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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) = m1*x+0.5*(m0-m1)*(fabs(x+1)-fabs(x-1))● fx(x,y,z) = k*a*(y-x-g(x))● fy(x,y,z) = k*(x-y+z)● fz(x,y,z) = k*(-b*y-c*z)

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Lyapunov Exponent

● 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

● Lyap[1] = x● Lyap[2] = y● Lyap[3] = z

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Changes in a: (b=31, c=-0.35, k=1, m0=-2.5, and m1=-0.5)

● a=5● Lyap[1] = -0.142045● Lyap[2] = -0.142055● Lyap[3] = -4.2604

● a=10● Lyap[1] = 6.10059● Lyap[2] = 0.0877721● Lyap[3] = 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● Lyap[1] = 64.3746● Lyap[2] = 1.24994● Lyap[3] = 1.17026

● K=-0.001● Lyap[1] =

0.00870778● Lyap[2] = -

0.00025575● Lyap[3] = -

0.000300807

● k=5● Lyap[1]= 26.4646● Lyap[2] =

0.032529● Lyap[3] = -

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)=m1*x+0.5*(m0-m1)*(abs(x+1)-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: a=10, b=31, c=-0.35, k=1, m0=-2.5, m1=-0.5

● Lyap[1] = 0.27213● Lyap[2] = 0.272547● Lyap[3] = -8.69594

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Increasing m1 and m0● M0 = -2.15● M1 = -0.2545● Lyap[1] = 0.197958● Lyap[2] = 0.197989● Lyap[3] = -12.0894

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● M0 = -1.8● M1 = -0.009● Lyap[1] = 0.111414● Lyap[2] = 0.111658● Lyap[3] = -15.4614

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● M1 = -0.9● Lyap[1] = -0.0108036● Lyap[2] = -0.0107962● Lyap[3] = -2.35885

Decreasing of m1:

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● M1 = -1● Lyap[1] = -0.257964● Lyap[2] = -0.339839● Lyap[3] = -0.33995

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● M1 = -1.01● Lyap[1] = -0.0393278● Lyap[2] = -0.376931● Lyap[3] = -0.377225

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● M1 = -1.0135● Lyap[1] = 0.0371617● Lyap[2] = -0.389859● Lyap[3] = -0.390291

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● M1 = -1.035● Lyap[1] = 11.567● Lyap[2] = -0.711636● Lyap[3] = -0.426731

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● M1 = -1.0351● Lyap[1] = 11.5797● Lyap[2] = -0.711924● Lyap[3] = -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● Lyap[1] = -0.0317025● Lyap[2] = -0.0312853● Lyap[3] = -22.6063

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Conclusions:

● 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.

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g(x)=m1*x+0.5*(m0-m1)*(abs(x*x+1)-abs(x*x-1))

● Lyap[1] = 0.27213● Lyap[2] = 0.272547● Lyap[3] = -8.69594