Dissipative systems with nonlocal delayed feedback control [original]

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Dissipative systems with nonlocal delayed feedback control
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New J. Phys. 20 ( 2018 ) 113010 https: // doi.org / 10.1088 / 1367-2630 / aae998
PAPER
Dissipative systems with nonlocal delayed feedback control
Josua Grawitter , Reinier van Buel , Christian Schaaf and Holger Stark
Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, D-10623 Berlin, Germany
E-mail: [email protected]
Keywords: time-delayed feedback, closed-loop control, soft matter, latency, nonlocal control, spatiotemporal dynamics, double delay
Abstract
We present a linear model, which mimics the respons e of a spatially e xtended d issipative medi um to a
distant perturbation, and inves tigate its dynamics under delayed feedback control. The time a perturbation
needs to propagate to a measurement point is captured by an inherent delay time ( or latency ) .Ad e t a i l e d
linear stability analysis demonstrates that a nonzero system delay acts to destabilize t he otherwise stable
ﬁ xed poi nt for suf ﬁ c iently large feedback strengths. The imagina ry part of the dominant eigenvalue is
bounded by twice the feedb ack strength. In the relevant p arameter sp ace it changes discont inuously alon g
speci ﬁ c lines when switching between eigenvalues. When the feedback control force is bounded by a
sigmoid function, a supercritical Hopf bifurcation occurs at the stability – instability transition. Perturbing
the ﬁ xed point, the frequency and amplitude of the resu lt ing limit cycles respon d to parameter changes like
the dominant e igenvalue. In particular, the y show similar discontin uities along spe ci ﬁ cl i n e s .T h e s er e s u l t s
are largely independent of the exact shape of the sigmoid function. Our ﬁ ndings match well with previously
reported resu lts on a feedback-induc ed instability of vortex diffusion in a rotationally driven Newtonian
ﬂ uid ( Zeitz et al 2015 Eur. P hys. J. E 38 22 ) . Thu s, our model captures the essential features of nonlocal
delayed fee dback control in sp atially extended dissipative s ystems.
1. Introduction
Many phenomena in soft matter science require exciting a dissipative material. From mixing two liquids [ 1 ] ,
sorting colloids [ 2 , 3 ] , controlling reaction rates [ 4 ] and heat transport in micro ﬂ uidic devices [ 5 ] ,t o ﬂ uid optics
[ 1 ] and spiral patterns in liquid crystals [ 6 ] , soft matter systems often display their most useful or interesting
properties under external stresses. Several control and driving schemes have already been applied to these
systems, including optimal control [ 7 ] , hysteresis control [ 8 ] , and time-delayed feedback [ 8 – 10 ] . These methods
sense the characteristic response of a material and adapt their control to it. Here, we focus on time-delayed
feedback control of a linear model system with internal delay, which captures the essential features of how soft
matter systems respond to such a feedback scheme.
As control strategies several time-delayed feedback schemes with one or multiple delay times have been
proposed [ 11 – 13 ] . Among these, the so-called Pyragas scheme is particularly well suited for chaotic systems and
to stabilize unstable periodic orbits [ 14 ] . It has since been applied to various dynamical systems, such as lasers
[ 15 – 17 ] and neural networks [ 18 ] . The method falls into the broader category of closed-loop control schemes
because its control force depends purely on the present and past states of the controlled system. Often, the
Pyragas scheme is called a noninvasive control scheme, as its stabilizing force ideally vanishes in a stabilized
state [ 19 ] .
Earlier theoretical studies [ 8 , 10 , 20 ] and experiments [ 9 , 21 ] applied delayed feedback to the spatiotemporal
dynamics of speci ﬁ c systems. One study showed that a system with two delay times can be mapped to the
complex Ginzburg – Landau equation [ 13 ] . While several investigations of purely temporal systems with an
intrinsic latency have focused on stabilizing unstable foci [ 22 – 24 ] , in one earlier study on vortex diffusion at low
Reynolds numbers an oscillating ﬂ uid ﬂ ow in a circular geometry was initiated by delayed feedback [ 8 ] . Thus,
time-delayed feedback used invasively can also destabilize stable ﬁ xed points and thereby potentially create novel
nonequilibrium states in soft matter systems.
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REVISED
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7 November 2018
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In spatially extended systems the response to a control force at location  F needs the system or intrinsic delay
time to propagate to a distant location  A. There, it is sensed and then fed back into the system at the ﬁ rst
location  F ( see the schematic in ﬁ gure 1 ) . To mimic this situation, we propose a linear dissipative model system,
with the inherent system delay time appearing in the response function. We then apply additional time-delayed
feedback control and investigate the response for conditions concentrate on the case, where the system remains
stable for vanishing system delay. We perform a detailed linear stability analysis and demonstrate how a nonzero
system delay acts to destabilize the otherwise stable ﬁ xed point. Typically, the response of a physical system to an
external stimulus is bounded by nonlinearities. To mimic such a behavior, we limit the strength of the feedback
control force with a sigmoid function. As a result, stable limit cycles appear in the otherwise unstable parameter
regions. While their amplitudes depend on the speci ﬁ c realization of the sigmoid as hyperbolic tangent, algebraic
sigmoid, and ramp function, their frequencies are similar. Both, the stability – instability transition and the
appearance of limit cycles match to the ﬁ ndings in [ 8 ] . This suggests that our model captures the essential
features of a spatially extended dissipative systems when subjected to nonlocal delayed feedback.
We present the linear dissipative model in section 2 , derive the characteristic function of its ﬁ xed point, and
describe our numerical methods. In section 3 we investigate and discuss the linear stability of the ﬁ xed point. In
section 4 we modify the feedback term by the sigmoid function such that its absolute value is limited and study
the stable limit cycles arising from this modi ﬁ cation. Finally, we summarize our ﬁ ndings and conclude in
section 5 .
2. Linear model with nonlocal delayed feedback
2.1. Derivation
We derive a simple model for a dissipative physical process A ( t ) , which is driven by an external, nonlocal force  F
( t ) . Generally, the linear response of A ( t ) to F ( t ) is written as
ò c =-
¢¢ ¢
-¥
¥
() ( ) ( ) ( ) At t t F t t d, 1
where the response function  χ ( t ) characterizes the physical system completely. If F acts at some distance from
the position where A is observed, there will be a system delay ( or latency ) quanti ﬁ ed by time
t
s
, before A is
affected by F . We describe this causal link in χ using the Heaviside function
t
Q

