Chua's Circuit and Conditions of Chaotic Behavior

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Chua's Circuit and Conditions of Chaotic Behavior
Caitlin Vollenweider
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Introduction

• Chua's circuit is the simplest electronic circuit
  exhibiting chaos.
• In order to exhibit chaos, a circuit needs
• at least three energy-storage elements,
• at least one non-linear element,
• and at least one locally active resistor.
• The Chua's diode, being a non-linear locally
  active resistor, allows the Chua's circuit to
  satisfy the last of the two conditions.

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• Chua's circuit exhibits properties of chaos
• 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.

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Chua's Equations

• g(x) m1x0.5(m0-m1)(fabs(x1)-fabs(x-1))
• fx(x,y,z) ka(y-x-g(x))
• fy(x,y,z) k(x-yz)
• fz(x,y,z) k(-by-cz)

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Lyapunov Exponent

• This is a tool to find out if something is chaos
  or not.
• L gt 0 diverging/stretching
• L 0 same periodical motion
• L lt 0 converging/shrinking

• Lyap1 x
• Lyap2 y
• Lyap3 z

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Changes in a (b31, c-0.35, k1, m0-2.5, and
m1-0.5)

• a5
• Lyap1 -0.142045
• Lyap2 -0.142055
• Lyap3 -4.2604

• a10
• Lyap1 6.10059
• Lyap2 0.0877721
• Lyap3 0.0873416

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Changes in a, b, and c

• Changing any of these three variables will have
  the same results.
• All three change the shape
• None of the three actually affect chaos
• There has been plenty of research on the changes
  for these three variables.

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Changes in k

• K-5
• Lyap1 64.3746
• Lyap2 1.24994
• Lyap3 1.17026

• K-0.001
• Lyap1 0.00870778
• Lyap2 -0.00025575
• Lyap3 -0.000300807

• k5
• Lyap1 26.4646
• Lyap2 0.032529
• Lyap3 -6.78771

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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.

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The Power Supply

• 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)m1x0.5(m0-m1)(abs(x1)-abs(x-1))

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Research

• How the power supply actually affects chaos and
  the graphs by
• 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.

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Results

• Parameters a10, b31, c-0.35, k1, m0-2.5,
  m1-0.5
• Lyap1 0.27213
• Lyap2 0.272547
• Lyap3 -8.69594

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Increasing m1 and m0

• M0 -2.15
• M1 -0.2545
• Lyap1 0.197958
• Lyap2 0.197989
• Lyap3 -12.0894

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• M0 -1.8
• M1 -0.009
• Lyap1 0.111414
• Lyap2 0.111658
• Lyap3 -15.4614

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Decreasing of m1

• M1 -0.9
• Lyap1 -0.0108036
• Lyap2 -0.0107962
• Lyap3 -2.35885

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• M1 -1
• Lyap1 -0.257964
• Lyap2 -0.339839
• Lyap3 -0.33995

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• M1 -1.01
• Lyap1 -0.0393278
• Lyap2 -0.376931
• Lyap3 -0.377225

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• M1 -1.0135
• Lyap1 0.0371617
• Lyap2 -0.389859
• Lyap3 -0.390291

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• M1 -1.035
• Lyap1 11.567
• Lyap2 -0.711636
• Lyap3 -0.426731

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• M1 -1.0351
• Lyap1 11.5797
• Lyap2 -0.711924
• Lyap3 -0.426757

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• M0 M1 -3
• L1 29.4742
• L2 -0.78322
• L3 -0.783714

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Positive m0 and m1

• Lyap1 -0.0317025
• Lyap2 -0.0312853
• Lyap3 -22.6063

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Conclusions

• Both m0 m1 have regions that arent as
  sensitive to changes
• For almost all positive ms, 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 supplys
  equation to see how it will change the graph's
  shape.

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g(x)m1x0.5(m0-m1)(abs(xx1)-abs(xx-1))

• Lyap1 0.27213
• Lyap2 0.272547
• Lyap3 -8.69594