Period Doubling Route to Chaos

Chaos is exhibited in the diode resonator circuit due to the unrecombined charges that cross the p-n junction when the diode is in the forward bias mode. When the diode switches biases, the charges diffuse back to their original plate making the diode act as a capacitor. The larger the forward current, the greater the amount of charges that cross the junction and the longer the system will need to return to its reverse bias equilibrium. If the reverse current is unable to reach equilibrium before the forward bias, then the next cycle depends upon the previous cycle. This may lead to different parameters for the beginning of each cycle. Under such conditions, the system may become chaotic. One of the routes to chaos is by period doubling. In this case, the period continues to double until there are no more stable states available. When driven at a frequency near the diode’s resonant frequency, the circuit can exhibit periodic behavior. As the driving amplitude is increased, the periodic state becomes unstable. The state divides into two frequencies dependent on the resonance. The harmonic frequency remains but a second frequency appears at half the harmonic. This is defined as period doubling. Further increase in the amplitude results in the splitting of the two periods, giving quadrupling, octupling, and finally chaos. A visual representation of this process can be seen in Fig. 2B. Although not drawn to scale, you can see in this diagram how the frequency splits repeatedly and with a set pattern. Periodicity splits into period doubling, which then splits into period quadrupling, and so forth. Hence we see that the system endures more and more period bifurcations. This will continue until the separation between the neighboring frequencies becomes indistinguishable. At this point, the system becomes unstable and chaotic. Further increase of the voltage will bring the system back to a linear state.