- (
)

t
s
. Furthermore, in
dissipative systems the response to some perturbation often decays exponentially with rate α and the response
function becomes
ct a t =Q - - - () ( ) [ ( ) ] ( ) tt t exp . 2
ss
Substituting χ ( t ) into equation ( 1 ) and differentiating both sides with respect to t ,w e ﬁ nd
at =- + -
() () ( ) ( )
At
t At F t
d
d .3
s
Following Pyragas [ 14 ] , we now implement for F ( t ) delayed feedback control with delay time t c , control strength
k

, and a constant external force F
0
,
kt =- - - () [ () ( ) ] ( ) Ft F A t A t ,4
0c
and substitute the expression into equation ( 3 ) :
ak t t t =- + - - - - -
() () [ ( ) ( ) ] ( )
At
t At F At At
d
d .5
0s s c
Figure 1. Schematic of the model with nonlocal delayed feedback. The process  A ( t ) measured at location  A feeds back to the time-
delayed control force  F ( t ) at location, the response of which then propagates during time 
t s
to location  A.
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New J. Phys. 20 ( 2018 ) 113010 J Grawitter et al

This equation has one ﬁ xed point  A
*
, where = At dd 0
, which is determined by the ﬁ rst two terms because
the delayed control term vanishes for constant solutions
*
a
= () A F .6
0
We nondimensionalize time  t and
k

with α , force  F
0
with
*
a A
, and A with A
*
to obtain
kt t t =- + - - - - -
() () [ ( ) ( ) ] ( )
At
t At At At
d
d 1. 7
ss c
In the following we study this form of the delay differential equation ( DDE ) .
Note that for
k < 0
the delayed feedback term acts to destabilize A , because when t >- () ( )
A

tA t c the
feedback term gives a positive contribution to A
t

dd
. Already for
t = 0
s
, this destabilizes the system for
suf ﬁ ciently large
k
∣∣

. In the following we take  k 0 to explore the role of
t
s
in the model.
2.2. Characteristic function
Within control theory, equation ( 7 ) is categorized as a closed loop system with delayed feedback. Because such
systems can become unstable [ 25 ] , we investigate the linear stability of the ﬁ xed point at A  =  1 by introducing
the perturbation ansatz
el =+ () ( ) ( ) At t 1e x p 8
with e 
∣

∣ 1 and complex eigenvalues
 l

Î into equation ( 7 ) , which gives a transcendental equation for λ ,
lk =- - -
lt lt --
() ( ) 1e 1 e . 9
sc
In the following we solve this equation numerically and study the properties of its solutions.
The eigenvalues  λ are roots of the characteristic function l (
) g

,
ll k =+ + -
lt lt --
() ( ) ( ) g 1e 1 e . 1 0
sc
Note that they cannot be expressed using the Lambert W function, as is usually possible for time-DDEs [ 19 ] , due
to the double delay terms. However, we can infer some properties of the roots from the structure of g . First, with
each complex λ also its complex conjugate
l

¯ is a root of l (
) g

since
k

,
t
s
, and t c are real parameters. Second, in
appendix B we show that any root λ of g with
l > () Re 0
has a nonzero imaginary part bounded by
k

2
,
lk <
∣

() ∣ Im 2 . Thus any eigenvalue associated with an unstable ﬁ xed point has a nonzero imaginary part, which
is bounded by control strength. Since the imaginary part is the oscillation frequency, we conclude that fast
oscillations require suf ﬁ ciently large control strengths.
2.3. Numerical methods
2.3.1. Root ﬁ nding
In general, the roots of the characteristic function l (
) g

cannot be determined analytically. Therefore, we use a
numerical root ﬁ nding algorithm to ﬁ nd the dominant eigenvalue in our stability analysis, i.e. the eigenvalue
with the largest real part for one set of system parameters. As a prerequisite we need to restrict the complex plane
to a ﬁ nite box, which is guaranteed to contain the dominant eigenvalue. In appendix B , we prove that the
dominant eigenvalue for any parameter combination is contained in the compact complex region
 

