CHAOS 16, 033119 共2006兲
Ultra-high-frequency chaos in a time-delay electronic device
with band-limited feedback
Lucas Illinga兲 and Daniel J. Gauthier
Department of Physics and Center for Nonlinear and Complex Systems, Duke University,
Durham, North Carolina 27708
共Received 24 April 2006; accepted 18 July 2006; published online 30 August 2006兲
We report an experimental study of ultra-high-frequency chaotic dynamics generated in a delaydynamical electronic device. It consists of a transistor-based nonlinearity, commercially-available
amplifiers, and a transmission-line for feedback. The feedback is band-limited, allowing tuning of
the characteristic time-scales of both the periodic and high-dimensional chaotic oscillations that can
be generated with the device. As an example, periodic oscillations ranging from 48 to 913 MHz are
demonstrated. We develop a model and use it to compare the experimentally observed Hopf bifurcation of the steady-state to existing theory 关Illing and Gauthier, Physica D 210, 180 共2005兲兴. We
find good quantitative agreement of the predicted and the measured bifurcation threshold, bifurcation type and oscillation frequency. Numerical integration of the model yields quasiperiodic and
high dimensional chaotic solutions 共Lyapunov dimension ⬃13兲, which match qualitatively the
observed device dynamics. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2335814兴
Time-delayed feedback occurs in many systems and is
particularly important at high-speeds, where the time it
takes signals to propagate through the device components
is comparable to the time scale of the fluctuations. Hence,
many fast systems are most accurately described by timedelay differential equations. A fascinating feature of such
systems is that seemingly simple devices can show exceedingly complex dynamics. This has motivated the use of
time-delay feedback devices in practical applications of
chaos, such as chaos communications and ranging, for
which microwave and radio-frequency oscillators are
needed. As an example of such a device, we explore the
dynamics of an electronic circuit that consists of a simple
transistor-based nonlinearity and time-delayed feedback.
Taking the band-limiting characteristics of the device
components into account allows us to develop a simple
model that captures many of the observed features. We
compare the observed dynamics to both theory and numerical calculations and find good agreement.
I. INTRODUCTION
The generation of chaotic signals is well understood for
low-speed circuits,1 i.e., for devices with characteristic frequencies in the Hz or kHz range. However, there is considerable interest in generating high-speed chaos because of
promising applications such as random signal radar/lidar,2–5
random number generation,6–8 and communications.9–14
Generating broadband chaotic signals in the radiofrequency 共rf兲 regime is challenging because electronic components used in low-speed electronic circuits, such as operational amplifiers, are not readily available above a few GHz.
Therefore, new techniques for chaos generation need to be
explored. Furthermore, design issues arise at rf frequencies
a兲
Electronic mail: illing@phy.duke.edu
1054-1500/2006/16共3兲/033119/10/$23.00
that do not exist at low speeds, such as proper circuit layout,
proper isolation of power supply and active circuit elements,
and non-negligible time delays due to signal propagation, to
name just a few.
These challenges are in part the reason that, to date, the
most successful schemes for generating broadband chaotic
signals at GHz-frequencies use optic,9–11 electro-optic,13 and
optoelectronic14,15 devices. However, the use of all-electronic
devices is desirable because electronic components are inexpensive and compact. Some recent efforts have started to
address the issue of all-electronic generation of chaos in
technologically relevant radio frequency bands.16–20 For example, low dimensional chaos in the very-high-frequency
range 共30– 300 MHz兲 has been demonstrated in a time-delay
system with a diode-based nonlinearity.18 It was also shown
that Colpitts oscillators can generate chaos at GHz
frequencies19 and a high-speed chaotic delay-system with a
transistor-based nonlinearity was reported.20
In this paper we describe an electronic time-delay feedback device that generates chaos in the very-high and ultrahigh 共0.3– 3 GHz兲 frequency band. It is built using inexpensive commercially-available components such as ac coupled
amplifiers and a transistor-based nonlinearity. Our device is
similar to the one described by Mykolaitis et al.20 However,
in contradistinction to their work, we do not neglect, in modeling, the fact that rf components are ac coupled and we
provide a detailed comparison of experimentally observed
dynamics and the dynamics of the deterministic model of the
device.
In our chaos generator, the complexity of the dynamics
is controlled by feedback strength and time delay, allowing
the device to be tuned from steady-state behavior, to periodic, quasiperiodic, and chaotic dynamics. The characteristic
time scale of the dynamics can be tuned by adjusting the
pass-band characteristics of the feedback loop. We chose to
16, 033119-1
© 2006 American Institute of Physics
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033119-2
L. Illing and D. J. Gauthier
work with feedback characteristics resulting in oscillations of
several hundred MHz to a few GHz because this allows detailed measurements and characterization of the device dynamics. However, the device can, in principle, operate at
higher frequencies, such as the 3 – 10 GHz frequency band,
by using readily available higher-bandwidth components.
In modeling the device, we pay special attention to the
fact that components are ac coupled, which means that lowfrequency signals are suppressed. The time-delayed feedback
is bandpass filtered because, in addition to the cutoff at low
frequencies, high frequencies are suppressed due to the finite
response time of device components.
The goal of this paper is to present details about the
experimental implementation and to compare measurements
and theory21 to demonstrate that many aspects of the device
dynamics can be explained if the band-limiting characteristic
of the feedback is taken into account.
