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Journal ArticleDOI

Parameter space of experimental chaotic circuits with high-precision control parameters

TL;DR: High-resolution measurements are reported that experimentally confirm a spiral cascade structure and a scaling relationship of shrimps in the Chua's circuit and confirm that their sizes decay exponentially with the period of the attractor, a result expected to be found in maps of the quadratic family.
Abstract: We report high-resolution measurements that experimentally confirm a spiral cascade structure and a scaling relationship of shrimps in the Chua's circuit. Circuits constructed using this component allow for a comprehensive characterization of the circuit behaviors through high resolution parameter spaces. To illustrate the power of our technological development for the creation and the study of chaotic circuits, we constructed a Chua circuit and study its high resolution parameter space. The reliability and stability of the designed component allowed us to obtain data for long periods of time (∼21 weeks), a data set from which an accurate estimation of Lyapunov exponents for the circuit characterization was possible. Moreover, this data, rigorously characterized by the Lyapunov exponents, allows us to reassure experimentally that the shrimps, stable islands embedded in a domain of chaos in the parameter spaces, can be observed in the laboratory. Finally, we confirm that their sizes decay exponentially with the period of the attractor, a result expected to be found in maps of the quadratic family.

Summary (2 min read)

Introduction

  • These factors are behind their motivation to propose an approach to obtain experimental parameter spaces that are not only reliable, autonomous, and reproducible but that can also reproduce nominal parameter values considered in numerical experiments.
  • With these potentiometers and a set of resistors, switches, and relays, the authors were able to autonomously obtain a high resolution parameter space of the Chua’s circuit (with resolutions of 400 562 and 1023 126 points, varying one resistor with step sizes of 0.200001 X and another with 0.10022 X), which could remarkably reproduce the numerically obtained parameter spaces.
  • In particular, the authors showed by calculating the Lyapunov exponents numerically and experimentally that this parameter space presents self-similar periodic structures, the shrimps, embedded in a domain of chaos.

II. EXPERIMENTAL AND NUMERICAL ASPECTS

  • Let us start describing the digital potentiometer.
  • The 10 pin left connector stands for the digital data coming from input/ output (I/O) digital ports of the DAQ board.
  • The five-fold piecewise linear element that provides the nonlinear character of the Chua’s circuit consists of two operational amplifiers and the resistances R1 to R6.
  • For each time series, the potentiometers R and rL were switched by a LabView routine with values previously determined and calibrated to give precise equivalent steps.
  • By starting at parameters leading to such attractors provides results as if the system was never switched off.

III. RESULTS AND DISCUSSION

  • The parameter space in Fig. 4(a) shows by colors the values of the largest Lyapunov exponent k, calculated by the method of Sano and Sawada25 from the 400 562 experimental time series with R and rL as the control parameters.
  • The simulated parameter space in Fig. 4(b) considered also the values of k obtained from time series generated by 1600 values of R and 562 values of rL in the same range of the experimental data.
  • The authors estimated from the measured time series that this noise is between 1 mV and 2 mV.
  • This result led us to conclude that their experimentally and numerically obtained periodic windows have a complex structure as expected in Refs. 1, 2, 9, and 27.
  • Fitting results in Fig. 7 indicate that the decay exponents, b ¼ 0:3060:04 (experimental results) and b ¼ 0:3960:09 (numerical results), are in the same order of magnitude of the largest positive Lyapunov exponent of the chaotic attractor in the chaotic regions surrounding the shrimps, as expected.

IV. CONCLUSIONS

  • The authors have successfully built autonomous, reliable, and reproducible digital potentiometers that allowed precise measurements of the set of experimental physical parameters of electronic circuits.
  • As an application of the power of their component, the authors obtained high-resolution experimental parameter spaces of Chua’s circuit that is remarkably similar to the one obtained by simulations using the same set of physical parameters values.
  • To the best of their knowledge, this work provides, for the first time, experimental Lyapunov exponent parameter spaces of a spiral cascade structure of several shrimps of electronic circuits.
  • Also, with respect to simulations, the authors show that considering features such as 5-fold idðxÞ and careful consideration of its equations drawn directly from measurements, it was performed simulations with results close to the ones measured experimentally with regard to the shape of the periodic structures, its exponential decay law, as well as the overall range of Lyapunov exponent values.
  • The very high precision and stability on the resistance steps improved the definition of periodic structure borders when compared with other methods of parameter variation.

