1. Deterministic Chaos and the Chao Circuit

2. Determinism and Randomness
Classical physics is deterministic!
If you know where you started you know where you are going
Randomness:
Quantum randomness is truly random and unpredictable
A lot of randomness is actually complexity and uncertainty

3. Deterministic Chaos If we have a system with two things present
A sensitivity to initial conditions
A non-linear response
Then we can get a system response that appears random but is actually chaotic
“Chaotic” in this case means complex and unpredictable but not truly random

4. Deterministic Chaos
This combination can produce a great deal of complexity in the response.
Fractals -- geometric chaos
“Butterfly effect” -- where we have unpredictable transient response due to a very large sensitivity to small disturbances.

5. Linear Dynamical Systems
Well behaved
Not chaotic
x and y follow smooth trajectories
Generally solvable, predictable and intuitive
“dot” indicates time derivative

6. RLC Driven Linear Oscillator
Linear circuit
VR = IR
VL = L dI/dt
I = C dVC/dt
Excite with a step
Oscillating response
State-space (plot of v(t), i(t)) shows a spiral

7. Non-linear systems Lorentz system
Simple non-linear system of 3 variables
Produces deterministic chaos
State-space plot shows the movement (trajectory) of x,y and z in time
The state-space description shows two “attractors” around which the “system” orbits

8. Double Pendulum
Another non-linear system is the double (jointed) pendulum
Also produces chaos
Trajectory is very sensitive to the initial starting point

9. Double Pendulum

10. Oscillator Circuits
Circuits are used to create “self starting” oscillators.
Use a transistor to provide non-linearity
Design to oscillate at a particular frequency.
Your first oscillator will be your first amplifier!
Need to make sure it is not chaotic

11. The lab: Chua Circuit
For your lab you will build and test a non-linear circuit that can oscillate and also produce chaos.
Circuit uses diodes and a opamp to produce a non-linear element (negative resistance).
By changing the value of R and R1 you can change the behavior from oscillation to chaos.

12. Non-linear elements Diode Opamp Simple semiconductor device
Exponential non-linearity
Opamp
Ideal amplifier
Very high gain
Complex circuit
Clamping at max/min

13. Other Chaotic Systems
Weather
Economics
History (cliodynamics)
Etc.