Download Chaos in a simple electronic circuit

NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS, NCNSD-2003
325
Chaos in a simple electronic circuit
Gyan Prakash and Kaushik Mitra
Abstract- An electronic circuit was built using a “jerk
equation” with no quadratic nonlinearity. The numerical
simulation of the phase space and the attractor obtained by the
circuit was compared. There was remarkable similarity between
the attractor produced by the circuit and that found by the
simulation. Also the Feigenbaum’s delta constant was estimated
from the circuit.
Keywords- Jerk equation, Period doubling, Feigenbaum
diagram, Feigenbaum delta.
I. INTRODUCTION
2
nonlinear terms (like |x| or sgn(x) rather than x etc.) the
schematics of these circuits are a lot simpler than those
proposed by Chua and Elwakil [5], [6].
A chaotic jerk equation was chosen from the list given by
Sprott so that a circuit may be built and compared with
simulation. The equation chosen has the form
...
T
HE qualitative solutions of ordinary differential
equations become more interesting as the dimension of
the flow increases. A flow is the entire pattern of
.
..
trajectories in the phase space (x, x , x ) .
Unlike systems of lower dimensions, systems in three or
higher dimensions can have complicated attractors and limit
cycles. Attractors may have a fractional dimension and
sensitive dependence to initial condition. Such behavior is
termed as chaos. Lorenz found an example of a system of
differential equations with as few as three variables and two
quadratic nonlinearities that leads to chaos [1]. Sprott [2]
later found several other simple equations that lead to chaos
with a single quadratic or cubic nonlinearity, including one
with three terms and a single quadratic nonlinearity:
...
simple functional forms of three-dimensional dynamical
systems that exhibit chaos [2], [3], [4]. Some of the Jerk
equations found have simple nonlinear functions that should
permit easy electronic implementation [2]. Due to simple
..
..
.
x = A x− x+ x − 1
(2)
Using a computer simulation the bifurcation diagram of
this equation was studied, with A as the bifurcation
parameter, varied between -0.8 and -0.5. Using the initial
..
.
conditions x = x = x = 0 , we obtained the bifurcation
diagram (Fig. 1) which shows the period doubling route to
chaos.
.2
x = −2.017 x + x − x
(1)
Such an equation is called a “jerk” equation because it
involves the third derivative of x (or the time derivative of
.
the acceleration). The first derivative of position x is called
..
velocity; the second derivative of position x
is called
...
acceleration. The third derivative of position x is called jerk.
II. THEORY
What is the simplest jerk equation that gives chaos?
Although we know that a chaotic flow must be at least a third
order ODE with minimum one nonlinear term, the sufficient
conditions for chaos in a system of autonomous ordinary
differential equations remain unknown. Sprott obtained a few
Gyan Prakash and Kaushik Mitra are final year Integrated MSc students
in the Department of Physics at IIT Kharagpur. (email:
[email protected] & [email protected] )
Fig 1: Bifurcation diagram for (2). The x axis is the value of A which is
varied from -0.8 to -0.5.
.
The phase space (‘ x ’ versus ‘x’) of the above equation
was plotted numerically, using the 4th order Runge-Kutta
method in C. Then it was analyzed and compared with the
phase space obtained by the circuit implemented to simulate
the above equation.
In order to implement (2) as a circuit we need three
INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 721302, DECEMBER 28-30, 2003
326
..
.
successive inverting integrators to generate x ’, − x and x
...
from − x . The weighted sum of these three signals and a
constant term generated with a dc voltage source are then fed
back to the input of the first integrator. Op-amps configured
as integrators were used for this purpose. The absolute value
part of (2) is implemented by two diodes acting as a full wave
rectifier and an inverting unity-gain amplifier [2]. The circuit
can be considered an oscillator with three 90
and non-linear positive feedback.
o
phase shifts
III. EXPERIMENT
Using the schematic shown below in Fig 2, and TL-082
op-amps, the circuit was built. The TL-082 was used to
minimize output drift due to offset and bias current.
Fig.2 : Chaotic circuit implementation of (2) using inverting op-amps. The
diodes are germanium, the battery is 1V, the capacitors are 0.1µF and the
resistors are 1kΩ except for the variable resistor, which should be adjustable
from 1 to 2 kΩ.
The following relation relates the variable resistor to the
parameter A in (2).
R =
1
kΩ
A
measuring the value of the variable resistance and using (3).
This value of A was entered in the simulation to check for
agreement. If there was no agreement, then values of A close
to the predicted value were tried until the desired attractor
was observed in the simulation.
The sequence of periodicity, chaos, and windows of
periodicity observed in the circuit were period-1, period-2,
period-4, period-8, chaos, period-3, period-6, chaos again. In
the simulation the sequences that were obtained were period1, period-2, period-4, period-8 chaos, period-5, period-10,
chaos and period-2. We have noticed some difference
between the experimental results and those obtained from
simulation. In the experiment there was a range of the
parameter where period-3 attractor was observed, but there
was not such range observed in the simulation. In contrast,
the simulation showed a range with period-5 attractor, which
was not observed in experiment. Apart from these, the
simulation and the experiment agree in all other parameter
ranges. Moreover, where the same behavior was observed in
the experiment and the simulation, the parameter values were
not exactly the same. These differences occurred because of
the parasitics that inevitably come into any experimental
setup, which are difficult to model.
