example, also needed for medical
imaging. We study the light emission dynamics of these devices under the impact of
optical time-delayed feedback. By means of numerical bifurcation analysis we determine
the different regimes of operation, ranging from stable mode-locking regimes with short
pulses to quasi-periodic and unstable pulse trains. The regularity of the emitted pulses,
which is deteriorated by the effect of spontaneous emission noise, is a key property for
applications and usually characterized by the timing jitter. We calculate the timing jitter
with a stochastic approach in the long term limit [1]. For a setup with two delay sections
a reduction of the timing jitter is observed, provided that one of the feedback cavities is
resonant with the laser cavity [2]. Maximal jitter reduction occurs when both feedback
cavities are resonant. The additional degree of freedom introduced by a second cavity
increases the locking ranges, as compared with single cavity feedback, and drastically
increases the tunability of the repetition rate of the pulse train.
[1] C. Otto, L. C. Jaurigue, E. Sch¨oll, and K. L¨udge, Optimization of timing jitter
reduction by optical feedback for a passively mode-locked laser, IEEE Photonics Journal
6, 1501814 (2014).
[2] L. C. Jaurigue, E. Sch¨oll, and K. L¨udge, Passively mode-locked laser coupled to two
external feedback cavities, Novel In-Plane Semiconductor Lasers XIV, SPIE Proc. 91342,

JAN SIEBER, University of Exeter, Extended time-delayed feedback and its odd-number limitation

Starting from the results of Amann and Hooton (2013), I prove an existence result for
extended time-delayed feedback (ETDF) stabilization as introduced by Gauthier et al
(94). If the uncontrolled periodic orbit is linearly feedback stabilizable with a single input
in the classical sense of linear control theory then one can find gains (periodic in time)
such that the ETDF stabilization with sufficiently long memory is also stabilizing. I will
also demonstrate automatic normal form computations in DDE-Biftool that have been
recently implemented by B Wage, Y Kuznetsov et al.

JIAN-QIAO SUN, University of California, Multi-objective Optimal Design of Feedback Controls for Nonlinear Dynamical Systems with Time Delay

The first and the most important step in the control design process for nonlinear dynamic
systems with time delay is to guarantee the stability. The performance of the closed-loop
system as a function of various system and control parameters is the next step, which
has received much less attention in the literature. When there are multiple parameters
and control objectives, such a quantitative design step is highly challenging. Very often,
the control performance objectives are conflicting to each other, meaning that the
improvement of one objective often causes other objectives to degrade. In this talk, we
present the recent results of quantitative design of controls for nonlinear dynamic systems
by using the advanced algorithms of multi-objective optimization. The controls can be of
linear PID type or nonlinear feedback such as sliding mode. The advanced algorithms of
multi-objective optimization consist of parallel cell mapping methods with sub-division
techniques. Interesting examples of linear and nonlinear controls will be presented with
both numerical simulations and experimental validations.

SERHIY YANCHUK, Weierstrass Institute for Applied Analysis and Stochastics
Multistable jittering in oscillators with pulsatile delayed feedback

Oscillatory systems with time-delayed interactions play important role in various applications,
especially in neuroscience. Here, we consider one of the simplest possible setup:
a system with pulsatile delayed feedback. For such a system, we report an unusual scenario
of destabilization of a periodic regular spiking regime. At the bifurcation point,
numerous solutions with non-equal interspike intervals emerge. We show that the number
of the emerging, so-called “jittering” solutions grows exponentially with the delay
value. Although this appears as highly degenerate from a dynamical systems viewpoint,
the “multi-jitter” bifurcation occurs robustly in a large class of systems. We observe it
not only in a paradigmatic phase-reduced model, but also in a simulated Hodgkin-Huxley
neuron model and in an experiment with an electronic circuit.

ANNA ZAKHAROVA, Technische Universit¨at Berlin
Time delay control of symmetry-breaking patterns: oscillation death and
chimera states
Symmetry breaking in a complex dynamical system is a universal phenomenon which
occurs in diverse fields such as physics, chemistry, and biology. Special attention has
recently been paid to oscillation death (inhomogeneous steady state) and chimera states
(coexisting incongruous coherent and incoherent domains) both implying the breakup of
symmetry. Using a paradigmatic model of coupled Stuart-Landau oscillators we study how
these patterns can be controlled by introducing time-delay in the system. In particular, we
show that time delay influences the stability of an inhomogeneous steady state, providing
the opportunity to modulate the threshold for oscillation death [1]. Moreover, time delay
allows to significantly increase the lifetime of transient amplitude chimera states [2].
[1] A. Zakharova, I. Schneider, Y. N. Kyrychko, K. B. Blyuss, A. Koseska, B. Fiedler, E.
Schll, Time delay control of symmetry-breaking primary and secondary oscillation death,
Europhys. Lett. 104, 50004 (2013)
[2] A. Zakharova, M. Kapeller, E. Schll, Chimera Death: Symmetry Breaking in Dynamical
Networks, Phys. Rev. Lett. 112, 154101 (2014)