Ì
  

ll
t kt
lk tl t l
=Î - -
-+ -
t
⎧
⎨
⎩
⎡
⎣
⎢ ⎤
⎦
⎥
⎫
⎬
⎭
∣( ) ( )
( ) [ ( )] { [ ( )]} ( )
W 1R e m a x 0 ,
1 2e 1 ,
0I m e x p R e 1e x p R e . 1 1
s
s
sc
s
We search a rectangular superset of


( see appendix B ) using an interval Newton method  [ 26 ] , which provably
ﬁ nds all complex roots of l (
) g

in


and is implemented in [ 27 ] .
2.3.2. Numerical continuation
To track individual eigenvalues through parameter space, we implement a simple numerical continuation
scheme for the eigenvalue equation l = ()
g

0 . We explain it for the general problem of ﬁ nding the roots
(
) u

p
of
a function
(
)

fu p ,
. Given a set of starting values 
(

) u p ,
0 0 with
= () fu p ,0
0 0 , the continuation scheme ﬁ rst
calculates an approximate step 
˜
u
1

along the tangent vector of the implicit function 
(
) u

p
at
u

0
.
=- D  ¶
¶
-
˜( ) ( )
()
uu f f
p p .1 2
u
u p
10 1
,
0 0
Here,
D p

is a small number ( in our calculations 10
− 2
) and
 f
u
is the Jacobian matrix of
f
w.r.t. 
u

. The
approximate value is then re ﬁ ned iteratively by solving
= () fu p ,0
1 1
for
u

1
using Newton ’ s method with
=+ D pp
p

10
and starting from
˜
u
1

. The procedure produces a new set of values
()

u p ,
1 1 and is repeated as
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New J. Phys. 20 ( 2018 ) 113010 J Grawitter et al

necessary to ﬁ nd approximate curves in parameter space. The derivatives are calculated using the library of [ 28 ]
and Newton ’ s method is implemented in [ 29 ] .
2.3.3. Trajectories
We also calculate the time evolution of A ( t ) to investigate its long-time behavior, such as limit cycles in the case of
a bounded control force. To do so, we use the method of steps based on a 5th-order Runge – Kutta method [ 30 ]
and the 4th-order Rosenbrock method RODAS [ 31 ] when using the ramp function to bound the control force
( see section 4 ) . Both methods are implemented in the software package DifferentialEquations.jl  [ 32 ] .
The method of steps treats delay terms by splitting the domain of integration into a sequence of time
intervals, so that on each interval the delayed values of the dynamic quantities are fully known. In our case ( see
equation ( 18 ) in section 4 ) the method initially requires a history function describing A ( t ) for all times
 tt
-

+< () t 0
sc . The history function is used to integrate over the interval  t < t
0

s because
t - ()
A

t
s
[ and
tt -- () At
sc
] are then known until
t =
t

s
. For remaining intervals
 tt <+ ()
n

tn 1
ss
( with  Î
n

)
integration continues using A ( t ) from the previous intervals.
2.3.4. Periodic solutions
To ﬁ nd time-periodic solutions with period T for the DDE
tt =- -
() [ ( )( )( ) ] ( )
At
t fA t A t A t
d
d ,, 1 3
12
with delays
t
1
and
t
2
for a given function  f and initial conditions, we use the following method. First, we
construct a truncated Fourier series 
˜ ()
A

t
with N real coef ﬁ cients corresponding to the N lowest-order sine and
cosine harmonics with periodicity T . Initially, the coef ﬁ cients are chosen such that
p =
˜ () ( )
A

tt T 0.25 sin 2
.
From the given DDE we introduce a residual function 
e

() t , which describes the error of approximation 
˜
A

,
et t =- - - () ˜ () [ ˜ () ˜ ()
˜ () ] ( ) t At
t fA t A t A t
d
d ,, . 1 4
12
We also truncate
e

() t in Fourier space with N real coef ﬁ cients ( corresponding to the same modes as before ) .W e
apply a root- ﬁ nding method to ﬁ nd the set of coef ﬁ cients for
˜ ()
A

t
, which minimizes the absolute values of the
coef ﬁ cients of
e

() t . When all coef ﬁ cients of
e

() t become vanishingly small, we have arrived at a good
approximation for A ( t ) . To discretize both
˜
A

and ε in Fourier space, we use the library of [ 33 ] with N  =  20. For
root ﬁ nding we use up to 10  iterations of a trust region method implemented in [ 29 ] .
Note that for many periods  T the only periodic solutions are ﬁ xed points. To distinguish ﬁ xed points from
limit cycles, we use the ratio of the amplitudes of
˜ ()
A

