Laminar chaos in systems with quasiperiodic delay David M uller-Bender1and G unter Radons1 2y 1Institute of Physics Chemnitz University of Technology 09107 Chemnitz Germany

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Laminar chaos in systems with quasiperiodic delay

David M¨uller-Bender1, ∗and G¨unter Radons1, 2, †

1Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany

2ICM - Institute for Mechanical and Industrial Engineering, 09117 Chemnitz, Germany

(Dated: January 18, 2023)

A new type of chaos called laminar chaos was found in singularly perturbed dynamical systems

with periodic time-varying delay [Phys. Rev. Lett. 120, 084102 (2018)]. It is characterized by nearly

constant laminar phases, which are periodically interrupted by irregular bursts, where the intensity

levels of the laminar phases vary chaotically from phase to phase. In this paper, we demonstrate

that laminar chaos can also be observed in systems with quasiperiodic delay, where we generalize

the concept of conservative and dissipative delays to such systems. It turns out that the durations

of the laminar phases vary quasiperiodically and follow the dynamics of a torus map in contrast to

the periodic variation observed for periodic delay. Theoretical and numerical results indicate that

introducing a quasiperiodic delay modulation into a time-delay system can lead to a giant reduction

of the dimension of the chaotic attractors. By varying the mean delay and keeping other parameters

ﬁxed, we found that the Kaplan-Yorke dimension is modulated quasiperiodically over several orders

of magnitudes, where the dynamics switches quasiperiodically between diﬀerent types of high- and

low-dimensional types of chaos.

I. INTRODUCTION

Processes that involve transport or evolution by a ﬁnite

velocity are characterized by time-delays. Time-delay

systems are widely applied to model such processes and

appear in many areas of science [1–5] and engineering

[3, 6, 7]. Beyond the well established mathematical the-

ory [8–10], an overview of recent advances of the theory

and applications of time-delay systems can be found in

the theme issues introduced by [11–13]. A review on

chaos in time-delay systems can be found in [14]. Since

the delay generating processes are in general inﬂuenced

by environmental ﬂuctuations or by the state of the con-

sidered system itself, the delays are in principle time- and

state-dependent. Generalizing the widely studied case of

a constant delay and simplifying the challenging case of

a state-dependent delay, a time-dependent delay can be

considered, which is realistic if the delay generating pro-

cess is nearly independent of the state of the system.

It is known that introducing a temporal delay variation

increases the complexity of time-delay systems [15, 16],

which can improve the security of chaos communication

[17–19]. A delay variation can induce diﬀerent types

of synchronization [20–26], it can stabilize [27–29] and

destabilize systems [30, 31], and inﬂuences mathemati-

cal properties such as the analyticity of solutions [32].

Also eﬀects on delayed feedback control [33–36] and on

amplitude death in oscillator networks [37] were studied.

While fast time-varying delays can be approximated by

constant distributed delays [38] and the stability of sys-

tems with slowly time-varying delays can be derived from

the stability of constant delay systems [29], the interme-

diate case induces features that are not known from con-

stant delay systems. In [39, 40] it was demonstrated that

∗david.mueller-bender@mailbox.org

†radons@physik.tu-chemnitz.de

there are basically two types of periodically time-varying

delays, where one of the types leads to large diﬀerences

from the known behavior of constant delay systems in

the tangent space dynamics such as the scaling of the

Lyapunov spectrum and the structure of the Lyapunov

vectors. Moreover in [41, 42] it was shown that this de-

lay type leads to a previously unknown type of chaotic

dynamics called laminar chaos, which is characterized

by nearly constant laminar phases, whose intensity levels

vary chaotically from phase to phase and the phases are

periodically interrupted by short irregular bursts. This

comparably low-dimensional behavior diﬀers drastically

from the high-dimensional chaotic dynamics observed in

the same systems for constant delay, which is charac-

terized by high-frequency oscillations. The ﬁrst experi-

mental observation of laminar chaos in an optoelectronic

system [43, 44], where its robustness against noise was

demonstrated, was followed by further experimental ob-

servations in electronic systems [45, 46]. The synchro-

nization of laminar chaotic systems was investigated in

[26], and in [47], for the ﬁrst time, laminar chaos was

found in a constant delay system that is coupled to a lam-

inar chaotic time-varying delay system. In this paper, we

generalize the theory on laminar chaos to systems with

quasiperiodically time-varying delay. Such delays are rel-

evant, for instance, in the analysis of quasiperiodic solu-

tions of systems with state-dependent delay [48, 49] or

can be viewed as an intermediate step to understand sys-

tems with randomly time-varying delay, which are com-

mon in many systems [50–54].

