**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

coupling.

**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*

adaptation

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,

(2015).

**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)** **