1. Chaos in a Diode observing period-doubling and chaotic behavior in a driven diode circuit Chris OsbornDaniel Richman

2. The Diode Passive (not self-powered) circuit component Restricts current flow to one direction Modern diodes are based on semiconductor p-n junctions – Situation when diode is not connected to a circuit (a voltage): Electron and hole recombination in vicinity of junction, depletion of mobile charges; remaining negative charges in p-type and positive charges in n-type cause potential difference across junction (field points from n-type to p-type) – Connection methods: Forward bias—positive emf terminal to p-type, negative to n-type Holes and electrons are pushed toward non-conducting center and oppose built-in potential barrier; current tunnels through the barrier Reverse bias: negative emf terminal to p-type, positive to n-type Reinforces built-in potential barrier; current is blocked

3. The Circuit

4. Characteristics (why nonlinearity) Diode has threshold voltage required for conduction – Forward voltage drop must reach this value for conduction to occur Diode has residual conduction during reverse-bias part of cycle (forward bias means diode “on”, reverse bias means diode “off”) – Exponential decay of current—recovery time – Frequency of cycle is faster than recovery time  Affects starting condition for next forward-bias part of cycle

5. Experimental Procedures Function generator output: sine – Frequency is circuit’s resonance frequency (73.7 kHz) – Amplitude adjustable from millivolts to tens of volts Inductor set to ten mH Oscilloscope reads generator voltage and diode voltage 1.Fix the driving frequency 2.Start at a low voltage amplitude 3.Increase amplitude in steps of tenths 4.Observe multiplying curves on diode voltage display

6. Observing the Circuit What are we looking at? Diode voltage versus time (# of periods is # of distinct peaks) Diode voltage versus driving voltage (# of periods is # of loops)

7. Chaos

8. Experimental Data [Diode voltage versus driving voltage bifurcation graph] brief chaos between red and blue chaos between purple and gold

9. Simulation Original computation in Matlab for the longtime convergence of x n produces a sequence of terms whose long-time convergence is the bifurcation diagram x o = 0.7, r = {2.5 : 0.015 : 4.0} x n+1 = r x n (1 – x n )