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=5 ● Lyap[1] = ● Lyap[2] = ● Lyap[3] = ● a=10 ● Lyap[1] = ● Lyap[2] = ● Lyap[3] =

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] = ● Lyap[2] = ● Lyap[3] = ● K= ● Lyap[1] = ● Lyap[2] = ● Lyap[3] = ● k=5 ● Lyap[1]= ● Lyap[2] = ● Lyap[3] =

● 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] = ● Lyap[2] = ● Lyap[3] =

Increasing m1 and m0 ● M0 = ● M1 = ● Lyap[1] = ● Lyap[2] = ● Lyap[3] =

● M0 = -1.8 ● M1 = ● Lyap[1] = ● Lyap[2] = ● Lyap[3] =

● M1 = -0.9 ● Lyap[1] = ● Lyap[2] = ● Lyap[3] = Decreasing of m1:

● M1 = -1 ● Lyap[1] = ● Lyap[2] = ● Lyap[3] =

● M1 = ● Lyap[1] = ● Lyap[2] = ● Lyap[3] =

● M1 = ● Lyap[1] = ● Lyap[2] = ● Lyap[3] =

● M1 = ● Lyap[1] = ● Lyap[2] = ● Lyap[3] =

● M1 = ● Lyap[1] = ● Lyap[2] = ● Lyap[3] =

● M0 = M1 = -3 ● L1 = ● L2 = ● L3 =

Positive m0 and m1 ● Lyap[1] = ● Lyap[2] = ● Lyap[3] =

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*(m0- m1)*(abs(x*x+1)-abs(x*x-1)) ● Lyap[1] = ● Lyap[2] = ● Lyap[3] =