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Figures (8)

Content maybe subject to copyright    Report

Parameter space of experimental chaotic circuits with high-precision control
parameters
Francisco F. G. de Sousa, Rero M. Rubinger, José C. Sartorelli
,
, Holokx A. Albuquerque, and Murilo S. Baptista
Citation: Chaos 26, 083107 (2016); doi: 10.1063/1.4960582
View online: http://dx.doi.org/10.1063/1.4960582
View Table of Contents: http://aip.scitation.org/toc/cha/26/8
Published by the American Institute of Physics

Parameter space of experimental chaotic circuits with high-precision control
parameters
Francisco F. G. de Sousa,
1
Rero M. Rubinger,
1
Jos
eC.Sartorelli,
2,a)
Holokx A. Albuquerque,
3
and Murilo S. Baptista
4
1
Instituto de F
ısica e Qu
ımica, Universidade Federal de Itajub
a, Itajub
a, MG, Brazil
2
Universidade de S
~
ao Paulo, S
~
ao Paulo, SP, Brazil
3
Departamento de F
ısica, Universidade do Estado de Santa Catarina, Joinville, SC, Brazil
4
Institute of Complex Systems and Mathematical Biology, SUPA, University of Aberdeen, Aberdeen,
United Kingdom
(Received 23 March 2016; accepted 25 July 2016; published online 8 August 2016)
We report high-resolution measurements that experimentally confirm a spiral cascade structure and
a scaling relationship of shrimps in the Chua’s circuit. Circuits constructed using this component
allow for a comprehensive characterization of the circuit behaviors through high resolution
parameter spaces. To illustrate the power of our technological development for the creation and the
study of chaotic circuits, we constructed a Chua circuit and study its high resolution parameter
space. The reliability and stability of the designed component allowed us to obtain data for long
periods of time (21 weeks), a data set from which an accurate estimation of Lyapunov exponents
for the circuit characterization was possi ble. Moreover, this data, rigorously characterized by
the Lyapunov exponents, allows us to reassure experimentally that the shrimps, stable islands
embedded in a domain of chaos in the parameter spaces, can be observed in the laboratory. Finally,
we confirm that their sizes decay exponentially with the period of the attractor, a result expected to
be found in maps of the quadratic family. Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4960582]
Electronic circuits provide a simple alternative to test
in the laboratory theoretical approaches developed to
characterize more complex systems. So far, however,
experiments were being carried out in circuits with
course-grained parameter values. In this work, we pre-
sent a novel electronic architecture for a potentiometer
that permits fine variations in control parameters. To
demonstrate the usefulness of this potentiometer to the
behavioral analysis of electronic circuits, we make very
long time-series measurements of this circuit for a fine
variation of its parameters and experimentally report,
for the first time, that stable islands embedded in a
domain of chaos in the parameter spaces indeed have
sizes that decay exponentially with the period of the
attractor. Periodicity with high period, confirmed by the
calculation of the Lyapunov exponents, thus requires fine
tuning of parameters to be experimentally observed.
I. INTRODUCTION
Only a few chaotic circuits have periodicity high resolu-
tion parameter spaces experimentally obtained.
17
The rele-
vance of studying parameter spaces of nonlinear systems is
that it allows us to understand how periodic behavior, chaos,
and bifurcations come about in a nonlinear system. In fact,
parameters leading to the different behaviors are strongly cor-
related. Chaotic and periodic regions appear side by side in
all scales in universal shapes and forms. Gallas
8
numerically
observed periodic structures embedded in parameter chaotic
regions, in the parameter space of the H
enon map. For some
classes of nonlinear systems, such as the one studied here, the
periodic structures appear aligned along spiral curves describ-
ing parameters for saddle-node bifurcations and super-stable
behavior and that cross transversely parameter curves con-
taining homoclinic bifurcations.
914
Experimentally, the difficulty in obtaining parameter
spaces resides in a reliable method to vary precisely a param-
eter, usually a resistance, in a controlled, autonomous, and
reproducible fashion. Parameters are not constant and suffer
time varying alterations. This factor becomes even more
severe when the experiment is done over long time spans.
Finally, the numer ical resolution of the parameters and their
nominal values cannot be achieved or reproduced experi-
mentally. Even very simple nonlinear electronic systems
cannot have their behaviors reproduced numerically; the
main reason is that the electronic components have non-ideal
characteristic curves. These factors are behind our motiva-
tion to propose an approach to obtain experimental parame-
ter spaces that are not only reliable, autonomous, and
reproducible but that can also reproduce nominal parameter
values considered in numerical experiments. High resolution
parameter space
7
allows one to reproduce experimentally the
self-similar topological character observed in numerically
obtained parameter spaces.
Previous works have proposed different strategies to
construct experimental parameter spaces. Maranh
~
ao et al.
considered a manual or step motor control of precision
multi-turn potentiometer.
1
Stoop et al.
5,6
used a proto-board
a)
Electronic mail: sartorelli@if.usp.br
1054-1500/2016/26(8)/083107/7/$30.00 Published by AIP Publishing.26, 083107-1
CHAOS 26, 083107 (2016)