II. EXPERIMENTAL SETUP
It is well known that time-delayed feedback of the output signal of a nonlinearity to its input can give rise to chaotic dynamics.13,15,16,20,22–24 Our rf-chaos generator is based
on the same principle and we describe and characterize the
device in this section. We discuss separately the passive nonlinearity, the setup of the rf-chaos generator, and the characteristics of the band-limited feedback with variable feedback
gain.
A. Nonlinearity
The transistor-based nonlinearity consists of a bias-T and
a modified Mini-Circuits ERA-SM test board 共ERA-TB兲.
The ERA-TB is a commercially available proto-board that is
meant to test Mini-Circuits ERA amplifiers. A photograph of
the modified ERA-TB is shown in Fig. 1共a兲 and the circuit
diagram of the nonlinearity is shown in Fig. 1共b兲.
To create a nonlinear input-output relationship, the
ERA-TB is modified by replacing the ERA-amplifier with a
transistor 共T1兲, capacitor 共C1兲, and resistors 共RNL and R1兲
关Fig. 1共b兲兴. For this initial experiment, these components are
placed using a “dead bug design,” i.e., keeping them as close
together as possible to avoid undesired feedback 关white box
in Fig. 1共a兲兴.
The dc bias capabilities of the ERA-TB are used to set
the power-supply voltage associated with the collector of
transistor T1 关Fig. 1共b兲兴 while ensuring, at the same time,
good isolation of the rf-output signals and the dc-voltage part
of the circuit. A variable bias voltage of the transistor base is
achieved by further modifying the ERA-TB through removal
of the ac-coupling capacitor at the input port and replacing it
by an external bias-T, which combines the rf-input signal and
the dc-bias voltage VB. The bias voltage is VB = 0.55 V for all
data shown in this paper, i.e. it is below the turn-on voltage
VT of the transistor 共nominally VT ⬃ 0.7 V兲.
To develop a simple model of the nonlinearity, consider
the generated output as a function of input signals of sufficiently high frequency such that the effects of the coupling
capacitors can be neglected. In this case, the output current is
given by the voltage drop across the resistor RNL for input
Chaos 16, 033119 共2006兲
voltages smaller than the difference of the turn-on voltage
and the bias voltage 共VT − VB兲 because the transistor draws
essentially no collector current. In contrast, the output current is given by the difference of the current flowing through
RNL and the collector current of the transistor for input voltages larger than 共VT − VB兲. This suggests a piecewise linear
I − V characteristic, which results in a corresponding piecewise linear input-voltage to output-voltage relationship assuming a standard 50 ⍀ load 共the input impedance of the
device coupled to the output port is assumed to be 50 ⍀兲.
To determine the input-output characteristic of the nonlinearity experimentally, we inject a sinusoidal signal 共
= 13 MHz兲 into the input port of the nonlinearity and simultaneously recorded input and output using an oscilloscope
共Agilent Infiniium, with 50 ⍀ input impedance, 2.25 GHz
bandwidth, and 8 GSa/ s兲. The measured tent-map-like inputoutput characteristic is shown as dots in Fig. 1共c兲. It is seen
that the output depends linearly on the input for large input
powers and that the nonlinearity has a smooth transition from
positive to negative slopes. That is, a strictly piecewise-linear
relation of input and output is a clear oversimplification of
the actual nonlinearity. Furthermore, device models in which
the nonlinearity is approximated as piecewise-linear do not
reproduce several dynamic features of the experiment such
as Hopf bifurcations, which are observed in the experiment
but do not exist in a model with strictly linear segments. For
this reason, we adopt the following phenomenological description of the nonlinearity:
F共v兲 = V0 − 冑F1共v兲2 + a2
=
再
with F1共v兲
Al共v − v 兲 if v 艋 v* ,
Ar共v − v*兲 if v ⬎ v* .
*
冎
共1兲
Here, v denotes the input voltage, v* = VT − VB is the threshold voltage, and V0 is an offset voltage. The slopes of F
approach, respectively, Al and Ar for large 兩v兩 and the parameter a determines the sharpness of the peak of F.
In Fig. 1共c兲, a fit of Eq. 共1兲 to the data is shown as a solid
line, with fitted parameters Al = 0.47 共±0.04兲, Ar = −0.62
共±0.05兲, a = 0.05 共±0.01兲 V, and v* = 0.12 ±0.04 V. Leastsquare minimization is used for fitting and a crude estimate
of the sensitivity of the fit 共values in brackets兲 is obtained by
varying each parameter value individually and determining
the size of parameter deviations above which the resulting
input-output curve was clearly inconsistent with the data.
The fit value of v* suggests a transistor turn-on voltage of
VT = 0.67 共±0.04 V兲, which is within the expected range.
Also, the measured values of the slopes agree reasonably
well with their expected values. As an example, one expects
Al ⬃ RL / 共RNL + RL兲 = 0.42, where RL = 50 ⍀ is the load impedance and RNL is nominally 68 ⍀.
The correct value for the remaining parameter V0 cannot
be obtained from the data because the measured output signal is high-pass filtered by the coupling capacitor C2 关Fig.