t
and ε ( t ) : because ﬁ xed points have exactly zero amplitude
but the residual nevertheless ﬂ uctuates, the amplitude ratio is small for ﬁ xed points and large for limit cycles. By
maximizing the ratio we obtain the periods of limit cycles in our system. To perform the numerical optimization
of T , we use a golden-section search implemented in [ 34 ] .
Finally, we test the stability of limit cycles numerically using the time-integration methods described in
section 2.3.3 by introducing them as history functions and initial conditions. Due to numerical error, unstable
limit cycles eventually diverge from their orbits while stable limit cycles remain there. We observe trajectories for
100 units of time to determine stability.
All numerical calculations are performed using the Julia programming language  [ 35 ] and all plots are
created using the matplotlib package [ 36 ] .
3. Linear stability analysis
3.1. Stability for vanishing delays
In the limiting case of vanishing system delay,
t  0
s
, the ﬁ xed point is always stable. To prove this, we set
t = 0
s
and take the real part of g :
ll k k t l =+ + -
tl -
[ ( )] ( ) [ ( )] ( )
()
g Re Re 1 e cos Im . 15
Re c
c
We prove by contradiction
1
: suppose that
 l () Re 0
together with
kt > ,0
c
. Then the ﬁ nal term in
equation ( 15 ) is the only ( potentially ) negative contribution necessary to obtain = () g Re 0 . However, because
the absolute value of this term is always smaller than or equal to
k

, roots with
 l () Re 0
cannot exist. Therefore,
all eigenvalues have
l < () Re 0
and the ﬁ xed point must be stable for
t  0
s
.
1
Alternatively, one could prove the same result using the analytic solution for
t = 0
s
, which is
l

kk t k t t =- + + + () { [ () ] } W 1e x p 1
0c c c
where W
0
is the 0th branch of the Lambert  W function.
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New J. Phys. 20 ( 2018 ) 113010 J Grawitter et al

In the limit of vanishing control delay, t  0
c , the ﬁ xed point is stable because the characteristic function
ll =+
t  () ( ) g lim 1 16
0
c
only has one ( real ) root at
l

=- 1
. This is also clear from equation ( 7 ) since the delay term vanishes completely.
3.2. Stability-to-instability transition
A ﬁ xed point is unstable if
l > () Re 0
for its dominant eigenvalue  λ . Based on numerical calculations of the
dominant eigenvalues over a wide set of parameter combinations, we make several observations, which we
report here. For each
t > 0
s
and t > 0
c , there is a critical control strength
kt t () ,
crit s c
determined by
l = () Re 0
for the dominant eigenvalue and
 l () Re 0
for all others beyond which the ﬁ xed point is unstable for all
kk > crit . The k crit form a manifold in parameter space; cross sections for various
t
s
are displayed in ﬁ gure 2 . For
t  0
c the critical value k crit diverges like
t
-
c
1

. In this limit l (
) g

is approximated to linear order in t c as
ll k t l =+ +
lt -
() ( ) g 1e , 1 7
c
s
from which follows that any root  λ stays the same as long as the product  kt c remains constant. For increasing  t c
speci ﬁ c values of k crit exist, where a different eigenvalue becomes dominant. We observe that the resulting cusps
become less prominent as t c increases and eventually k crit as a function of t c approaches a constant value for
each
t
s
. Notably, these cusps imply that for
k

close to the cusp value alternating regions of stability and instability
occur as t c increases. We also observe that for small
t
s
the cusps are closer to each other w.r.t.  t c and k crit
diverges with decreasing
t
s
. From these observations and the ﬁ ndings of section 3.1 , we conclude that only the
combination of both nonzero delays causes the instability.
The dominant eigenvalue displays some general features ( shown in ﬁ gure 3 ) : its real part is smallest ( i.e. close
to − 1 ) in regions with either
k

or t c close to zero, where the control term becomes negligible. Conversely, this
implies that delayed feedback slows down the decay of individual modes. The imaginary part of the dominant
eigenvalue is nonzero and displays clear discontinuities w.r.t.  t c and
k

in the unstable region and close to the
transition curve k crit ( see ﬁ gure 3 ( b )) . Since also
l = () Re 0
at the transition, there is a Hopf bifurcation.
Furthermore, as t c increases the imaginary part repeatedly decreases and then jumps to a larger value when a
different eigenvalue becomes dominant. Figure 3 ( a ) shows that the jump lines correspond to valleys in the real
part of the dominant eigenvalue.
More generally, kt ()
crit c is the envelope of a family of Hopf curves , i.e.  lines in parameter space
corresponding to
l =

() Re 0
for each conjugate pair of complex eigenvalues  λ
±
. We determine the Hopf
curves numerically as implicit functions t (
) u

c de ﬁ ned by t = () u
g

,0
c with
lk = (( ) )
u

Im ,
and with constant
t
s
( note that g is complex and therefore constrains both coordinates of
u

) . Six Hopf curves for successive
eigenvalues are displayed in ﬁ gure 4 with t = 0.2
s ( corresponding to ﬁ gure 3 ) . Extending the Hopf curves to
large
k