We consider systems deﬁned by the scalar delay diﬀer-

ential equation

Θ˙z(t) + z(t) = f(z(R(t))),with R(t) = t−τ(t),(1)

where τ(t) is the time-varying delay. Systems with

this structure and various nonlinearities f(z) of the

delayed feedback appear in many applications: The

arXiv:2210.04706v4 [nlin.CD] 16 Jan 2023

τ0−AAs

τ0

τ0+AAs

τ(t)

(a) periodic delay

012345678910

time t

τ0−AAs

τ0

τ0+AAs

τ(t)

(b) quasi-periodic delay

FIG. 1. Exemplary time-varying delays according to Eq. (4).

(a) Periodic delay (N= 1) with frequency ν1= 1 and (b)

quasi-periodic delay with N= 2 incommensurate frequen-

cies ν1= 1 and ν2=√π. The delays are parameterized by

the mean delay τ0and the amplitude parameter A∈[0,1],

which determines the largest bound A Asof the delay varia-

tion, where we have As= (1/N)PN

n=1(2πνn)−1(see Eq. (4)).

Mackey-Glass equation [55], which is a model for blood-

production, is given by f(z) = µ z/(1 + z10), a sinu-

soidal nonlinearity, f(z) = µsin(z), gives the Ikeda

equation [56, 57], which ﬁrst appeared as a model for

light dynamics in a ring cavity with a nonlinear op-

tical medium and also well describes certain optoelec-

tronic oscillators [43, 58, 59], and the quadratic nonlin-

earity f(z) = µ z(1 −z) was used to analyze general

properties of such types of systems [60]. Chaotic dif-

fusion can be observed with the climbing-sine nonlinear-

ity f(z) = z+µsin(2π z) [61, 62]. Studies with further

nonlinearities can be found in [4]. If the parameter Θ is

large, as we assumed in this paper, Eq. (1) belongs to the

class of singularly perturbed delay diﬀerential equations

and large delay systems, which both are widely studied

for constant delay [60, 62–75] and also results on state-

dependent delay are available [76–80]. The separation

into a large delay timescale and a small internal timescale

plays a crucial role in the spatio-temporal representa-

tion of time-delay systems, which enables the observa-

tion of spatio-temporal phenomena in time-delay systems

[81–84]. Using the concept of singularly perturbed time-

delay systems, potentially high-dimensional systems can

be easily implemented by opto-electronic systems, which

is interesting for applications such as chaos communi-

cation [85–88], random number generation [89–91], and

reservoir computing [92–95]. According to the concept

of strong and weak chaos introduced in [96], chaotic dy-

namics generated by Eq. (1), including the types of chaos

considered here, can be classiﬁed as weak chaos since the

linear instantaneous term z(t) on the left hand side leads

to a negative instantaneous Lyapunov exponent.

To generalize the theory on laminar chaos, we consider

delays that are quasiperiodic in the sense of being an al-

most periodic function (cf. [97]) instead of the periodic

delay considered in [41]. Such type of delays can be de-

ﬁned by

τ(t) = τ0+A g(ν1t, ν2t, . . . , νNt),(2)

where g:RN→Ris 1-periodic in all arguments and N

is the number of frequencies. As in the original theory

on laminar chaos, we assume that ˙τ(t)<1 for almost

all t, which avoids several mathematical problems and

often can be motivated by physical arguments [98, 99].