to do the experiment and varied by hand 490 162 values of
a negative resistor and an inductor to produce parameter
spaces. In Ref. 2, a Keithley power source controlled by
LabView
V
R
was used as one parameter and a precision poten-
tiometer manually controlled as the second parameter. The
use of a digital potentiometer would provide a reliable,
autonomous, and reproducible way to obtain parameter
spaces. However, the available digital potentiometers on the
market are limited to resistances above 1 kX and usually not
more than 256 steps are possible. In addition, their control is
not easy to carry out with typical data acquisition (DAQ)
systems or LabView
V
R
. The novelty in our experimentally
obtained parameter spaces is that they are constructed by
calculating the spectrum of Lyapunov exponents from recon-
structed attractors, considering long-time series of measured
trajectories.
The technological novelty presented is the design of a dig-
ital potentiometer with precisely calibrated small resistance
steps(aslowas0.10022X) that allows 1024 steps (or even
more) to change the resistance. With these potentiometers and
a set of resistors, switches, and relays, we were able to autono-
mously obtain a high resolution parameter space of the Chua’s
circuit (with resolutions of 400 562 and 1023 126 points,
varying one resistor with step sizes of 0.200001 X and another
with 0.10022 X), which could remarkably reproduce the
numerically obtained parameter spaces. This new electronic
component allowed us to carry out a detailed experimental
investigation of the parameter space of a modified Chua’s cir-
cuit, namely, we have characterized this circuit by varying the
resistance (R) and the inductor resistance (r
L
).
15,16
Among
other accomplishments, we have demonstrated that even
higher period periodic windows and the complex topological
structure of the scenario for the appearance of periodic
behaviors can be observed experimentally. Simulations, con-
sidering a normalized equation set that models the Chua’s cir-
cuit and also the normalized version of the experimental i(V)
curve, were carried out in order to demonstrate that the occur-
rence of periodic structures observed in the high-resolution
experimental parameter space could also be numerically
observed. In particular, we showed by calculating the
Lyapunov exponents numerically and experimentally that this
parameter space presents self-similar periodic structures, the
shrimps, embedded in a domain of chaos.
18,10,11,13,14
We also
show experimentally that those self-similar periodic regions
organize themselves in period-adding bifurcation cascades,
and whose sizes decrease exponentially as their period
grows.
1,9,17,18
We also report on malformed shrimps on the
experimental parameter space, result of tiny nonlinear devia-
tions close to the junction of two linear parts from a symmetric
piecewise linear i(V) curve.
We have considered Chua’s circuit,
19
with 5 linear parts,
to perform our study experimentally and numerically
because this circuit has been studied in many applications
such as in chaos control,
20,21
synchronization
22,23
and others,
but the use of a higher precision potentiometers here pro-
posed can be used to characterize, study, and precisely con-
trol the behavior of any electronic equipment.
II. EXPERIMENTAL AND NUMERICAL ASPECTS
Let us start describing the digital potentiometer. Its dia-
gram is presented in Fig. 1, but only 3 of the set of 10 resis-
tors in series are shown to illustrate its structure. The 10 pin
left connector stands for the digital data coming from input/
output (I/O) digital ports of the DAQ board. A 60 Ah 12.0 V
car battery was used to drive the digital potentiometers. The
digital potentiometer idea was captured from the structure of
FIG. 1. Schematics of the designed
adjustable digital potentiometer. Only
three of ten resistor circuits of the in
series association are shown in order to
clearly present their components. On
the right connector, t1 and t2 represent
the output resistance to be connected
to the Chua’s circuit. The circuit is
feed by 60 Ah 12.0 V car battery. See
text for component and respective
function description.
083107-2 de Sousa et al. Chaos 26, 083107 (2016)

