Experimental Realization and Synchronization of a Chua Circuit Daniella Masante. Department of Physics. University of California, Davis. Physics Building, 1 Shields Avenue, Davis, CA 95616 USA. [email protected]Abstract A Chua circuit, known to exhibit chaotic behavior, has been constructed and studied. The circuit was built with a potentiometer as a control resistor, and chaotic dynamics such as limit cycles and attractors were measured and compared to a numerical simulation. The circuit showed intermittent chaos from 1307Ω to 1777Ω, and were in agreement with the numerical simulation within an 11.81% to 18.81% relative error range. Two Chua Circuits were then successfully synchronized using the bidirectional method. 1. Introduction This work was motivated by the question “Can you synchronize chaos?” which was made by me during PHY256A. The physical implementation of a chaotic system was motivated by section 9.5 “Using Chaos to Send Secret Messages” of the book Nonlinear Dynamics and Chaos by Steven H. Strogatz. In this particular section, a brief discussion on how to construct a chaotic mask using an electronic implementation of the Lorentz equations is held. A Chua Circuit was then preferred over the Lorentz both for simplicity and historical reasons. When Edward Lorentz provided his idealized model of the atmosphere [1], some scientists questions the physical nature of his result. Their argument was that no real experimental confirmation could ever be made because his model was very crude and simplified; that his model could only be exhibited by abstract mathematical models but had no connection to reality. In order to settle the ongoing dispute, Takashi Matsumoto’s research group decided to build an electrical circuit that would mimic the equations [2]. However, they had trouble implementing the multiplications that appear on the equations of motion, so their circuit ended up being very complex. After almost three years of work, the circuit was finally completed in October 1983. Unfortunately, the premier was a spectacular disaster due to the failure of one of the integrated circuits [3]. It was then when Leon Chua, a visiting professor at the time, wondered if one could construct a circuit that would not be governed by the Lorentz equations, but would still give rise to chaotic behavior. The Chua circuit was born after he figured out exactly how to remove all the unnecessary components of the Lorentz circuit while still preserving the chaos, and Chua himself was among the first persons who showed that chaos can be easily constructed and observed. The first goal of this work was to build a Chua circuit with a variable resistor as a bifurcation parameter. Secondly, a numerical simulation was used to compare the experimental measurements. The measurements were compared both visually and in terms of the variable resistor. Lastly, two Chua circuits were synchronized using the bidirectional synchronization method discussed on the already mentioned Strogatz book. Section 2 on this work introduces the Chua circuit. In section 3 a fixed point analysis is performed on the equations of motion of the Chua circuit. Section 4 details the experimental realization of the Chua circuit. Section 5 goes through the numerical simulation methodology and parameters. In section 6 all of the results are presented and a discussion of them is done on section 7. Finally, the conclusions and a brief proposal of future work are discussed in section 8.
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Experimental Realization and Synchronization of a Chua Circuit
Daniella Masante. Department of Physics. University of California, Davis.
Physics Building, 1 Shields Avenue, Davis, CA 95616 USA. [email protected]
Abstract
A Chua circuit, known to exhibit chaotic behavior,
has been constructed and studied. The circuit was built
with a potentiometer as a control resistor, and chaotic
dynamics such as limit cycles and attractors were
measured and compared to a numerical simulation. The
circuit showed intermittent chaos from 1307Ω to 1777Ω,
and were in agreement with the numerical simulation
within an 11.81% to 18.81% relative error range. Two
Chua Circuits were then successfully synchronized using
the bidirectional method.
1. Introduction
This work was motivated by the question “Can you
synchronize chaos?” which was made by me during
PHY256A. The physical implementation of a chaotic
system was motivated by section 9.5 “Using Chaos to
Send Secret Messages” of the book Nonlinear Dynamics
and Chaos by Steven H. Strogatz. In this particular
section, a brief discussion on how to construct a chaotic
mask using an electronic implementation of the Lorentz
equations is held. A Chua Circuit was then preferred
over the Lorentz both for simplicity and historical
reasons.
When Edward Lorentz provided his idealized model
of the atmosphere [1], some scientists questions the
physical nature of his result. Their argument was that no
real experimental confirmation could ever be made
because his model was very crude and simplified; that
his model could only be exhibited by abstract
mathematical models but had no connection to reality.
In order to settle the ongoing dispute, Takashi
Matsumoto’s research group decided to build an
electrical circuit that would mimic the equations [2].
However, they had trouble implementing the
multiplications that appear on the equations of motion,
so their circuit ended up being very complex.
After almost three years of work, the circuit was
finally completed in October 1983. Unfortunately, the
premier was a spectacular disaster due to the failure of
one of the integrated circuits [3]. It was then when Leon
Chua, a visiting professor at the time, wondered if one
could construct a circuit that would not be governed by
the Lorentz equations, but would still give rise to chaotic
behavior.
The Chua circuit was born after he figured out
exactly how to remove all the unnecessary components
of the Lorentz circuit while still preserving the chaos,
and Chua himself was among the first persons who
showed that chaos can be easily constructed and
observed.
The first goal of this work was to build a Chua
circuit with a variable resistor as a bifurcation parameter.
Secondly, a numerical simulation was used to compare
the experimental measurements. The measurements were
compared both visually and in terms of the variable
resistor. Lastly, two Chua circuits were synchronized
using the bidirectional synchronization method discussed
on the already mentioned Strogatz book.
Section 2 on this work introduces the Chua circuit.
In section 3 a fixed point analysis is performed on the
equations of motion of the Chua circuit. Section 4 details
the experimental realization of the Chua circuit. Section
5 goes through the numerical simulation methodology
and parameters. In section 6 all of the results are
presented and a discussion of them is done on section 7.
Finally, the conclusions and a brief proposal of future
work are discussed in section 8.
2. Background
Back when he was figuring out how to create his
circuit, Leon Chua had a lot of decisions to make. He
intuitively knew that his circuit didn’t have to be as
complicated as the Lorentz prototype to get chaos.
However, he also knew going into his search that he
needed to reach a certain minimal complexity (Poincaré-
Bendixson theorem) of at least three degrees of freedom
to not lose the behavior [4].
That was the first question that Leon Chua had to
ask himself: what is a degree of freedom on an electrical
circuit? If the system contained only resistive
components all currents and voltages could be computed
directly, without any additional knowledge. If the
information about this current is provided, it can be
treated just like an ordinary current source and the
resulting resistive equations can be solved. Therefore,
such system would have zero degrees of freedom.
The answer is that a degree of freedom is any
component in which the relation between voltage and
current depends on previous history (i.e. a differential
relation). This is the case because the number of degrees
of freedom describes how many scalar quantities (at each
moment in time) are needed to fully classify the state of
the circuit, meaning that every voltage and current can
be computed from such information. This is achievable
by energy-carrying components, such as inductors or
capacitors. And it also meant that by default Chua’s final
circuit would be autonomous, which means dependent
on time. So Chua decided to use one of the former and
two of the later and moved on.
The second problem that Chua encountered is that a
linear system cannot exhibit chaotic behavior [5]. He
knew that he needed a non-linear component because a
linear system is independent of scale and therefore
whatever happens on a small scale happens at a larger
scale; there is no element of surprise on a linear system.
Chua opted for a purely resistive non-linear component,
but that didn’t reduce his list of possibilities a great
much. He explored a lot of possibilities with elements
such as transistors, varistors, diodes and many more to
see what worked best. At the end he decided to use
operational amplifiers, and the first version of the Chua