1共b兲兴. As a result of the filtering, the measured output signal
has a time average of zero, which, in terms of the filter-free
input-output description of Eq. 共1兲, implicitly determines the
value of V0 for a given input, thereby making the fit value of
V0 dependent on the properties of the input signal. For fur-
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033119-3
UHF chaos in a time-delay device
Chaos 16, 033119 共2006兲
FIG. 1. Transistor-based nonlinearity.
共a兲 Picture of the modified ERA-SM
test board 共modifications highlighted
by the white box兲. 共b兲 Details of the
implementation of the nonlinearity,
which consists of a bias-T 共MiniCircuits ZFBT 6GW, 0.1 MHz–6 GHz
bandwidth兲 and a modified ERA-SM
test board 共Mini-Circuits ERA-TB兲.
The component values on the
ERA-TB
are:
C1 = 47 nF;
C2
= 0.39 F; C3 = 0.1 F; RNL = 68 ⍀;
R1 = 47 ⍀; R2 = 4.75 ⍀; R3 = 70 ⍀;
L1 is a rf choke 共MCL Model AdcH80A兲; D1 is a 10 V Zener diode; and
T1 is an NPN 9 GHz wideband transistor BFG520. 共c兲 Measured nonlinearity 共dots兲 and fit of Eq. 共1兲 共line兲.
ther modeling purposes, the knowledge of a precise value for
V0 is not necessary because the model also includes a highpass filter 关see Eq. 共5兲兴. We set V0 = 6 V in numerical simulations.
To test the frequency dependence of the nonlinearity, we
vary the frequency of the sinusoidal injection signal. For
frequencies up to 100 MHz, the input-output characteristics
are essentially identical to that shown in Fig. 1共c兲. Beyond
100 MHz noticeable distortions arise mainly due to low-pass
filtering in the nonlinearity and the measurement setup. Nevertheless, we believe that Eq. 共1兲 is a valid description of the
input-output characteristic for all experimentally relevant
frequencies 共3 MHz–1 GHz兲 in particular since the dynamic
device model also includes a low-pass filter 关see Eq. 共5兲兴.
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033119-4
L. Illing and D. J. Gauthier
Chaos 16, 033119 共2006兲
FIG. 2. An electronic generator of rf
chaos. 共a兲 Schematic of the device. 共b兲
Transfer function 兩H共兲兩 for the case of
a 10 pF capacitor in the feedback loop
共쎲: data; — fit兲 and without capacitor
共⫻: data; - - -: fit兲. 共c兲 Linear-scale
plot of 兩H兩 for the case of a 10 pF capacitor in the feedback loop 共쎲: data;
—: fit兲.
B. rf-chaos generator
To generate chaos, linear and amplified time-delayed
feedback from the nonlinearity to itself is implemented as
shown schematically in Fig. 2共a兲. In this device, the signal
coming from the feedback loop passes through a voltagetunable attenuator 共Mini-Circuits ZX73-2500, bandwidth
10 MHz–2.5 GHz兲, a 12 dB fixed attenuator 共JFW Industries
Inc. 50HF-012, bandwidth dc-18 GHz兲, and a fixed-gain amplifier 共Mini-Circuits ERA 5SM, bandwidth 60 kHz–4 GHz兲,
which, in combination, results in a tunable amplification of
the signal. The amplified signal passes through the nonlinearity, another fixed-gain amplifier 共Mini-Circuits ERA 3SM,
bandwidth 60 kHz–3 GHz兲 and is subsequently high-pass
filtered by a capacitor 共e.g., 10 pF兲. The filtered signal is then
inverted and split by a hybrid magic-tee 共M/A-COM H-9,
bandwidth 2 MHz–2 GHz兲. Half the signal power is used for
feedback and the other half is used as output. The output is
recorded with an oscilloscope 共Agilent Infiniium, with 50 ⍀
input impedance, 2.25 GHz bandwidth, and 8 GSa/ s兲. All
connections are made using SMA connectors and sufficient
isolation of dc and ac signals is provided by the wideband
rf-chokes 共MCL Model ADCH-80A兲 contained in the biasing
circuits for the amplifiers 共Mini-Circuits ERA-TB兲. The total
feedback-delay in our experiment is between 6 and 25 ns
and can be varied on a coarse scale by changing the length of
the coaxial-cable connecting the magic-tee and the attenuator
关see Fig. 2共a兲兴 or on a finer-scale by using a line stretcher
共Microlab/FXR SR-05N 500 MHz− 4 GHz兲 in the feedback
共not shown兲.
C. Band-limited feedback with variable gain
To determine the transfer characteristics of the feedback,
the coaxial cable that is attached to the output of the magic-
tee 关Fig. 2共a兲兴 is disconnected and a small-amplitude sinusoidal signal is injected into the cable. Simultaneous measurement of input and output signals for different frequencies
allows the determination of the transfer function 兩H 兩
= 兩Vout 兩 / 兩Vin兩. The tent-map-like nonlinearity does not adversely affect this measurement because small-amplitude input signals 共Vrms
in ⬍ 100 mV兲 only probe the “left” linear segment of the nonlinearity 关see Fig. 1共c兲兴.
The result of such a measurement of 兩H兩 is shown in Fig.
