Deterministic Chaos and the Chao Circuit

Deterministic Chaos and the Chao CircuitDeterminism and Randomness Classical physics is deterministic!If you know where you started you know where you are goingRandomness:Quantum randomness is truly random and unpredictableA lot of randomness is actually complexity and uncertainty

Deterministic ChaosIf we have a system with two things presentA sensitivity to initial conditionsA non-linear responseThen we can get a system response that appears random but is actually chaoticChaotic in this case means complex and unpredictable but not truly random

Deterministic ChaosThis combination can produce a great deal of complexity in the response. Fractals -- geometric chaosButterfly effect -- where we have unpredictable transient response due to a very large sensitivity to small disturbances.

Linear Dynamical SystemsWell behavedNot chaotic x and y follow smooth trajectoriesGenerally solvable, predictable and intuitive

dot indicates time derivativeRLC Driven Linear OscillatorLinear circuitVR = IRVL = L dI/dtI = C dVC/dtExcite with a stepOscillating responseState-space (plot of v(t), i(t)) shows a spiral

Non-linear systemsLorentz systemSimple non-linear system of 3 variablesProduces deterministic chaosState-space plot shows the movement (trajectory) of x,y and z in timeThe state-space description shows two attractors around which the system orbits

Double PendulumAnother non-linear system is the double (jointed) pendulumAlso produces chaos Trajectory is very sensitive to the initial starting point

Double Pendulum

Oscillator CircuitsCircuits are used to create self starting oscillators. Use a transistor to provide non-linearityDesign 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 elementsDiodeSimple semiconductor deviceExponential non-linearityOpampIdeal amplifierVery high gainComplex circuitClamping at max/min

Other Chaotic SystemsWeatherEconomicsHistory (cliodynamics)Etc.