Download Travis Hoggard, Katharina Ochterbeck, Katie M. Reynolds, Faculty

Chaos in Electric Circuits
Travis Hoggard, Katharina Ochterbeck, Katie M. Reynolds, Faculty Mentor: Stephen R. Addison
Department of Physics and Astronomy,
University of Central Arkansas
Abstract
UCH recent work in experimental chaos has focused on developing classroom
demonstrations. The work performed in the development of these classroom demonstrations can be combined with the identification of simple sets of chaotic differential equations and communications theory to investigate the control and onset of chaos in electronic
circuits, and to exploit that control to develop methods of encrypting and decrypting communications. We have concentrated on the Sprott family of circuits, rather than the more
frequently used Chua circuit, as it is easier to control the operating frequency of these
circuits. It is also easier to model the behavior of the Sprott circuits. MatLab was used
to code and solve several chaotic differential equations and investigate their suitability for
realization as circuits.
dx3
= −x3
dt
dv3
= −v3
dt
da3
= −a3
dt
There are numerous choices for D(x), and thus numerous possibilities for possible synchronization. There are many conditions possible that yield synchronization, i.e. make
x3, v3, and a3 to approach zero as time increases. Thus, by taking the connection points
between the two circuits to be any of these nodes, where x = x1, v = v1, a = V2, and da
dt =
v3, appropriate values of a(t), β(t), and γ(t) can be obtained in order to establish synchronization between the two circuits.
1. Synchronization
2. Circuits
M
The following examples are theoretical plots from two particular systems.
N the late 1980s Lou Pecora developed the idea of synchronizing chaotic circuits. The
thrust of his thought was in driving chaos with chaos. In other words, letting the output
of one chaotic circuit be the input of the other. Thus, with a continuous signal, with each
point of the output signal, the second circuit would reset itself to the initial conditions of
that point. As explained in Steven Strogatzs Sync, one circuit is a transmitter and the other
the receiver, with one-way communication. To synchronize the systems, send the continuously changing numerical value of one of the driver variables to the receiver and use it to
replace the corresponding receiver variable moment by moment. Under the right circumstances the other receiver variables snap into sync with the counterpart driver. Resulting
in all of the variables being matched and the two systems are completely synchronized.
Now, consider the synchronization of two nonlinear circuits with equations described by
I
Figure 1: Chua’s Circuit
dx1
= v1
dt
dv1
= a1
dt
da1
= −Aa1 − v1 + G(x1)
dt
dx2
= v2 + α(t)
dt
dv2
= a2 + γ(t)
dt
da2
= −Aa2 − v2 + H(x2) + γ(t)
dt
where again, the slave is denoted by the subscript 2, and the master by the subscript
1. These equations correspond to the circuit under study. Notice that the nonlinear element of the slave does not have to be identical to that of the master. The variables
α(t), β(t), and γ(t) are the nonlinear control variables that have to be determined. The
difference of the three variables satisfies
dx3
= v3 + α(t),
dt
dv3
= a3 + β(t),
dt
da3
= −Aa3 − v3 + H(x2) − G(x1) + γ(t)
dt
HILE Chuas circuit (1) has been widely used in experiments, it contains an inductor
to generate nonlinearities. The inductor makes the circuit difficult to model, and also
makes it difficult to change the operating frequencies of the circuit. These difficulties can
be eliminated by using recently discovered circuits. These new circuits correspond to relatively simple third-order differential equations. These circuits can be used in combinations
to investigate synchronization and secure communication. The corresponding differential
equations are readily solvable allowing a concurrent theoretical analysis that guides the
experimental studies. The circuits under investigation can be described by the equation
W
...
x = −Aẍ − ẋ + D(x) − α
Where x represents the voltage at a particular node in a circuit, A and a are constants, the
dots represent differentiation with respect to (dimensionless) time, and D(x) represents
nonlinearities in the circuit. A schematic representation of the circuit is shown in Figure 2.
This particular circuit (designed by Kiers) is described by the following version of (2).
R
R
...
x =−
ẍ − ẋ + D(x) −
V0
Ro
Ro
Varying the sub-circuit represented by D(x) enables the investigation of many different
chaotic systems.
The control variables are similarly defined as
!
+
α(t) = Va(t),
β(t) = Vb(t),
γ(t) = −H(x2) + G(x1) + Vc(t)
3. Conclusions and Further Research
IRCUITS corresponding to Sprott’s third-order differential equation provide an excellent series of increasingly complex circuits, which can be used in schemes in the
development of analog computers for use in encryption and decryption in communications devices. However, the resulting system should not be considered as secure, instead
it constitutes a masking system. We will continue to develop these chaotic circuits so that
we can investigate the properties of chaotic systems. We will study the effect of power
supply impedance to examine situations where the power supply has inductive and or
capacitive components. The study of such systems provides useful results and suitable
research experiences for undergraduates.
C
4. Acknowledgements
!
!
!
+
+
+
Then, the differences will satisfy
Figure 2: Kier’s Circuit
Student Research Symposium, College of Natural Sciences and Mathematics, 21 April 2006
This research was done at the University of Central Arkansas. Support was provided by
the University of Central Arkansas Research Counsel. Special thanks to John Grey of the
Naval Surface Warfare Center, Dahlgren Division for support and consultation.