Chua's Circuit and Conditions of
Chaotic Behavior
Caitlin Vollenweider
Introduction
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Chua's circuit is the simplest electronic circuit
exhibiting chaos.
In order to exhibit chaos, a circuit needs:
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at least three energy-storage elements,
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at least one non-linear element,
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and at least one locally active resistor.
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The Chua's diode, being a non-linear locally
active resistor, allows the Chua's circuit to
satisfy the last of the two conditions.
Chua's circuit exhibits properties of
chaos:
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It has a high sensitivity to initial
conditions
Although chaotic, it is bounded to
certain parameters
It has a specific skeleton that is
completed during each chaotic
oscillation
The Chua's circuit has rapidly became
a paradigm for chaos.
Chua's Equations:
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g(x) = m1*x+0.5*(m0-m1)*(fabs(x+1)-fabs(x-1))
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fx(x,y,z) = k*a*(y-x-g(x))
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fy(x,y,z) = k*(x-y+z)
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fz(x,y,z) = k*(-b*y-c*z)
Lyapunov Exponent
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This is a tool to find
out if something is
chaos or not.
L>0=
diverging/stretching
L = 0 = same
periodical motion
L<0=
converging/shrinking
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Lyap[1] = x
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Lyap[2] = y
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Lyap[3] = z
Changes in a: (b=31, c=-0.35, k=1,
m0=-2.5, and m1=-0.5)
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a=10
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a=5
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Lyap[1] = 6.10059
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Lyap[1] = -0.142045
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Lyap[2] = 0.0877721
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Lyap[2] = -0.142055
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Lyap[3] = 0.0873416
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Lyap[3] = -4.2604
Changes in a, b, and c
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Changing any of these three variables will have
the same results.
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All three change the shape
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None of the three actually affect chaos
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There has been plenty of research on the
changes for these three variables.
Changes in k:
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k=5
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Lyap[1]= 26.4646
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Lyap[2] =
0.032529
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Lyap[3] = 6.78771
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K=-0.001
Lyap[1] =
0.00870778
Lyap[2] = 0.00025575
Lyap[3] = 0.000300807
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K=-5
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Lyap[1] = 64.3746
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Lyap[2] = 1.24994
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Lyap[3] = 1.17026
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Unlike the variables a, b, and c,
k does affect chaos
The closer k gets to zero, the
less chaotic; however, the father
k gets from zero (in either
direction) the more chaotic it
becomes.
The Power Supply
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Every Chua circuit
has its own special
power supply. To the
right is what and ideal
power supply graph
should look like.
The equation for the
power supply is:
g(x)=m1*x+0.5*(m0-m1)*(abs(x+1)-abs(x-1))
Research:
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How the power supply actually affects chaos
and the graphs by:
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Going from reference point to increasing m1 and
m0 heading towards zero
Decreasing m1, m0 will stay the same
Using Lyapunov Exponent to show whether or not
its chaotic
Other fun graphs done by changing the power
supply equation.
Results:
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Parameters: a=10, b=31, c=0.35, k=1, m0=-2.5, m1=-0.5
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Lyap[1] = 0.27213
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Lyap[2] = 0.272547
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Lyap[3] = -8.69594
Increasing m1 and m0
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M0 = -2.15
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M1 = -0.2545
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Lyap[1] = 0.197958
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Lyap[2] = 0.197989
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Lyap[3] = -12.0894
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M0 = -1.8
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M1 = -0.009
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Lyap[1] = 0.111414
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Lyap[2] = 0.111658
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Lyap[3] = -15.4614
Decreasing of m1:
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M1 = -0.9
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Lyap[1] = -0.0108036
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Lyap[2] = -0.0107962
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Lyap[3] = -2.35885
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M1 = -1
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Lyap[1] = -0.257964
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Lyap[2] = -0.339839
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Lyap[3] = -0.33995
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M1 = -1.01
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Lyap[1] = -0.0393278
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Lyap[2] = -0.376931
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Lyap[3] = -0.377225
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M1 = -1.0135
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Lyap[1] = 0.0371617
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Lyap[2] = -0.389859
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Lyap[3] = -0.390291
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M1 = -1.035
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Lyap[1] = 11.567
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Lyap[2] = -0.711636
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Lyap[3] = -0.426731
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M1 = -1.0351
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Lyap[1] = 11.5797
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Lyap[2] = -0.711924
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Lyap[3] = -0.426757
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M0 = M1 = -3
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L1 = 29.4742
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L2 = -0.78322
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L3 = -0.783714
Positive m0 and m1
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Lyap[1] = -0.0317025
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Lyap[2] = -0.0312853
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Lyap[3] = -22.6063
Conclusions:
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Both m0 & m1 have regions that aren’t as
sensitive to changes
For almost all positive m’s, the graph converges
Out of all the parts of Chua's Circuit, it is the
power supply that has the most obvious affect
on Lyapunov Exponent and Chaos.
For future research: changing the power
supply’s equation to see how it will change the
graph's shape.
g(x)=m1*x+0.5*(m0m1)*(abs(x*x+1)-abs(x*x-1))
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Lyap[1] = 0.27213
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Lyap[2] = 0.272547
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Lyap[3] = -8.69594