Presentation on theme: "Chua's Circuit and Conditions of Chaotic Behavior Caitlin Vollenweider."— Presentation transcript:
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.
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 = x ● Lyap = y ● Lyap = z
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.
● 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.
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.