a switched resistor digital to analog converter which contain s
a parallel resistor network. The calibration was done using a
Keithley digital multimeter model 2001 in the four wire
mode, i.e., in order to subtract leads resistance. The relays of
the series association were switched by transistor driver cir-
cuits connected to I/O digital ports of the data acquisition
board used for control and data acquisition. Thus, it was pos-
sible to write a ten digit binary number in order to select one
of the 1024 possible combinations. The digital potentiometer
is used to provide the resistances R and r
L
in the circuit,
which take up values R ¼ r and r
L
¼ r by following the
equation:
r ¼ step ðbit0 2
0
þ bit1 2
1
þþbit 9 2
9
Þ: (1)
For R we used two-step values: 0.20001 X and 2.00002 X
and for r
L
just one value for step, i.e., 0.10022 X.
The Chua’s circuit scheme is presented in Fig. 2,
constructed in a single face circuit board with the same
scheme of Ref. 24, i.e., C
1
¼ 23:50 nF, C
2
¼ 235:0 nF, and
L ¼ 42.30 mH. Thes e values wer e obtained from the co mbi -
nation of passive commercial available components and
measured with a Keithley digital multimeter model 2001 or
an impedance analyzer for the reactive components. The
measurement of component s allowed the choice of the clos-
est possible values to components, better than the factory
precision. We evaluate the oscillation main frequency as
a rough approximation by 1=ð2pðLC
2
Þ
1=2
Þ which gives
1596 Hz. Further increase on frequency, i.e., by reducing
passive component values, seems to destroy periodic struc-
tures that are observed in this circuit, thus this oscillation
frequency was the best choice. It was built with TL084
Operational Amplifiers (OPAMPs) and was fed by two
12.0 V, 7 Ah no-break batteries.
The five-fol d piecewise linear element that provides the
nonlinear character of the Chua’s circuit consists of two
operational amplifiers (OPAMP) and the r esistances R
1
to
R
6
.Itsið V
C1
Þ characteristic curve was defined and normal-
ized by the scheme x ¼ V
C1
=B
P
and i
d
ðxÞ¼iðxÞ=ðm
0
B
P
Þ
with m
0
¼4:156315 mS and B
P
¼ 1.38501 V. Here, S
stands for the inverse resistance unity. This curve is pre-
sented in Fig. 3, with 5-fold linear fittings used for simula-
tions with the signi ficant digits limited by the fitting
accuracy, given by
i
d
ðxÞ
¼
32:51240 4:76600xx< 5:43000;
x 0:82999 5:43000 x <1:00000;
1:84957x jxj1:00000;
x þ 0:86378 1:00000 < x 5:92900;
37:35590 5:15200xx> 5:92900:
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
(2)
The electronic inductor is defined by two OPAMPs con-
nected to the resistors R
7
, R
8
, R
9
, R
10
, and r
L
together with
the capacitor C
3
. This is a gyrator circuit with inductance as
L ¼ðC
3
R
7
R
9
R
10
Þ=R
8
.
The points in the circuit where we take measurements
can be seen in Fig. 2 by the probe points x, y, and P. They
correspond to the voltages across the capacitors C
1
and C
2
and the third variable, the current I
L
, is obtained from the
relation I
L
¼ðV
P
V
C2
Þ=ðR
7
þ r
L
Þ, where V
P
is the voltage
indicated by the P probe. The voltage across C
1
, passed by a
simple OPAMP buffer, was measured by a National
FIG. 2. Chua’s circuit using the elec-
tronic inductor and indicating the mea-
suring points x, y and the P point. The
current through the inductor is defined
as I
L
¼ðV
P
V
C2
Þ=ðR
7
þ r
L
Þ. Here,
V
ccþ
¼þ12 V and V
cc
¼12 V.
FIG. 3. i
d
ðxÞ characteristic of the circuit presented in Fig. 2. The linear fit-
tings are presented in the figure. The equations corresponding to linear fit-
tings are presented in the experimental section. The normalization
considered equations x ¼ V
C1
=B
P
and i
d
ðxÞ¼iðxÞ=ðm
o
B
P
Þ and the parame-
ters m
o
¼4:156315 mS and B
P
¼ 1.38501 V.
083107-3 de Sousa et al. Chaos 26, 083107 (2016)