2共b兲 on a log-log scale for a setup with a 10 pF capacitor in
the feedback loop 共쎲兲 and for a setup without a capacitor in
the feedback loop 共⫻兲. In the latter case, the high-pass filtering is entirely due to the fact that the rf components in the
device are ac coupled. It is seen that the high-frequency cutoff is not affected by the inclusion of the 10 pF capacitor,
whereas the high-pass filtering due to the capacitor shifts the
low-frequency cutoff considerably.
To make contact with theory,21 we fit the data shown in
Fig. 2共b兲 to a model, where, for simplicity, the transfer characteristics of the feedback is approximated by a two-pole
bandpass filter,
兩H共兲兩 =
␦
冑共20 − 2兲2 + 2␦2 .
共2兲
Here, 0 denotes the frequency of maximal transmission and
␦ denotes the bandwidth. These parameters are given in
terms of the 3 dB cutoff frequencies + and − 共兩H共±兲兩2
= 1 / 2兲 by 20 = +− and ␦ = + − −. For the case of a 10 pF
capacitor in the feedback loop, the fit yields 0 ⬃ 236 MHz
and ␦ ⬃ 391 MHz 关− = 110 MHz, + = 502 MHz, solid line in
Figs. 2共b兲 and 2共c兲兴. For the case without an additional capacitor, 0 ⬃ 41 MHz and ␦ ⬃ 547 MHz is obtained 关−
= 3 MHz, + = 550 MHz, dashed line in Fig. 2共b兲兴. In fitting,
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033119-5
UHF chaos in a time-delay device
Chaos 16, 033119 共2006兲
we add more weight to data points with large values of 兩H兩
because a good fit of the peak of 兩H兩 is important for reproducing the observed dynamics.
Figures 2共b兲 and 2共c兲 show that a two-pole bandpass
filter description of the transfer function is not a perfect approximation of the measured transfer characteristics. Clearly,
allowing higher order filters would result in a better fit of the
data. However, a description by higher-order filters would
increase the model-complexity and we believe that the twopole-filter approximation is a good compromise between
model accuracy and model simplicity. Indeed, we will show
below that quantitative agreement of theory and experiment
is possible with this approximation.
The measured high-frequency 3 dB cutoff of
500– 600 MHz is mainly due to the band-limitations of the
nonlinearity 共Fig. 1兲, for which a 3 dB cutoff frequency of
⬃1 GHz was measured. The bandwidth is further suppressed
due to the combined effect of the nonideal behavior of the
remaining device components 关Fig. 2共a兲兴, all of which exhibit a slow roll-off of the transfer function for frequencies
below their 3 dB cutoff 共艌2 GHz兲. Higher bandwidths could
be achieved in a future design by reducing the number of
components and by implementing the nonlinearity as a microstrip circuit.
Another important parameter that characterizes the device is the feedback gain. It is varied, in the experiment, by
changing Vcon, the control voltage of the variable attenuator
关Fig. 2共a兲兴. The dependence of the small-signal feedback
gain on Vcon was determined using the same setup that was
used to measure 兩H兩, only this time the frequency of the input
signal is fixed at the frequency of maximal transmission 0
and Vcon is varied. What is measured in this way is the smallsignal effective gain at 0, denoted by b
b共Vcon兲 = − ␥共Vcon兲F⬘共0兲.
共3兲
Here, ␥共Vcon兲 is the total feedback gain 共non-negative兲, the
minus sign accounts for the signal inversion at the magic-tee,
and F⬘ is the slope of the nonlinearity. The measured mapping of Vcon to b is approximately linear for the control voltages used in the experiment 共2 – 4 V兲 and is given by
b共Vcon兲 = − 共Vcon − V0兲e−ᐉ
共4兲
−1
with  = 1.195± 0.002 V and V0 = 1.15± 0.02 V. The exponential factor on the right-hand side of Eq. 共4兲 takes into
account signal attenuation in the coaxial cable, where ᐉ is the
length of the RG-174/U cable and = 0.05 m−1.
III. MODEL AND THEORETICAL PREDICTIONS
In this section, we present a model and recall some theoretical results concerning steady-state bifurcations in delaysystem with band-limited feedback.
Dynamics arises in the device due to the interaction of
the nonlinearity and the linear time-delayed feedback. The
complexity of models for such delay-feedback systems increases as more details about the feedback characteristics are
included. For example, assuming a linear feedback of infinite
bandwidth results in map-models, whereas taking into
account the finite response time of device components
through a single-pole low-pass filter results in scalar delay-
differential equations.22,23 It has been shown that for rf devices, such as the one discussed in this paper, it is essential to
account for both the finite-bandwidth of the devicecomponents 共low pass filter兲 and the fact that rf components
are ac coupled 共high pass filter兲, in order to reproduce the
observed dynamics at least qualitatively.15,21,24 Thus, we
adopt a model consisting of an integro-differential equation
with time delay, where, for simplicity, it is assumed that the
transfer characteristics of the feedback can be approximated
by a two-pole bandpass filter:
v共t兲 +
v̇共t兲 20
+
⌬
⌬
冕
t
0
冋
dlv共l兲 = ␣F −
册
␥
v共t − 兲 .