The following are some of the printouts from the
oscilloscopes and the related screenshots of the numerical
simulation. They are presented in the sequence of their
observation by increasing the value of A from about -0.8 to
about -0.5 as suggested by the bifurcation diagram. This is
done by increasing the variable resistance. Note that the
oscilloscope printouts are on the left and the simulation
screen shot is on the right, unless otherwise noted. Also
observe the remarkable similarity between the experiment
and simulation.
(3)
With the circuit built the voltage of the battery is set to 1 V,
and the variable resistor is set so that A is below the region of
chaos according to Linz and Sprott’s Feigenbaum diagram
for the system [4]. The power is then turned on and off, this
is done many times until the capacitors are charged to the
steady-state conditions. Also the circuit has a basin of
attraction outside of which the dynamics are unbounded,
which manifests itself in the saturation of the Op-amps. If the
Op-amps saturate, it is necessary to restart the circuit.
Fig 3: Experimentally obtained Period-1 at A=-0.67 and the simulated plot
showing the same attractor at A=-0.74
To observe the attractor, the horizontal input of the
oscilloscope is connected to the x output and the vertical
..
input to the − x output of the circuit. When this is done the
oscilloscope needs to be set to the x-y plot mode for the
phase space to be observed. By increasing the resistance of
the variable resistor, period doubling, chaos, and windows of
periodicity were observed in the circuit. The value of A for
each range of periodicity or chaos was determined by
Fig 4: Experimentally obtained Period-2 at A= -0.63 and simulated plot
showing the same attractor at A= -0.71.
NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS, NCNSD-2003
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Fig 5: Experimentally obtained Period-4 at A= -0.56 and simulated plot
showing the same attractor at A= -0.65.
Fig 10: Experimentally obtained chaos reappears at A= -0.505.
IV. ANALYSIS AND INTERPRETATION
Fig 6: Experimentally obtained period-8 attractor at A= -0.55, and the
simulated plot showing a similar attractor at A= -0.643.
Due to the many chaotic regimes in the bifurcation
diagram, it was impossible to pinpoint all the periodicities
observed in the circuit. Looking at the attractors produced by
both the simulation and the experiment, similarities can be
seen in their pattern. The resistance of the variable resistor
was measured just before period doubling occurred. This was
done so that Feigenbaum’s δ constant could be calculated.
Feigenbaum’s delta can be calculated by using the
following relation
δ = lim
n →∞
Fig 7: Experimentally obtained Chaos at A= -0.54 and the simulated plot
showing a similar behavior at A= -0.62
An −1 − An − 2
An − An −1
(4)
The subscript in A denotes the A value during transition
from n to n+1 period. For n=8 (corresponding to period-16),
we got 4.545. The accepted value is 4.5515.
V. CONCLUSION
The experiment validated the predictions obtained from
simulation of the circuit. Looking at Fig. 1 we see that in
simulation period-1 exists until around A= -0.71, and the
orbit bifurcates to period-2, while in the experiment the
bifurcation was observed at around A= -0.63. The bifurcation
sequence tallied till the first occurrence of chaos. Afterwards
some dissimilarity was observed.
Fig 8: Experimentally obtained Period-3 at A= -0.512, not observed in
simulation
This experiment shows that chaos can be observed with
remarkably simple electronic circuits. Our circuit represents a
generic class of jerk equations of the form
...
.. .
x + A x + x = G ( x)
Fig 9: Experimentally obtained Period-6 at A= -0.515, not observed in
simulation.
(5)
Here G (x ) is the nonlinear term and A is a constant. The
nonlinear mathematical operation G (x ) for simplest cases
can be performed with operational amplifiers and ideal diode
[2] and the derivatives of x can be implemented using three
op-amps configured as integrators. Integrating each term in
(5) reveals that this system is a damped harmonic oscillator
driven by a nonlinear memory term that involves the integral
of G (x ) . Such an equation can also arise in the feedback
INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 721302, DECEMBER 28-30, 2003
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control of an oscillator in which the experimentally
accessible variable is a transformed and integrated version of
the fundamental dynamical variable.
Our circuit is similar in spirit to Chua’s circuit [5], which
uses two capacitors, an inductor, and diodes with op-amps or
transistors to provider a piecewise linear approximation to a
cubic nonlinearity. Chua’s circuit however has a more
complicated jerk representation with many more than four
terms, involving step functions, delta functions, and their
products with derivatives of x. It is more difficult to
construct, scale to arbitrary frequencies, and analyze because
of the inductor with its frequency dependant resistive losses.
A future course of work might be to apply this idea in
generating pseudo-random sequences which find frequent
applications in many areas like encryption, spread-spectrum
communication etc.
ACKNOWLEDGEMENTS
We would like to thank Dr. Soumitro Banerjee for giving
us valuable suggestions to perform the experiment. We
would also like to thank Dr. Sayan Kar for the idea and for
showing us the direction to work on this experiment.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130141 (1963).
J.C. Sprott, “Simple chaotic systems and circuits,” Am. J. Physics. 68,
758-763 (2000)
J.C. Sprott, “A new class of chaotic circuit”, Physics Letters A. 266 1923 (2000)
S. J. Linz and J. C. Sprott,” Elementary chaotic flows”, Physics Letters
A. 259 240-245 (1999.
A. S. Elwakil and A. Soliman, “Chaos from a family of minimumcomponent oscillators”, Chaos, Solitons, and Fractals, 8 , 335-356
(1997).
T. Matsumoto, L.O. Chua, and M. Komoro, “Birth and death of the
double scroll,” IEEE Trans. Circuits Syst. CAS-32, 797-818 (1985).