Deterministic Chaos and the Chao Circuit

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

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

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.

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

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

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

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

Double Pendulum

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

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.

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

Other Chaotic Systems Weather Economics History (cliodynamics) Etc.