,w e ﬁ nd that they approach constant values for t c . The cusps visible in ﬁ gures 2 and 3 correspond to
intersecting Hopf curves, i.e.  two pairs of eigenvalues, where the real part switches from negative to positive at
the same set of parameters. The pairs of eigenvalues have positive real parts on the right-hand side of their
respective curves and, therefore, crossings imply that several eigenvalues can have positive real parts for the same
Figure 2. Stability – instability transition curves in the
k
– t c plane for various
t s
with the stable region to the left of each curve.
5
New J. Phys. 20 ( 2018 ) 113010 J Grawitter et al

set of parameters. Furthermore, when we project the numerically calculated curves
kt ()
c
onto the
l ()
I

m
–
k

plane, they all fall onto the same line ( see inset of ﬁ gure 3 ) . This suggests there is a global minimum of critical
control strength
k
crit
min
for the envelope kt ()
crit c below which no instability can occur. Lastly, we observe that
successive Hopf curves 
kt ()
c
become ﬂ atter, which means their envelope kt ()
crit c approaches a straight line for
large t c .
We close with two comments. First, we note the similarities between our dominant eigenvalues and
eigenvalues described in [ 8 ] for the speci ﬁ c case of delayed feedback control applied to vortex diffusion in a
circular geometry. The characteristic function for that system is derived by solving the spatiotemporal problem
explicitly. It contains Bessel functions, which play a similar role as the exponentials in equation ( 10 ) .
Furthermore, the diffusive response function ( compare equation ( 20 ) in appendix A with a  0 ) , has an initial
increase and then decays to zero, just as in our model. We consider these similarities a strong indication that
some spatiotemporal systems map well on our simpli ﬁ ed model response functions approximately given by
equation ( 2 ) .
Second, when Pyragas feedback is applied to a system with spatial components like the Swift – Hohenberg
equation [ 37 , 38 ] or an reaction-diffusion system [ 39 ] , the relation kt = 1
c marks the instability boundary,
where spontaneous motion of spatial patterns occurs. In our simple model we do not ﬁ nd an explicit expression
for the transition line  kt ()
crit c but a similar scaling in case of small control delays t  0
c ( see above ) .
4. Bounded control force
Typically, the response of a physical system to an external stimulus is bounded by nonlinearities because
ultimately no control mechanism can exert an in ﬁ nite force. We mimic this behavior here by implementing a
bounded control force. Bounded delayed feedback was previously studied, e.g.  to suppress overshooting due to
Figure 3. Dominant eigenvalue λ color-coded in t c –
k
plane together with stability – instability transition curve 
k
crit
( dashed red /
white lines ) for system delay
t = 0.2
s
: ( a ) real part and ( b ) imaginary part.
Figure 4. Hopf curves of six successive eigenvalues λ ( solid colored lines ) in the t c –
k
plane for system delay
t = 0.2
s
. Inset: in the
l (
) I

m
–
k
plane all six Hopf curves project onto the same curve ( solid black line ) . As a result, they all share the same minimum of
critical control strength
k
crit
min
( dashed gray line ) .
6
New J. Phys. 20 ( 2018 ) 113010 J Grawitter et al

overcompensating control forces [ 40 ] , or for hydrodynamic vortex diffusion in a circular geometry to stabilize
unstable modes [ 8 ] .
4.1. Model and setup
We start by modifying the DDE of equation ( 7 ) such that the time-delayed control term is limited by a
monotonically increasing odd function
s

() x
with
s

¥ =  () 1 and
s

¢= () 01
:
sk t t t =- - - - - -
() ( ) { [ ( ) ( )]} ( )
At
t At At At
d
d .1 8
ss c
Compared to equation ( 7 ) we set the constant force to zero, which shifts the ﬁ xed point of the modi ﬁ ed DDE
with the bounded feedback to
* =
A

0
. Linearizing around this ﬁ xed point, one recovers equation ( 7 ) without
the term + 1.
To realize upper and lower bounds for the control forces, we consider three sigmoid functions: the
hyperbolic tangent
s

= () (
)

xx tanh
, which approaches ± 1 exponentially, an algebraic sigmoid
s

=+
-
() ( ∣ ∣ ) xx x 1
1
, which approaches ± 1 like a power law, and the nonsmooth ramp function
s

== - () () [ ( )
]

xx x ramp min 1 , max 1 ,
. All three are displayed in ﬁ gure 5 .
We solve equation ( 18 ) numerically
2
as described in section 2.3 with the history function
<= ()
A

t 00
and
a small initial perturbation
3
, A ( t  =  0 )  =  10
− 3
. If the ﬁ xed point is unstable, the perturbation will grow over
time, otherwise it will decay to zero. Using this setup, we study the long-time behavior of the bounded system
and its relationship to the unstable ﬁ xed point studied in section 2 by calculating the time evolution of A ( t ) until
time  T  =  10
3
and by examining the frequencies and amplitudes of the occurring limit cycles for times
0.8 T  <  t  <  T . We chose these initial conditions because we are speci ﬁ cally interested in the long-time behavior
following a simple perturbation from the ﬁ xed point without any prescribed mode or frequency.
4.2. Long-time dynamics
For all three sigmoid functions σ , the bounded system displays a Hopf bifurcation, where the ﬁ xed point A
( t )  =  0 becomes unstable and a limit cycle emerges ( see ﬁ gure 6 ) . It is stabilized by the upper and lower bound of
the control force, which would otherwise grow to in ﬁ nity. The limit cycle does not correspond to an unstable
periodic orbit of the uncontrolled system as envisaged in Pyragas ’ original idea [ 14 ] , because the feedback term
does not vanish when the limit cycle is reached. Generally, the limit cycles for all sigmoid functions should
converge for
k ¥
because in this limit they all approach the discontinuous step function. In the studied
parameter region, the frequencies of the observed limit cycles are determined by the imaginary part of the
dominant eigenvalue at the unstable ﬁ xed point. This becomes evident when comparing the frequency plots in
the left column of ﬁ gure 6 with ﬁ gure 3 ( b ) . In particular, the jumps from one mode to the other agree well for
both frequencies. However, the amplitude 
D