For instance, in systems, where the delay is caused by a

transport process, this assumption means that the dis-

tance a signal has to travel does not change faster than

the velocity of the signal. The function gmust fulﬁll

1>˙τ(t) =

n=1

νng(n)(ν1t, ν2t, . . . , νNt) (3)

for almost all t, where g(n)is the partial derivative of

gwith respect to the nth argument. For our numerical

investigations, we consider delays of the form

τ(t) = τ0+A

n=1

cos(2πνnt)

2πνn

(4)

with the mean delay τ0and the amplitude parameter A,

where Eq. (3) is fulﬁlled for almost all tif we assume

A∈[0,1]. An exemplary periodic and a quasiperiodic

delay generated by Eq. (4) is shown in Fig. 1(a) and (b),

respectively.

The paper is structured as follows. In Sec. II the the-

ory of laminar chaos is reviewed. First numerical experi-

ments with a quasiperiodic delay variation in Eq. (1) are

documented in Sec. III A, where it is demonstrated that

laminar chaos exists in such systems. In Sec. III B, the

theory of laminar chaos is generalized to systems with

quasiperiodic delay by a rigorous analysis of the limiting

system obtained for Θ = ∞, which is needed to under-

stand the numerical results, especially the diﬀerences to

systems with periodically time-varying delay. The eﬀec-

tive dimension of the chaotic dynamics resulting from

Eq. (1) with quasiperiodic delay is considered in Sec. IV,

where the Kaplan-Yorke dimension [100, 101] is numeri-

cally computed as a function of the mean delay τ0. The

diﬀerences of the structure in parameter space compared

to periodically time-varying delay systems are elaborated

and an outlook to systems with random delay is given.

II. REVIEW OF LAMINAR CHAOS

First we shortly recall the theory of laminar chaos

from [41, 42]. In principle, our system of interest is a

feedback loop, where a signal z(t) is delayed and fre-

quency modulated by a time-varying delay τ(t) and the

function values of the signal are modiﬁed by the nonlin-

earity f, which is represented by the right hand side of

Eq. (1). According to the left hand side, the resulting

signal f(z(R(t))) is then ﬁltered by a low-pass ﬁlter with

a cutoﬀ frequency Θ before the next roundtrip inside the

feedback loop begins. Mathematically, this process can

be described by the so-called method of steps, which is

an iterative procedure for solving DDEs with constant

[102] and time-varying delays [103]. For that the solu-

tion is divided into suitable solution segments zk(t), with

t∈(tk−1, tk] = Ik, which represent the memory of the

system at time t=tk. If a signal ended a roundtip inside

the feedback loop at time tk, it began the roundtrip at

time tk−1=R(tk) = tk−τ(tk). Therefore the bound-

aries tkof the so-called state intervals Ikof the solution

segments are connected by the so-called access map given

t0=R(t) = t−τ(t),(5)

whose dynamics also plays a crucial role in mathematical

properties of systems with time-varying delays [32, 49]. A

solution segment zk+1(t) can be computed from the pre-

ceding segment zk(t) using the solution operator deﬁned

zk+1(t) = zk(tk)e−Θ(t−tk)+

dt0Θe−Θ(t−t0)f(zk(R(t0))),

(6)

which can be derived from Eq. (1) by substituting the in-

stantaneous terms z(t) and ˙z(t) with zk+1(t) and ˙zk+1(t),

respectively, as well as the delayed term z(R(t))with

zk(R(t)), and solving the resulting ordinary diﬀerential

equation for zk+1(t). In the limit Θ → ∞, which means

that the cutoﬀ frequency of the low-pass ﬁlter is sent to

inﬁnity, the integral kernel in Eq. (6) converges to a delta

distribution [66] and we obtain the singular limit map

zk+1(t) = f(zk(R(t))),(7)

which can be used to approximate the dynamics of

Eq. (1) for large Θ given that the derivative ˙z(t) is much

smaller than Θ, i.e., that the characteristic timescale of

the temporal structures of z(t) must be coarser than the

width Θ−1of the kernel in Eq. (6) for a good approxi-

mation. Equation (7) can be interpreted as iteration of

the graph (t, zk(t)) under the two-dimensional map

xk=R−1(xk−1) (8a)