␣
共5兲
In this equation, is the delay, is the bandpass filtered
signal that is measured in the experiment, the parameter 0
= 20 is the angular-frequency of maximal transmission,
and ⌬ = 2␦. For the case of a 10 pF capacitor in the feedback, 0 = 1.5⫻ 109 rad/ s and ⌬ = 2.5⫻ 109 rad/ s. The nonlinearity F is given by Eq. 共1兲 and the minus sign in the
argument of F in Eq. 共5兲 accounts for the inversion of the
feedback signal in the magic-tee. That is, the device has
negative feedback. The parameter ␥ denotes the total feedback gain, whereas ␣ ⬃ 9 is the fixed gain a signal experiences between the output of the nonlinearity and the output
of the magic-tee that is connected to the oscilloscope 关see
Fig. 2共a兲兴.
Of the model parameters, 0 and ⌬ are known, the delay
can be measured directly, and the total gain ␥ is determined
in terms of the attenuator control voltage Vcon by Eqs. 共3兲 and
共4兲 for the case of small amplitude dynamics. Thus, there are
no free-parameters in the model and, consequently, quantitative predictions are possible.
In a previous publication, we analyzed the steady-state
bifurcations for time-delay systems with band-limited feedback and arbitrary nonlinearity.21 This analysis applies directly to the device model Eq. 共5兲. The theory predicts that
oscillations arise through Hopf bifurcations of the steady
state and that the frequency of the oscillations at onset
roughly scale as = n / 共2兲, where n is an odd integer for
negative feedback. As a consequence of ac coupling, there
exist jumps in the value of as a function of the delay, a
feature that distinguishes bandpass feedback from low-pass
feedback 共Ikeda-type systems22兲. Furthermore, the theory
predicts that the stability boundary of the steady state shows
a lobe-like structure when plotted in the b − parameter
plane.
The Hopf-bifurcation-type can also be determined. To
that avail, note that the effective gain b is negative due to the
negative feedback 关Eq. 共3兲兴 and that the first and third derivative of the nonlinearity evaluated at the steady state satisfy F⬘F ⬍ 0 关Eq. 共1兲兴. We show in the Appendix 共Corollary
1兲 that, under these conditions, it follows from Illing and
Gauthier21 that the Hopf bifurcation is supercritical for delays satisfying
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033119-6
L. Illing and D. J. Gauthier
Chaos 16, 033119 共2006兲
FIG. 3. Characterization of the Hopf bifurcations for a setup with a 10 pF
capacitor. 共a兲 Supercritical Hopf bifurcation: oscillation amplitude as a function of the attenuator control voltage Vcon for = 10.4 ns. Increased Vcon
corresponds to increased 共negative兲 feedback gain. 共b兲 The measured dependence of the Hopf frequency on the time delay is shown 共쎲兲. The dashed
lines are the approximate n / 共2兲 scaling. The solid line is from theory 共see
text兲.
艌
C1 共2/3兲 1 ␦
=
共冑3 + 420/␦2 − 冑3兲,
12 20
2␦
共6兲
as long as 20 / ␦2 艋 6 holds. In particular, for the case of a
10 pF capacitor in the feedback loop, the model predicts the
Hopf bifurcations to be supercritical for ⬎ 0.22 ns, i.e., for
all experimentally accessible delays.
IV. HOPF BIFURCATIONS
In this section, we compare experiment and theoretical
predictions regarding the steady state bifurcation.
In experiment, we observe the dynamics of the device as
we vary the feedback gain while the delay is held fixed. The
feedback gain is increased by increasing the control voltage
Vcon of the tunable attenuator, i.e., by decreasing the attenuation. For small values of the feedback gain, the system resides in a steady state. As the gain crosses a critical value, a
supercritical Hopf bifurcation occurs and gives rise to periodic oscillations.
An example of such a measurement is shown in Fig.
3共a兲, where it is seen that the oscillation amplitude grows
smoothly as Vcon is increased beyond a critical value of
⬃2.1 V. In contrast, the measured oscillation frequency remains roughly constant 共 ⬃ 240 MHz兲. In combination,
these measurements indicate that the steady state loses sta-
bility through a supercritical Hopf bifurcation. We find that
the steady state becomes unstable through supercritical Hopf
bifurcations for all delays 共6 – 25 ns兲, in agreement with the
theoretical predictions.
An important advantage of chaos generators that use
band-limited feedback is the ability to tune the main time
scale of the device dynamics. In our device, the main time
scale can be varied by more than a factor of 10 simply by
changing the capacitor in the feedback loop 共see Fig. 2兲. For
example, at a fixed feedback delay of = 10.5 ns, the frequency is = 48 MHz for a setup without an additional capacitor in the feedback loop, = 238 MHz for a ⬃ 10 pF capacitor, and = 913 MHz for a ⬃ 2 pF capacitor. Here, we
are using the frequency of the oscillations at the Hopf bifurcation 共Hopf frequency兲 as a proxy for the main time scale.
Experimentally, the Hopf frequency is the quantity most
readily measured and therefore ideally suited for comparison
of experiment and theory. For all quantitative comparisons,
we use an experimental setup with a 10 pF capacitor in the
feedback. As a result, the main time scale of the dynamics is
⬃4 ns 共=−1
0 兲, which allows high-resolution measurements
in the time and frequency domain with the available equipment. Choosing 0 does not entirely fix the oscillation frequency at the Hopf bifurcation, which also depends on the
time delay, as is shown in Fig. 3共b兲. It is seen that jumps
occur between oscillation modes and that each mode scales
as = n / 共2兲 with n odd 共dashed lines兲, as predicted by the
theory. The solid line in Fig. 3共b兲 is the result from a linear
stability analysis of Eq. 共5兲.21 The agreement of theory and
experimental data 共쎲兲 in terms of the values of the Hopf
frequency and the location of the mode jumps is excellent.