Instruments data acquisition (DAQ) interface, model PCI-
6259 with 16 bit resolution, maximu m sampling rate of 1.25
Msamples/s, for data storage. Also, LabView was used to
data acquisition and analysis.
2,3
A Keithley 2400 voltage/
current source in series with the Chua’s diode was applied to
obtain the i(V) data.
For each time series, the potentiometers R and r
L
were
switched by a LabView routine with values previously deter-
mined and calibrated to give precise equivalent steps. After
calibration, the 1024 values of each potentiometer were
tested with a linear fitting, giving slopes equal to steps up to
four significant digits. We have carried out experiments with
the Chua’s circuit recollecting time series for the calculation
of the Lyapunov exponents. Time series were generated with
a 50 Ksamples/s and a 7 s length. As transient, after setting
the pair of parameters R and r
L
, a 50 s waiting time was con-
sidered. It is worth commenting that data acquisition to
obtain 562 400 meshes of time series lasted 21 weeks.
The fact that the experimental circuit could reproduce
many relevant structures obtained numerically implies that
the general bifurcation scenario of these periodic windows of
the parameter space is robust to external perturbations and
should be expected to be observed in nature. The circuit is
not switched off between subsequent measurements, only at
the highest R value when recharge was necessary. Each
restart of the system has changed the attractor initial condi-
tions. However, because the parameters were varied from
high R to lower R values and from low r
L
to higher r
L
values,
the trajectory at restart was always going towards the same
fixed points or simple periodic orbits. By starting at param e-
ters leading to such attractors provides results as if the sys-
tem was never switched off. Thus, this form of covering the
parameter space allows best reproducibility since the next
attractor for a renewed set of parameters has its initial condi-
tion from the state of the circuit set with parameters very
close to those parameters. This allows the experiments and
the simulations to be performed without any experimental or
numerical discontinuity in the state variables.
Experimental characteristic curves of the Chua’s circuit
are often asymmetric. However, some authors have done
simulations by considering a symmetric piecewise-linear
function. For us, successful reproduction of the experimental
parameter spaces is also a consequence of the fact that our
correspondent simulations were perfo rmed by considering
the non-symmetric i
d
ðxÞ give in Eq. (2) to integrate Chua’s
differential equations. We have considered the same set of
differential equations and normalized parameters presented
in full detail in Ref. 14.
III. RESULTS AND DISCUSSION
The parameter space in Fig. 4(a) shows by colors the
values of the largest Lyapunov exponent k, calculated by the
method of Sano and Sawada
25
from the 400 562 experi-
mental time series with R and r
L
as the control parameters.
The simulated parameter space in Fig. 4(b) considered also
the values of k obtained from time series generated by 1600
values of R and 562 values of r
L
in the same range of the
experimental data. In the case of simulations, k was obtained
from the tangent space method, and the values are in units
of integration step. This form of calculating k allows faster
simulations but produces absolute values of the exponents dis-
tinct from the corresponding experimental values. The time
unit of the experimental Lyapunov exponent was rescaled to
match the exponents from numerical data. A larger range
parameter space is presented in Fig. 4(a).InFig.4(b),we
present the correspondent simulations, generated with a
0.10022 X step, as described in Section II. The color scales
were defined for the experimental data with a smooth color
variation from white to black to represent the range of values
of k,withk < 0, corresponding to fixed point and periodic
attractors, and from black to red to represent the range of
values k 0; 0:2. The transition between periodic to chaotic
orbits occurring through saddle-node bifurcations or through
a period doubling cascade is characterized by the shift of
colors between yellow and orange. In both parameter spaces
of Fig. 4, it is possible to identify the occurrence of complex
periodic structures embedded in a chaotic domain and orga-
nized in a spiral structure.
8,13,26
We can see, in Fig. 5, high-resolution amplifications of
the inner region of the red boxes shown in Fig. 4. Figure 5(a)
shows the experimental data, obtained with a 0.20001 X res-
olution in R. This parameter space has ten times the R reso-
lution as compared with that used in Fig. 4(a). In Fig. 5(b),
we present a parameter space constructed by considering a
600 600 mesh of values for R and r
L
for the corresponding
FIG. 4. Lyapunov parameter spaces of the Chua’s circuit. White to black
stands for periodic orbits and for fixed points; yellow to red color for chaotic
orbits. (a) Experimental parameter space diagram associating color scale to
k. Resolution of parameters R and r
L
is 2 X and 0.1 X, respectively, and we
have considered a mesh with 400 values for R and 562 values for r
L
. (b)
Corresponding simulated parameter space obtained from using the model of
Ref. 15 but with a 5-fold piece-wise i
d
ðxÞ Eq. (2). Resolution of parameters
R and r
L
is 0.5 X and 0.1 X, respectively, and we have considered a mesh
with 1600 values for R and 562 values for r
L
.
083107-4 de Sousa et al. Chaos 26, 083107 (2016)