A
of the limit cycle
(
) A

t
lc
de ﬁ ned as
D

=- [ { () } { () } ] AA t A t max min 2
tt lc lc is generally not related to the real part of the dominant eigenvalue
( compare amplitudes in the right column of ﬁ gure 6 with ﬁ gure 3 ( a )) . It rather behaves like the frequency of the
Figure 5. Three realizations of a sigmoid function for implementing delayed control forces with upper and lower bounds plotted
versus x . Three asymptotic scalings for large
∣

∣ x
are realized: ﬂ at ( ramp ) , exponential ( tanh ) , and power law.
2
It might be possible to ﬁ nd analytic approximations for limit cycles ( observed in the following ) and their Floquet exponents using the
Poincaré – Lindstedt method [ 41 ] .
3
The discontinuity at t  =  0 is equivalent to a force proportional to the Dirac delta function. Thus it is in ﬁ nitely large and exerted over an
in ﬁ nitesimally short period of time such that its integral over time ( impulse ) is ﬁ nite.
7
New J. Phys. 20 ( 2018 ) 113010 J Grawitter et al

limit cycle with one difference: it increases as t c grows and drops to a smaller value when the long-time dynamics
switches to a different limit cycle. So, at the discontinuity lines the sudden increase in frequency is accompanied
by a sudden decrease in amplitude. This is explicitly demonstrated in ﬁ gure 7 .
Using the method described in section 2.3.4 we analyzed the two limit cycles displayed in ﬁ gure 7 in more
detail. We ﬁ nd that they co-exist for both sets of parameters ( with slightly shifted amplitudes and frequencies ) .
For t = 1.0
c only the limit cycle with the larger frequency is stable, while for
t = 1.2
c
both limit cycles are stable
and the long-time dynamics depends on initial conditions. Apparently, in the bistable case the high-frequency
limit cycle attracts trajectories departing from the unstable ﬁ xed point, which where attracted to the low-
frequency limit cycle for t = 1.0
c . Notably, the transition to bistability occurs close to one of the Hopf curves
displayed in ﬁ gure 4 , which suggests that the second limit cycle becomes stable as an additional pair of complex
eigenvalues acquires positive real parts. We, therefore, expect that for larger t c even more than two stable limit
cycles co-exist, depending on the number of complex eigenvalues with positive real part
4
.
Figure 6. Frequencies and amplitudes of the limit cycles resulting from the DDE of equation ( 18 ) color-coded in the
k
– t c parameter
space at
t = 0.2
s
for different realizations σ of the bounded control force: ( a ) ramp sigmoid, ( b ) hyperbolic tangent, and ( c ) algebraic
sigmoid. Zero frequency ( dark blue ) indicates the region of stable ﬁ xed point ( no oscillations ) . The dashed white line indicates the
transition from stable to unstable ﬁ xed point.
4
Furthermore, we point out the possibility that the unstable limit cycles could be stabilized with an additional ( noninvasive ) time-delayed
feedback term with a delay time matching its period.
8
New J. Phys. 20 ( 2018 ) 113010 J Grawitter et al

There are some notable differences between the limit cycles generated by the three sigmoid functions
bounding the control term. Most obviously, their amplitudes ( right column of ﬁ gure 6 ) behave differently at the
bifurcation line: while they increase smoothly from zero for the hyperbolic tangent and algebraic sigmoid
functions as in a supercritical Hopf bifurcation ( rows ( b ) and ( c )) , they jump to a nonzero value for the ramp
sigmoid function ( row ( a )) . The ramp function is a special case because it is linear up to the bounding values.
This causes the control amplitude to always reach its maximum value in the unstable region, once the transition
line is crossed. The step-like behavior could, for example, be used in experiments to accurately locate the
transition line. Furthermore, for the ramp function the amplitudes of the limit cycles are largest close to the
bifurcation line, while they increase for the smooth functions when moving away from the bifurcation line with
increasing
k

. Finally, a closer inspection shows that the discontinuity lines of the limit cycle frequencies for the
ramp function accurately track the corresponding lines in the imaginary part of the dominant eigenvalue in
ﬁ gure 3 ( b ) . However, there are slight deviations for the smooth sigmoid functions.
4.3. Transient pulse trains
In particular, for large control times  t c we observe transient pulse trains at the beginning of the dynamic
evolution of our system. They occur for stable and unstable ﬁ xed points with decaying or growing amplitude,
respectively. One example, when the ﬁ xed point is unstable, is displayed in ﬁ gure 8 ( a ) . These pulse trains grow
into regular limit cycles, as shown in ﬁ gure 8 ( b ) , while for stable ﬁ xed points their amplitude decays to zero.
Generally, we observe that pulse trains repeat with a periodicity given by  t c and their oscillation frequency is
close to the imaginary part of the dominant eigenvalue  lp ()
I

