chua's circuit and conditions of chaotic behavior
DESCRIPTION
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, - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/1.jpg)
Hirophysics.com
Chua's Circuit and Conditions of Chaotic Behavior
Caitlin Vollenweider
![Page 2: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/2.jpg)
Hirophysics.com
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.
![Page 3: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/3.jpg)
Hirophysics.com
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.
![Page 4: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/4.jpg)
Hirophysics.com
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)
![Page 5: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/5.jpg)
Hirophysics.com
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
![Page 6: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/6.jpg)
Hirophysics.com
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
![Page 7: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/7.jpg)
Hirophysics.com
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.
![Page 8: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/8.jpg)
Hirophysics.com
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
![Page 9: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/9.jpg)
Hirophysics.com
● 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.
![Page 10: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/10.jpg)
Hirophysics.com
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))
![Page 11: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/11.jpg)
Hirophysics.com
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.
![Page 12: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/12.jpg)
Hirophysics.com
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
![Page 13: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/13.jpg)
Hirophysics.com
Increasing m1 and m0● M0 = -2.15● M1 = -0.2545● Lyap[1] = 0.197958● Lyap[2] = 0.197989● Lyap[3] = -12.0894
![Page 14: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/14.jpg)
Hirophysics.com
● M0 = -1.8● M1 = -0.009● Lyap[1] = 0.111414● Lyap[2] = 0.111658● Lyap[3] = -15.4614
![Page 15: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/15.jpg)
Hirophysics.com
● M1 = -0.9● Lyap[1] = -0.0108036● Lyap[2] = -0.0107962● Lyap[3] = -2.35885
Decreasing of m1:
![Page 16: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/16.jpg)
Hirophysics.com
● M1 = -1● Lyap[1] = -0.257964● Lyap[2] = -0.339839● Lyap[3] = -0.33995
![Page 17: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/17.jpg)
Hirophysics.com
● M1 = -1.01● Lyap[1] = -0.0393278● Lyap[2] = -0.376931● Lyap[3] = -0.377225
![Page 18: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/18.jpg)
Hirophysics.com
● M1 = -1.0135● Lyap[1] = 0.0371617● Lyap[2] = -0.389859● Lyap[3] = -0.390291
![Page 19: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/19.jpg)
Hirophysics.com
● M1 = -1.035● Lyap[1] = 11.567● Lyap[2] = -0.711636● Lyap[3] = -0.426731
![Page 20: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/20.jpg)
Hirophysics.com
● M1 = -1.0351● Lyap[1] = 11.5797● Lyap[2] = -0.711924● Lyap[3] = -0.426757
![Page 21: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/21.jpg)
Hirophysics.com
● M0 = M1 = -3● L1 = 29.4742● L2 = -0.78322● L3 = -0.783714
![Page 22: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/22.jpg)
Hirophysics.com
Positive m0 and m1● Lyap[1] = -0.0317025● Lyap[2] = -0.0312853● Lyap[3] = -22.6063
![Page 23: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/23.jpg)
Hirophysics.com
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.
![Page 24: Chua's Circuit and Conditions of Chaotic Behavior](https://reader035.vdocuments.site/reader035/viewer/2022062310/5681634c550346895dd3e235/html5/thumbnails/24.jpg)
Hirophysics.com
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