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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

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