m2
.
5. Conclusions
Motivated by the growing interest in applying delayed feedback to soft matter systems, we have studied a linear
dissipative model, which mimics nonlocal delayed feedback coupling. To do so, we introduced an inherent
system delay in the response function, which represents the time a perturbation needs to propagate to a distant
location. Our results provide an indication how nonlocal delayed feedback in a spatially extended system
determines its dynamics and helps to classify the observed spatiotemporal response.
In our investigations we concentrated on the case where the system remains in its stable ﬁ xed point when the
intrinsic delay is not present. Turning on the intrinsic delay, the feedback-driven system becomes unstable for
suf ﬁ ciently large control strengths. We are able to show that the absolute imaginary part of the dominant
eigenvalue is bounded from above by twice the feedback strength. Stability – instability transition curves and the
imaginary part of the dominant eigenvalue match well with the ﬁ ndings reported in [ 8 ] , where delayed feedback
was applied to vorticity diffusion of a Newtonian ﬂ uid in a circular geometry at low Reynolds numbers. This
demonstrates that our simple model system captures the essential properties of the spatiotemporal dynamics in a
speci ﬁ c system.
To further study the feedback-induced instability, we examined the long-time dynamics of our model with
bounded feedback, which in linearized form yields the original system. Realizing the bounded feedback by
smooth sigmoid functions, the stability – instability transition occurs via a supercritical Hopf bifurcation. Thus,
the ﬁ xed point becomes unstable and a stable limit cycle evolves continuously. In contrast to Pyragas ’ original
idea [ 14 ] , this limit cycle does not correspond to an unstable periodic orbit of the uncontrolled system, but is
Figure 7. Limit cycles of a system with feedback control bounded by the hyperbolic tangent as sigmoid function at two different
control delays t c for k = 5 and
t = 0.2
s
. The system with larger t c has larger periodicity but smaller amplitude.
9
New J. Phys. 20 ( 2018 ) 113010 J Grawitter et al

stabilized by the control force. For all three sigmoid functions and across many parameter combinations, the
frequencies of the limit cycle, which attracts trajectories starting from the ﬁ xed point, match well the imaginary
part of the ﬁ xed point ’ s dominant eigenvalue. Discontinuity lines are visible, which occur when the long-time
behavior switches to a different limit cycle causing frequency to jump up and amplitude to drop sharply. The
results for the nonsmooth ramp function differ from the other sigmoid functions, because at the Hopf
bifurcation the amplitude of the limit cycle jumps to a nonzero value. In addition, the discontinuity lines of the
limit cycle frequencies accurately track the corresponding lines for the imaginary part of the dominant
eigenvalue of the linearized system. Both features predestine the ramp sigmoid function to accurately determine
the stability – instability transition in an experimental system.
The linear dissipative model presented in this article with its intrinsic delay time helps to classify and
understand the spatiotemporal response of a spatially extended system subject to nonlocal delayed feedback.
Our work demonstrates that this model already exhibits complex and nontrivial behavior. It also provides an
example for double delay systems, which have recently attracted attention [ 42 – 44 ] . In future studies we will
apply delayed feedback to speci ﬁ c nonlinear soft matter systems such as photoresponsive ﬂ uid interfaces [ 45 ]
and viscoelastic ﬂ ow in Taylor – Couette geometry [ 46 ] . The work presented in this article will help us to
categorize the observed spatiotemporal dynamics.
Acknowledgments
We acknowledge ﬁ nancial support from DFG ( German Research Foundation ) via International Research
Training Group  1524 and Collaborative Research Center  910. We further acknowledge support from the DFG
and the Open Access Publication Funds of TU Berlin.
Appendix A. A physical system with intrinsic delay
We consider a diffusion-reaction equation of a substance with density
r

() r t ,
, which decays at a rate α in two-
dimensions:
rr a r ¶=  - () () ( ) ( ) rr tD t x t ,, , , 1 9
t 2
where D is the diffusion constant. As initial condition at t  =  0 we choose a delta peak located at
= r 0
,
r

d = () ( ) rr N ,0 . The solution of this problem is given by
r p a =- -
⎛
⎝
⎜ ⎞
⎠
⎟
() ( ) r t N
Dt t r
Dt
, 4 exp 4 ,2 0
2
Figure 8. Dynamic response of the system to feedback control bounded by the hyperbolic tangent function for t = 0.
4

s
, t = 25
c , and
k = 2.1
: ( a ) transient pulse trains and ( b ) limit cycle oscillations ( note the different scales ) .
10
New J. Phys. 20 ( 2018 ) 113010 J Grawitter et al

where = ∣
∣

r
r

is the distance from the perturbation. The density at any point
¹ r 0
0
increases up to the time
a
a
=+ -
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟ () t r
D
1
2 11 2 1
0 0
2
and then decreases to zero. Note that for pure diffusion ( a  0 ) the maximum is reached at  = -
()
t

rD 4
0 0
2 1 .
Thus, a disturbance initiated at r  =  0 needs the intrinsic delay time t
0
to reach r 0 . To approximate this behavior
in a response function of the form given in equation ( 2 ) , we assume a step function where the substance ρ jumps
to its maximum value
r

