A GALLERY OF CHUA ATTRACTORS - Università della Calabria

Connecting Great Minds
World Scientific Series on Nonlinear Science, Series A – Vol. 61
A GALLERY OF CHUA ATTRACTORS
(With DVD-ROM)
is considered as one of the most important
Chaos concepts in modern science. It originally appeared
only in computer simulation (the famous Lorenz equation of 1963),
but this changed with the introduction of Chua’s oscillator (1986)
— a simple electronic circuit with the ability to generate a vast
range of chaotic behaviors. With Chua’s circuit, chaos became
a physical phenomenon, readily understood and represented in
mathematical language. Yet, even so, it is still difficult for the nonspecialist
to appreciate the full variety of behaviors that the system
can produce.
Eleonora Bilotta & Pietro Pantano
Università della Calabria, Italy
This book aims to bridge the gap. A gallery of nearly 900 “chaotic
attractors” — some generated by Chua’s physical circuit, the majority
through computer simulation of the circuit and its generalizations
— are illustrated as 3D color images, time series and fast Fourier
transform algorithms. For interested researchers, also presented is
the information necessary to replicate the behaviors and images.
Finally, how the fractal richness can be plied to artistic ends in
generating music and interesting sounds is shown; some examples
are included in the DVD-ROM which comes with the book.
The contents have also appeared in the International Journal of
Bifurcation and Chaos (2007).
Contents: Chua’s Oscillator and Its Generations; The Physical
Circuit; Dimensionless Equations; The Cubic Function; Single-Scroll
Systems; Multiscrolls Systems.
Readership: Advanced undergraduate and graduate students in
mathematics and physics, engineers, and non-experts interested
in the theory of complexity and chaos.
Key Features
• Includes amazing images of chaos for the expert and the nonspecialist
reader
• Contains new insights into the nature of chaos
• Introduces easy-to-perform experiments in chaotic behavior
• Serves as a new source of inspiration for electronic music and
art
608pp Jul 2008
978-981-279-062-0 US$155 £79
981-279-062-4
Preferred Publisher for Leading Thinkers

CONTENTS of A Gallery of Chua Attractors
Nonlinear Book Series A Vol.61
Chapter 1. Chua’s Oscillator and Its
Generalizations
1. Introduction
2. Chua Oscillator
3. Diode with Cubic Function
4. n-Scroll Attractors
5. From Chaos to Hyperchaos
6. A Gallery of Chua Attractors
7. Conclusions
Chapter 2. The Physical Circuit
1. Introduction
2. A Gallery of Attractors
3. Visualization Issues
4. Conclusions
Chapter 3. Dimensionless Equations
1. Introduction
2. Information Seeking in Chaos Domain
3. The Main Features of the Gallery
4. Representing Parameter Space
5. Conclusions
Chapter 4. The Cubic Equation
1. Introduction
2. A Gallery of Attractors
3. Representations of Parameter Space
4. Laws of Morphogenesis
5. Shape Distance in Phase Space
6. Conclusions
Chapter 5. Single-Scroll Systems
1. Introduction
2. The Gallery of Attractors
3. Representing Attractors in Parameter Space
4. Visualizing the Parameter Space: The Inertial Ellipsoid Method
5. Visualizing Parameter Space: The Hausdorff Distance Method
6. Conclusions
Chapter 6. Multiscrolls Systems
1. Introduction
2. Formal Aspects of n-Scroll, Hyper-chaotic and Synchronized
Systems
3. The Gallery
4. Computational Tools
5Th 5. The
Virtual Museum and the Navigable Galleries
6. Conclusions

CD Contents
Together with the book, readers will find an audio CD with thirteen pieces, obtained by
editing the original music generated by Chua’s circuit. In what follows, we provide additional
details on the edited pieces.
Track 1
This piece was produced from a selection
of tunes produced by the Chua attractor.
Track 2
Here a single time series is subjected to
a sequence of transformations, each of
which doubles the previous tempo.
Track 3
This piece explores the possibility of
creating traditional musical forms, using
material produced by the Chua attractor.
Track 4
This piece is based on two different time
series representing similar melodies
which have both been produced in the
same way.
Track 5
Here, the three coordinates of the Chua
attractor are used to code the pitch,
volume and duration of the notes.
Track 6
This piece attempts to recreate classical
feelings of “lightness”. A single time
series is subjected to various time
transformations, with each version being
played on a different instrument.
Track 7
The raw material for this piece was
produced by filtering the data from the
attractor to produce scales in different
keys.
Track 8
Different parts were obtained by
transforming note durations in a single
original time series.
Track 9
In this piece, the aim was to create musical
structures similar to those we find in
serious music from the beginning of the
XX century.
Track 10
This piece uses three distinct time series to
create a complex orchestrated structure.
Whenever the values lie within a specific
range they are assigned to specific
instruments.
Track 11
This piece presents the chaotic
expressiveness of a Chua attractor, without
any arrangement required by the user. The
three time-series of the system play three
different musical instruments.
Track 12
This fragment shows the expressive
capabilities of the circuit without any
external arrangement. Sample times are
very narrow, causing a perceptive cohesion
and the creation of new timbres.
Track 13
This is a collection of many pieces
obtained from different Chua attractors
and using different codification processes.
In each composition, the three time-series
of the system play three different musical
instruments.

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