Another useful way to compare experiment and theory is
to study the values of the feedback gain for which the steady
state loses stability. Experimentally, an indirect measure of
the critical feedback gain is obtained by measuring the attenuator control voltage Vcon at the bifurcation for different
delays . Figure 4共a兲 shows the result of such a measurement
where the region of stability is shaded and an indication of
the expected lobe-like structure of the stability boundary is
discernible.
Theoretically, it is possible to predict the values of the
feedback gain for which Hopf bifurcations occur by using
the fitted values of 0 and ⌬ and linear-stability analysis of
Eq. 共5兲.21 The result of such an analysis is shown in Fig.
4共b兲, where the solid line represents the critical values of the
effective gain b plotted as a function of the delay . For
comparison, we use Eq. 共4兲 to map the measured values of
Vcon onto the corresponding b values and find very good
quantitative agreement of theory and experiment.
However, we note that comparing the theory and experiment using frequency measurements is far more robust than
the comparison based on the critical feedback gain because
the frequency can be measured directly and with high precision whereas the effective gain b is determined indirectly
through measuring Vcon and using Eq. 共4兲. Indeed, the error
bars in Fig. 4共b兲 are mainly due to the ⬃2% uncertainty of
the fit parameter V0 in Eq. 共4兲. Another indication of the
precision to which the mapping of Vcon to b has to be known
is the need to include the exponential factor in Eq. 共4兲. This
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033119-7
UHF chaos in a time-delay device
Chaos 16, 033119 共2006兲
V. HIGH-DIMENSIONAL CHAOS
FIG. 4. Stability boundary of the steady state. 共a兲 Experiment: measured
attenuator control voltage Vcon vs the feedback-delay . 共b兲 Comparison of
experiment and theory: values of Vcon from panel 共a兲 are mapped onto the
effective slope b 共쎲兲. The line indicates the theoretical prediction.
factor takes into account the small changes in signal attenuation that are the consequence of being varied by changing
lengths of cable. Even these small changes in attenuation
have noticeable effects, such as the increase, on average, of
the critical value of Vcon for increased delays 关Fig. 4共a兲兴.
In this section we have compared the experimentally
measured device dynamics to analytic predictions based on a
parameter-free model consisting of a simple integrodifferential equation with time delay. On the level of bifurcations from the steady state, such a simple model turns out
to be sufficient to reproduce the observed dynamics quantitatively.
Beyond the Hopf bifurcation, successively more complex dynamics develop as the gain is increased. Experimental
results are presented in Fig. 5, where time traces are displayed in the first column and the corresponding spectra in
the second column. In the third column, we show the location where the measured trajectory crosses unidirectionally a
Poincaré plane in a three-dimensional delay-embedding
space with embedding-delay ⌬ = 1 ns. At feedback gains
slightly above the critical value, the oscillations are sinusoidal 关Fig. 5共a兲兴, resulting in a spectrum that is dominated
by the main frequency 关Fig. 5共b兲兴 and a single point in the
Poincaré plane 关Fig. 5共c兲兴. At higher feedback gains, the oscillations begin to square off and prominent odd harmonics
appear in the spectrum 共not shown兲. As the gain is increased
further, frequency components that are noncommensurate
with the fundamental frequency appear and grow in power,
resulting in a dense comb of frequencies in the spectrum.
This signature of quasiperiodic dynamics is clearly seen in
Fig. 5共e兲. The existence of a torus attractor is confirmed by
the appearance of a closed curve in the Poincaré section 关Fig.
5共f兲兴. For even larger gains, the broad background in the
spectrum rises and the power in the fundamental frequency
peak shrinks, as can be seen in Fig. 5共h兲. Furthermore, no
structure can be discerned in the Poincaré section. This is
indicative of high-dimensional chaos.
We use numeric simulations to show that model 共5兲 reproduces the observed dynamics of the device qualitatively
and we substantiate the claim of high-dimensional chaos by
demonstrating that the deterministic model is chaotic.
Figure 6 shows time series, power spectra, and Poincaré
sections obtained by numerical simulation25 of Eq. 共5兲 共compare with Fig. 5兲. We find sinusoidal oscillations 关Figs.
6共a兲–6共c兲兴 for ␥ = 2.8 共b = −1.01兲. This value of ␥ corresponds
to Vcon = 2.07 V and is obtained by using Eqs. 共3兲 and 共4兲.
For the large amplitude oscillations that arise in the experiment for Vcon = 2.6 V 共Vcon = 3.6 V兲 saturation effects start to
FIG. 5. Experimentally measured time series as well as
the corresponding power spectra and Poincaré sections
are shown for 共a兲–共c兲 Vcon = 2.07 V, 共d兲–共f兲 Vcon
= 2.6 V, and 共g兲–共i兲 Vcon = 3.6 V. The feedback loop
contained a ⬃10 pF capacitor, = 10.5 ns, and ⌬
= 1 ns.
