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Chua's Circuit and Conditions of Chaotic Behavior

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,

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Chua's Circuit and Conditions of Chaotic Behavior

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  1. Chua's Circuit and Conditions of Chaotic Behavior Caitlin Vollenweider

  2. 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.

  3. 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.

  4. 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)

  5. 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

  6. 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

  7. 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.

  8. 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

  9. 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.

  10. 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))

  11. 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.

  12. 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

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

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

  15. M1 = -0.9 Lyap[1] = -0.0108036 Lyap[2] = -0.0107962 Lyap[3] = -2.35885 Decreasing of m1:

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

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

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

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

  20. M1 = -1.0351 Lyap[1] = 11.5797 Lyap[2] = -0.711924 Lyap[3] = -0.426757

  21. M0 = M1 = -3 L1 = 29.4742 L2 = -0.78322 L3 = -0.783714

  22. Positive m0 and m1 Lyap[1] = -0.0317025 Lyap[2] = -0.0312853 Lyap[3] = -22.6063

  23. 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.

  24. 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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