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Cites background or methods from "Parameter space of experimental cha..."

  • ...In some systems these structures organize themselves with specific bifurcation cascades, for example, period-adding cascades, and along preferred directions on the parameter-space [Gallas, 2015; de Sousa et al., 2016; Gallas, 2016]....

    [...]

  • ...We have chosen vary bias [Rocha & Medrano-T., 2009; Medrano-T & Rocha, 2014] instead of resistors [de Sousa et al., 2016; Tahir et al., 2016; Viana et al., 2010, 2012], which are commonly used in circuitry implementations of dynamical systems, for a better fine tuning that we can obtain in the…...

    [...]

  • ...…chaotic dynamics can arise [Strogatz, 2001] and a recent interest is on the description of its dynamics in parameter-planes simultaneously varying two control parameters [Celestino et al., 2011, 2014; Gallas, 2015; Meucci et al., 2016; de Sousa et al., 2016; da Costa et al., 2016; Gallas, 2016]....

    [...]

  • ...For example, these SPSs are observed in piecewise-linear [de Sousa et al., 2016; Hoff et al., 2014], polynomial [Celestino et al., 2014; Silva et al., 2015; Gallas, 2016], and exponential [Cardoso et al., 2009] nonlinearities....

    [...]

  • ...As far as we know, commercial or home-made (as the digital [de Sousa et al., 2016]) potentiometers, are noise sources in circuitry implementations, and in nonlinear dynamical systems, noise signal with a large enough strength may perturb the periodic attractors [Viana et al., 2010, 2012], leading…...

    [...]

Journal ArticleDOI
TL;DR: In this article, the effect of applying an asymmetric periodic continuous-feedback signal across one of the capacitors in the classical Chua's circuit model is examined, and the results indicate that the methodology used here will have a wide spectrum of applications in generic nonlinear complex dynamical systems presenting multistability, related to both theoretical and experimental issues.
Abstract: The effect of applying an asymmetric periodic continuous-feedback signal F(x), which is a function of voltage x, across one of the capacitors in the classical Chua’s circuit model is examined. We have performed a numerical investigation on the dynamics of the Chua’s circuit model to discuss a procedure of distorting, suppressing and moving attractors and stable periodic structures (SPSs) in phase and parameter spaces, respectively, by accordingly choosing the control parameters in the function F(x). Increasing the asymmetry strength in F(x), we are able to track and to move overlapped SPSs related to multistability phenomenon in phase space, and to suppress SPSs related to one single attractor. Our results indicate that the methodology used here will have a wide spectrum of applications in generic nonlinear complex dynamical systems presenting multistability, related to both theoretical and experimental issues.

9 citations

References
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Journal ArticleDOI
TL;DR: A new method is proposed to determine the spectrum of several Lyapunov exponents (including positive, zero, and even negative ones) from the observed time series of a single variable.
Abstract: The exponential divergence or convergence of nearby trajectories (Lyapunov exponents) is conceptually the most basic indicator of deterministic chaos. We propose a new method to determine the spectrum of several Lyapunov exponents (including positive, zero, and even negative ones) from the observed time series of a single variable. We have applied the method to various known model systems and also to the Rayleigh-B\'enard experiment, and have elucidated the dependence of the Lyapunov exponents on the Rayleigh number.