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033119-8
L. Illing and D. J. Gauthier
Chaos 16, 033119 共2006兲
FIG. 6. Numerical time series, power spectra, and
Poincaré sections 共⌬ = 1 ns兲 are shown for 共a兲–共c兲 ␥
= 2.8, 共d兲–共f兲 ␥ = 3.7, 共g兲–共i兲 ␥ = 4.5 共DL ⬃ 13兲.
play a role and based on measurements we estimate the corresponding gain to be ␥ ⬃ 3.7 共␥ ⲏ 6兲. Increasing the feedback gain in the simulations to ␥ = 3.7 confirms that the limit
cycle bifurcates to a torus-attractor as shown in Fig. 6共f兲 by
the closed curve in the Poincaré section. The power spectrum
关Fig. 6共e兲兴 exhibits a comb-like structure due to the two incommensurate frequencies of the quasiperiodic oscillation.
For even larger gains the system is chaotic, with a very
broadband spectrum 关Fig. 6共h兲兴 and no discernible structure
in the Poincaré section 关Fig. 6共i兲兴. We computed Lyapunov
exponents for the model with ␥ = 4.5 and find that the largest
few exponents are positive.26 This proves that the model dynamics shown in Fig. 6共g兲 are on a chaotic attractor. We
estimate the attractor’s Lyapunov dimension to be DL ⬃ 13.
In this section, we have presented evidence that our device exhibits high-dimensional chaos. We have shown that
the device can be tuned from steady state behavior to quasiperiodic dynamics, and aperiodic oscillations with a broad
power spectrum. This behavior is reproduced by the deterministic model and we have quantified the dimensionality of
the chaotic attractor.
VI. SUMMARY AND DISCUSSION
Time delays are present and signals are bandpass filtered
in circuits that operate in the ultra-high frequency band and
consist of commercially available prepackaged rf components. Our approach is to exploit inherent time delays and
band-limiting component characteristics for the generation of
complex dynamics. We discuss in this paper a device resulting from this approach, an easy to implement all-electronic
generator of chaotic oscillations at GHz frequencies. The device has the advantage that the characteristic time scale and
the bandwidth of the chaotic output can be tuned by changing the properties of the feedback loop. Further positive aspects of the device are that it is built using commerciallyavailable components and that it can, in principle, be
operated at higher frequencies such as the 3 – 10 GHz band
in which the operation of ultra-wideband transmission systems is allowed.27
Comparison of the measured device dynamics to theory
shows that a simple integro-differential equation with delayed feedback 共equivalent to a second-order delaydifferential equation兲 captures many of the observed dynamic features. Indeed, good quantitative agreement is found
on the level of bifurcations from the steady state and the
observed quasiperiodic and chaotic behavior is qualitatively
reproduced by the model.
In this context, it is important to note that the device was
operated in a “weak” feedback regime for all data shown in
this paper. In the experimental device, the amplifiers will
exhibit saturation effects for strong feedback, which introduces additional feedback nonlinearities. In the strongfeedback regime, we observe chaotic oscillations with a
nearly featureless power spectrum 共not shown兲. In contrast,
the saturation-free model 关Eqs. 共1兲 and 共5兲兴 becomes unstable
for large values of the feedback parameter ␥ 共␥ ⲏ 5兲. However, omitting saturation effects in the model allows us to
show conclusively that high-dimensional chaos arises in the
weak-feedback regime due to the interaction of the linear
band-limited time-delayed feedback with the transistor-based
nonlinearity.
It is well known that dynamical systems with delayed
feedback often have very rich dynamics and bifurcation diagrams. This is also true for delay systems with band-limited
feedback. For example, we observe coexisting stable limit
cycles over wide parameter ranges, demonstrating the system’s multistability. Furthermore, periodic windows are
found within the quasiperiodic and chaotic regimes, which is
indicative of the complex bifurcation scenario in this system.
It would be interesting to investigate in more detail the bifurcations leading to quasiperiodicity and to the appearance
of periodic windows.
Numerical studies of time-delay systems suggest28 that
the Lyapunov dimension of the chaotic attractor grows with
increased delay while the metric entropy stays roughly con-
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033119-9
UHF chaos in a time-delay device
Chaos 16, 033119 共2006兲
stant. In our device, one would therefore expect to find lowdimensional chaotic behavior for small delays with a
Lyapunov spectrum that has few but large positive Lyapunov
exponents. This is confirmed in numerical simulations where
we find that, for a short delay of = 0.5 ns, the Lyapunov
dimension never rises above DL = 3.5 for all feedback
strengths where model 共5兲 is stable 共␥ ⬍ 7.8兲. Furthermore,
the Lyapunov spectrum has at most a single positive exponent with typical values of ⬃1 ns−1. As a comparison, six
positive exponents of size ⱗ10−2 ns−1 are found for the
large-delay case 共 = 10.5 ns, DL ⬃ 13兲 shown in Fig. 6共g兲.
Experimentally, we cannot access the short-delay regime
with our current setup, which was designed to provide independent control over system parameters such as feedbackgain, time-delay, and transistor bias-voltage. As a result, our
device consists of many components with the consequence
that ⲏ 6 ns. In a future setup, it should be possible to operate the circuit in the short-delay regime by reducing the
number of components and by combining them onto a single
microstrip circuit.