yk=f(yk−1),(8b)

which consists of two independent one-dimensional iter-

ated maps, where Eq. (8a) describes the frequency mod-

ulation of the signal z(t) due to the delay variation and

Eq. (8b) describes the inﬂuence of the nonlinearity, which

modiﬁes the function values of z(t). Given that the

map deﬁned by Eq. (8b) shows chaos and Θ is suﬃ-

ciently large, it was demonstrated in [41, 42] that Eq. (1)

with a periodically time-varying delay shows two types of

chaotic dynamics, which depend on the dynamics of the

access map, Eq. (5). For periodically time-varying delay

this map is a lift of the monotonically increasing circle

map (cf. [104])

θ0=ν1R(θ/ν1) mod 1.(9)

This map shows two types of dynamics, which naturally

induce two classes of time-varying delays, which lead to

two types of chaos that are observed in Eq. (1). For

so-called conservative delay, the circle map and its in-

verse deﬁned by Eq. (8a) show quasiperiodic dynam-

ics with a Lyapunov exponent λ[R] = 0. In this case,

Eq. (8a) only leads to a quasiperiodic frequency modu-

lation. Stretching and folding of the chaotic map given

by Eq. (8b) causes strong ﬂuctuations, where the charac-

teristic frequency is bounded due to the low-pass ﬁlter of

the feedback loop. The resulting dynamics is called tur-

bulent chaos adapted from the term “optical turbulence”

which was introduced in [57], where a optical feedback

loop with the structure of Eq. (1) with a constant de-

lay was investigated. An exemplary time series is shown

in Fig. 3(a). For so-called dissipative delays, where the

circle map associated to the access map, Eq. (5), shows

stable periodic dynamics with a negative Lyapunov expo-

nent λ[R]<0. This means we have a resonance between

the average roundtrip frequency of the signal inside the

feedback loop and the frequency of the time-varying de-

lay. This resonant Doppler eﬀect leads to periodically

alternating phases with low and high frequencies in the

solution z(t). If the condition

λ[f] + λ[R]<0 (10)

is fulﬁlled, the low-frequency regions degenerate to nearly

constant laminar phases, which are periodically inter-

rupted by irregular bursts at stable periodic points of

R−1mod 1 corresponding to drifting stable orbits in

the lift, Eq. (8a). This type of dynamics is called lam-

inar chaos. The intensity levels of the laminar phases

are determined by the dynamics of the one-dimensional

map given by Eq. (8b). The criterion for laminar chaos,

Eq. (10), can be derived by a stability analysis of square

wave solutions of the limit map, Eq. (7), cf. [41].

III. LAMINAR CHAOS IN SYSTEMS WITH

QUASIPERIODIC DELAY

A. Numerical experiments

A good starting point for the generalization of the the-

ory of laminar chaos to systems with quasiperiodic delay

is Eq. (10). We will argue below that the Lyapunov ex-

ponent λ[R] of the access map is well deﬁned also for

quasiperiodic delay and we naively assume that laminar

chaos can be observed for large enough Θ if Eq. (10) is

fulﬁlled. As an exemplary system we choose Eq. (1) with

the nonlinearity f(z)=3.8z(1−z), and the quasiperiodic

delay shown in Fig. 1(b) with A= 0.9 and τ0= 1.135.

1000 1002 1004 1006 1008 1010

time t

0.00

0.25

0.50

0.75

1.00

z(t)

(a)

1000 1002 1004 1006 1008 1010

time t

0.00

0.25

0.50

0.75

1.00

z(t)

(b)

0.4 0.6

δk

0.4

0.5

0.6

0.7

δk+1

(c)

0.4 0.6

δk

0.4

0.5

0.6

0.7

δk+1

(d)

0.0 0.5 1.0

0.00

0.25

0.50

0.75

1.00

zk+p

(e)

0.0 0.5 1.0

0.00

0.25

0.50

0.75

1.00

zk+p

(f)

