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Chua's Circuit and Conditions of Chaotic Behavior Caitlin Vollenweider 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. 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. 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) 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 Changes in a: (b=31, c=-0.35, k=1, m0=-2.5, and m1=-0.5) ● a=10 ● a=5 ● Lyap[1] = 6.10059 ● Lyap[1] = -0.142045 ● Lyap[2] = 0.0877721 ● Lyap[2] = -0.142055 ● Lyap[3] = 0.0873416 ● Lyap[3] = -4.2604 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. Changes in k: ● k=5 ● ● Lyap[1]= 26.4646 ● ● ● Lyap[2] = 0.032529 ● Lyap[3] = 6.78771 ● K=-0.001 Lyap[1] = 0.00870778 Lyap[2] = 0.00025575 Lyap[3] = 0.000300807 ● K=-5 ● Lyap[1] = 64.3746 ● Lyap[2] = 1.24994 ● Lyap[3] = 1.17026 ● ● 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 ● ● ● 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: ● 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. 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 Increasing m1 and m0 ● M0 = -2.15 ● M1 = -0.2545 ● Lyap[1] = 0.197958 ● Lyap[2] = 0.197989 ● Lyap[3] = -12.0894 ● M0 = -1.8 ● M1 = -0.009 ● Lyap[1] = 0.111414 ● Lyap[2] = 0.111658 ● Lyap[3] = -15.4614 Decreasing of m1: ● M1 = -0.9 ● Lyap[1] = -0.0108036 ● Lyap[2] = -0.0107962 ● Lyap[3] = -2.35885 ● M1 = -1 ● Lyap[1] = -0.257964 ● Lyap[2] = -0.339839 ● Lyap[3] = -0.33995 ● M1 = -1.01 ● Lyap[1] = -0.0393278 ● Lyap[2] = -0.376931 ● Lyap[3] = -0.377225 ● M1 = -1.0135 ● Lyap[1] = 0.0371617 ● Lyap[2] = -0.389859 ● Lyap[3] = -0.390291 ● M1 = -1.035 ● Lyap[1] = 11.567 ● Lyap[2] = -0.711636 ● Lyap[3] = -0.426731 ● M1 = -1.0351 ● Lyap[1] = 11.5797 ● Lyap[2] = -0.711924 ● Lyap[3] = -0.426757 ● M0 = M1 = -3 ● L1 = 29.4742 ● L2 = -0.78322 ● L3 = -0.783714 Positive m0 and m1 ● Lyap[1] = -0.0317025 ● Lyap[2] = -0.0312853 ● Lyap[3] = -22.6063 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. g(x)=m1*x+0.5*(m0m1)*(abs(x*x+1)-abs(x*x-1)) ● Lyap[1] = 0.27213 ● Lyap[2] = 0.272547 ● Lyap[3] = -8.69594