() r t ,
00
and then decays exponentially
cr a =Q - - - () ( ) ( ) [ ( ) ] ( ) rr tt t t t t ,, e x p . 2 2
00 0 0 0
Thus, t = t
s0
is the intrinsic delay time and the physical decay rate α of our substance enters directly the response
function. Because the step function prevents any response for <
t

t 0 , t
0
is an effective propagation time.
Appendix B. Search region for dominant eigenvalues
We ﬁ nd the roots of the characteristic function g numerically. Because a numerical search on the in ﬁ nite
complex plane is unfeasible, we restrict our search to a bounded region which is guaranteed to contain the
dominant eigenvalue, i.e. the root of g with the largest real part.
First, we ﬁ nd an upper bound to the real part of all eigenvalues using l = [( ) ] g Re 0 :
l
k tt l tt l t l t l
+ =- + + --
() [ ( ) ( )] [( ) ( )] [ ( )] [ ( )] ( )
Re 1 exp Re cos Im exp Re cos Im . 23
sc sc s s
The cosines may at most change the signs of the terms such that both contribute positively. Thus

l
k tl t t l
+ -+ - +
() [ ( )] [ ( ) ( )] ( )
Re 1 exp Re exp Re . 24
ss c
Assuming
 l () Re 0

l
k tl
+ -
() [( ) ] ( )
Re 1 2e x p R e , 2 5
s
and we solve for
l () Re
using Lambert ’ s W function
 l t kt -
t
() ( ) ( ) W Re 1 2 e 1. 26
s
s
s
With our previous assumption we have
 l t kt -
t
⎡
⎣
⎢ ⎤
⎦
⎥
() ( ) ( ) W Re max 0 , 1 2e 1 . 2 7
s
s
s
Notably the upper bound is independent of t c .
Second, we ﬁ nd a lower bound for
l () Re
such that our search region contains at least one eigenvalue. To
simplify our search, we concentrate on
 l

Î , which is determined by
lt l t l = - -- -- () [ ( ) ] ( ) 1e x p 1e x p . 2 8
sc
On the interval -¥
(]

,0 the lhs  of this equation goes continuously from
-

¥
to 0 and the rhs  goes continuously
from
+¥
to − 1. Because both sides are continuous and strictly monotonic for tt > ,0
cs , they share exactly one
value in the overlap, i.e. 
 l
-

10
. Therefore, a search region with
 l - () Re 1
, which also contains the real
axis, will always contain at least one ( real ) eigenvalue.
Third, we ﬁ nd an upper bound for the imaginary parts of all eigenvalues using l = [( ) ] g
I

m0
lk tl t l k t t l t t l =- -- + + ( ) [ ( )] [ ( )] [ ( ) ( )] [( ) ( )] ( ) Im exp Re sin Im exp Re sin Im . 29
s s sc sc
Here, we simply drop the sines, assuming all terms contribute positively
 lk tl t l -+ - ( ) [ ( )] { [ ( )]} ( ) Im exp Re 1 exp Re . 30
sc
The rhs  depends on
l () Re
and on the interval determined for
l () Re
. It is maximal for
l =- () Re 1
implying
 lk +
tt
() [ ]
I

me 1 e
sc
. Note that the symmetry ll = ( ¯ ) ¯ ()
g

g implies that we only need to search the positive
half of the complex plane, i.e.  l ()
I

m0
, because all complex roots appear in conjugate pairs.
In summary, we have shown that the following set of complex numbers


must always contain the
eigenvalue with the largest real part:
11
New J. Phys. 20 ( 2018 ) 113010 J Grawitter et al

  

ll
t kt
lk tl t l
=Î - -
-+ -
t
⎧
⎨
⎩
⎡
⎣
⎢ ⎤
⎦
⎥
⎫
⎬
⎭
∣( ) ( )
( ) [ ( )] { [ ( )]} ( )
W 1R e m a x 0 ,
1 2e 1 ,
0 Im exp Re 1 exp Re . 31
s
s
sc
s
For simplicity and safety we search a bounded rectangular region which is a superset of


:
  

ll
t kt
lk
ÍÎ - -
-+
t
tt
⎧
⎨
⎩
⎡
⎣
⎢ ⎤
⎦
⎥
⎫
⎬
⎭
∣( ) ( )
() ( ) ( )
W 1.1 Re m a x 0 , 1.1 2e 1 . 1 ,
0.1 Im 1. 1 e 1 e . 32
s
s s
sc
Both regions are displayed in ﬁ gure B1 .
Note that the shape of


immediately implies that any eigenvalue λ with
l > () Re 0
has lk <
∣

() ∣ Im 2 .
Furthermore, it can be shown that
l > () Re 0
implies l ¹
∣

() ∣ Im 0 because equation ( 28 ) has no solutions for
l

> 0
. In this case the lhs  of equation ( 28 ) is always positive while the rhs  is always smaller than − 1 and
therefore no solution with
l > () Re 0
and l =
∣

() ∣ Im 0 exists.
ORCID iDs
Josua Grawitter https: / / orcid.org / 0000-0003-2944-7052
Reinier van Buel https: / / orcid.org / 0000-0002-4025-9063
Christian Schaaf https: / / orcid.org / 0000-0002-4154-5459
Holger Stark https: / / orcid.org / 0000-0002-6388-5390
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