We believe that this chaotic electronic device is ideally
suited for both experimental investigation of the nonlinear
dynamics of time-delay systems and technological application of high-speed chaos such as ranging, intrusion detection,
tracking, and random number generation.
ACKNOWLEDGMENTS
We thank W. L. Ebenstein for helping us with the measurements. This work was supported by the U.S. Army Research Office Grant No. W911NF-05-1-0228.
APPENDIX: HOPF BIFURCATION TYPE
In a previous publication,21 we have derived simple criteria that determine whether the Hopf bifurcation of the
steady state is supercritical or subcritical. In our analysis we
focus on large-bandwidth systems, i.e., systems where the
quality factor Q 艋 1 / 2 共Q = 0 / ⌬兲. However, the rf device
that is discussed in the current paper produces Hopf bifurcations and chaos for both large and small bandwidths. As an
example, when there is no capacitor in the feedback loop the
quality factor is Q ⬃ 0.07, whereas Q ⬃ 0.6 with a 10 pF capacitor in the feedback loop. The purpose of this section is to
derive a simple condition for the Hopf bifurcation type that
is valid for Q ⬎ 1 / 2.
To that avail, we first show that Eq. 共5兲 is equivalent to
the second order delay-differential equation studied in Ref.
21. To do so, we introduce new variables x = −共␥ / ␣兲v and
y = −␥F关0兴 + 共20␥兲 / 共⌬␣兲兰t0v共l兲dl, rescale time through s = t⌬
共ˆ = ⌬兲, and define r = 20 / ⌬2 as well as f共x兲 = F共x兲 − F共0兲. As
a result, we obtain
ẋ共s兲 = − x共s兲 + y共s兲 − ␥ f关x共s − ˆ 兲兴,
ẏ共s兲 = − rx共s兲,
共A1兲
which is precisely the model discussed in Ref. 21.
The main result concerning the Hopf-bifurcation-type,
i.e. Proposition 2 in Ref. 21, is derived without using the
assumption r = Q2 艋 1 / 4. Therefore, the following holds:
Proposition 1: For r , ˆ 苸 R+ and n = 0 , 1 , 2 , . . . :
If f ⬘共0兲f 共0兲 + f ⬙共0兲2Cn共ˆ , r兲 ⬍ 0共⬎0兲, the Hopf bifurcation is supercritical (subcritical).
Here, Cn共ˆ , r兲 is a function that is given by Eq. 共40兲 in
Ref. 21 共the label n denotes the mode兲. Proposition 1 shifts
the difficulty of determining the Hopf-bifurcation-type to
knowing the value of Cn共ˆ , r兲 and it is clearly useful to derive conditions that determine the sign of Cn共ˆ , r兲. Inspection
of the proof of Proposition 4 in Ref. 21 reveals that the
following is true:
Proposition 2: For r , ˆ 苸 R+ and n = 0 , 1 , 2 , . . . :
If b 苸 关−2 , −1兴, Cn共ˆ , r兲 ⬍ 0; if b 苸 共1 , 2兴 and
ˆ ⬎ 共n兲 / 冑r, Cn共ˆ , r兲 ⬍ 0; and if b = 1, Cn共ˆ , r兲 = 0.
In terms of the experiment, we are mainly interested in
the Hopf bifurcations of the stable steady state, i.e., in the
bifurcation type at the stability boundary. A simple criterion
that guarantees supercritical bifurcations at the stability
boundary is obtained by determining the values of r for
which the stability boundary is contained in the interval 兩b兩
苸 关1 , 2兴 for all ˆ .
The stability boundary in the b − ˆ parameter plane is
composed of pieces of Hopf curves 关the parameterized
curves 兵bCn共·兲 , Cn共·兲其 are given by Eqs. 共14兲 and 共15兲 in Ref.
21兴. The steady state is stable for all delays, if 兩b兩 ⬍ 1. For
positive feedback 共b ⬎ 0兲, the maximum value of b for which
the steady state is stable is given by the point in the b − ˆ
parameter plane where the Hopf curves of mode n = 0 and
n = 2 intersect 共double Hopf point兲. As r increases, this point
moves to larger values of b and crosses b = 2 for r = 15/ 16.
Similarly, for negative feedback and ˆ 艌 C1 共2 / 3兲, the minimum b for which the steady state is stable is given by either
bC1 共2 / 3兲 = −2 or the point where the Hopf curves of modes
n = 1 and n = 3 intersect. As r increases, the double Hopf
point moves to smaller values of b and crosses b = −2 for r
= 6. Thus, we obtain the following simple criteria for the
Hopf-bifurcation-type 共extending Corollary 5 in Ref. 21兲:
Corollary 1: At the stability boundary of the steady state,
the following holds for the Hopf bifurcation: If b ⬍ 0, ˆ
艌 C1 共2 / 3兲, r 艋 6, and f ⬘共0兲f 共0兲 ⬍ 0, the bifurcation is supercritical. If b ⬎ 0, ˆ 艌 n / 冑r, r 艋 15/ 16, and f ⬘共0兲f 共0兲
⬍ 0, the bifurcation of the nth mode from the steady state is
supercritical.
关Note the typographical error in Ref. 21, Corollary 5,
point 共3兲, the condition should read 艌 C1 共2 / 3兲.兴
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