903 citations

Journal ArticleDOI
TL;DR: A new chaotic system of three-dimensional quadratic autonomous ordinary differential equations, which can display two 1-scroll chaotic attractors simultaneously, with only three equilibria, is introduced, of particular interest is the fact that this chaotic system can generate a complex 4- scroll chaotic attractor or confine two attractors to a 2-scroll Chaos attractor under the control of a simple constant input.
Abstract: This article introduces a new chaotic system of three-dimensional quadratic autonomous ordinary differential equations, which can display (i) two 1-scroll chaotic attractors simultaneously, with only three equilibria, and (ii) two 2-scroll chaotic attractors simultaneously, with five equilibria. Several issues such as some basic dynamical behaviors, routes to chaos, bifurcations, periodic windows, and the compound structure of the new chaotic system are then investigated, either analytically or numerically. Of particular interest is the fact that this chaotic system can generate a complex 4-scroll chaotic attractor or confine two attractors to a 2-scroll chaotic attractor under the control of a simple constant input. Furthermore, the concept of generalized Lorenz system is extended to a new class of generalized Lorenz-like systems in a canonical form. Finally, the important problems of classification and normal form of three-dimensional quadratic autonomous chaotic systems are formulated and discussed.

286 citations

Journal ArticleDOI
TL;DR: The parameter space of the Henon map is reported to contain a regular structure-parallel-to-structure sequence of shrimp-shaped robust isoperiodic domains that appear densely concentrated on a neighborhood along a main α direction, extending across both orientation-preserving and -reversing domains.
Abstract: The parameter space of the H\'enon map is reported to contain a regular structure-parallel-to-structure sequence of shrimp-shaped robust isoperiodic domains. They appear densely concentrated on a neighborhood along a main \ensuremath{\alpha} direction, extending across both orientation-preserving and -reversing domains. There is also a secondary \ensuremath{\beta} direction, roughly perpendicular to a very dense ``foliation of legs'' emanating from the isoperiodic domains. Familiar bifurcation phenomena observed in unimodal maps correspond to particular cuts along the \ensuremath{\beta} direction. The \ensuremath{\alpha} direction is rich in new phenomena. The topology along \ensuremath{\alpha} is conjectured to be typical of bimodal maps.

239 citations

Journal ArticleDOI
TL;DR: An overview of timely publications on Chua’s circuit attempts to present an overview of these timely publications, almost all within the last six months, and to identify four milestones of this very active research area.
Abstract: More than 200 papers, two special issues (Journal of Circuits, Systems, and Computers, March, June, 1993, and IEEE Trans. on Circuits and Systems, vol. 40, no. 10, October, 1993), an International Workshop on Chua’s Circuit: chaotic phenomena and applica tions at NOLTA’93, and a book (edited by R.N. Madan, World Scientific, 1993) on Chua’s circuit have been published since its inception a decade ago. This review paper attempts to present an overview of these timely publications, almost all within the last six months, and to identify four milestones of this very active research area. An important milestone is the recent fabrication of a monolithic Chua’s circuit. The robustness of this IC chip demonstrates that an array of Chua’s circuits can also be fabricated into a monolithic chip, thereby opening the floodgate to many unconventional applications in information technology, synergetics, and even music. The second milestone is the recent global unfolding of Chua’s circuit, obtained by adding a linear resistor in series with the inductor to obtain a canonical Chua’s circuit— now generally referred to as Chua’s oscillator. This circuit is most significant because it is structurally the simplest (it contains only 6 circuit elements) but dynamically the most complex among all nonlinear circuits and systems described by a 21-parameter family of continuous odd-symmetric piecewise-linear vector fields. The third milestone is the recent discovery of several important new phenomena in Chua’s circuits, e.g., stochastic resonance, chaos-chaos type intermittency, 1/f noise spectrum, etc. These new phenomena could have far-reaching theoretical and practical significance. The fourth milestone is the theoretical and experimental demonstration that Chua’s circuit can be easily controlled from a chaotic regime to a prescribed periodic or constant orbit, or it can be synchronized with 2 or more identical Chua’s circuits, operating in an oscillatory, or a chaotic regime. These recent breakthroughs have ushered in a new era where chaos is deliberately created and exploited for unconventional applications, e.g. secure communication.

187 citations

Journal ArticleDOI
TL;DR: Based on a suitable separation of chaotic systems, Lyapunov stability theory and matrix measure, the complete synchronization and anti-synchronization for chaotic systems are investigated in this paper.

127 citations

Frequently Asked Questions (1)
Q1. What contributions have the authors mentioned in the paper "Parameter space of experimental chaotic circuits with high-precision control parameters" ?

In this paper, the authors used two digital potentiometers to obtain high-resolution experimental parameter spaces of Chua 's circuit that is remarkably similar to the one obtained by simulations using the same set of physical parameters values.