FIG. 2. Features of laminar chaotic time series in systems

with (a) periodic delay and (b) quasiperiodic delay. In (c)

and (d) the duration δk+1 of the (k+ 1)th laminar phase

is plotted over the duration δkof the kth laminar phase for

a periodic and quasiperiodic delay, respectively. For periodic

delays the points (δk, δk+1 ) accumulate at discrete points, i.e.,

the durations δkvary periodically, whereas for quasiperiodic

delay the points (δk, δk+1 ) densely ﬁll a circle, which indicates

quasiperiodic dynamics. The dynamics of the levels zkof

the laminar phases is governed by the map zn+p=f(zn) as

shown in (e) and (f) with p= 3 and p= 2 for periodic and

quasiperiodic delay, respectively, where the points (zk, zk+p)

(dots) resemble the nonlinearity f(dashed line). The time

series were obtained from Eq. (1) with Θ = 100 and f(z) =

3.8z(1 −z), where the delays were chosen as in Fig. 1 with

A= 0.9 as well as τ0= 1.5 for periodic and τ0= 1.135 for

quasiperiodic delay. The durations δkand the levels zkwere

estimated directly from the time series. We ﬁrst detected the

bursts times. Then the durations δkare simply the waiting

times between two subsequent bursts and the levels zkare the

values of the time series in the middle between two bursts.

1000 1002 1004 1006 1008 1010

time t

0.00

0.25

0.50

0.75

1.00

z(t)

(a)

1000 1002 1004 1006 1008 1010

time t

0.00

0.25

0.50

0.75

1.00

z(t)

(b)

1.71.61.51.4

τk

1.7

1.6

1.5

1.4

τk+1

(c)

1.2 1.3 1.4

τk

1.2

1.3

1.4

τk+1

(d)

FIG. 3. Features of turbulent chaotic time series in systems

with (a) periodic and (b) quasiperiodic delay. They are char-

acterized by chaotic ﬂuctuations with a large frequency, which

is bounded by the cutoﬀ-frequency Θ of the low-pass ﬁlter

in Eq. (1). In (c) and (d) the length τk+1 of the (k+ 1)th

state interval Ik+1 = (tk, tk+1 =R−1(tk)] according to the

method of steps, Eq. (6), is plotted over the duration τkof

the kth one for a periodic and a quasiperiodic delay, respec-

tively, where we set t0= 0. For periodic delay, the dynamics

of the state intervals is conjugate to a rotation on a circle,

whereas for quasiperiodic delay it is conjugate to a transla-

tion on the torus. The time series were computed from Eq. (1)

with Θ = 100 and f(z) = 3.8z(1−z). We have chosen the de-

lays as in Fig. 1 with A= 0.9 as well as τ0= 1.54 for periodic

and τ0= 1.32 for quasiperiodic delay, respectively.

For these parameters, Eq. (10) is fulﬁlled and the re-

sulting time series, which is shown in Fig. 2(b), clearly

shows features of laminar chaotic behavior. There are

nearly constant laminar phases, whose intensity levels

are determined by the map given by Eq. (8b) as illus-

trated in Fig. 2(f), where the intensity level zk+pof the

(k+p)th laminar phase is plotted over the level of the

kth laminar phase for a large number of intensity lev-

els, where p > 0 was set to the smallest natural number

such that the points (zk, zk+p) resemble a line. Thus

the time series passes the test for laminar chaos intro-

duced in [44]. The same behavior is observed for peri-

odic delay as shown in Fig. 2(e). While for a periodic

delay the durations δkof the laminar phases vary peri-

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LaminarchaosinsystemswithquasiperiodicdelayDavidMuller-Bender1,andGunterRadons1,2,y1InstituteofPhysics,ChemnitzUniversityofTechnology,09107Chemnitz,Germany2ICM-InstituteforMechanicalandIndustrialEngineering,09117Chemnitz,Germany(Dated:January18,2023)Anewtypeofchaoscalledlaminarchaoswasfoundinsing...

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Laminar chaos in systems with quasiperiodic delay David M uller-Bender1and G unter Radons1 2y 1Institute of Physics Chemnitz University of Technology 09107 Chemnitz Germany

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