effect will be negligible at low frequencies. Frequency The period of oscillation can be calculated from the breakdown and extinction voltage thresholds of the lamp used. During the charging period, the bulb has high resistance and can be considered an open circuit, so the rest of the oscillator constitutes an RC circuit with the capacitor voltage approaching VS exponentially, with time constant RC. If v(t) is the output voltage across the capacitor Solving for the time Although the first period is longer than the others because the voltage starts from zero, the voltage waveforms of subsequent periods are identical to the first between Ve and Vb. So the period T is the interval between the time when the voltage reaches Ve, and the time when the voltage reaches Vb This formula is only valid for oscillation frequencies up to about 200 Hz; above this various time delays cause the actual frequency to be lower than this. Due to the time required to ionize and deionize the gas, neon lamps are slow switching devices, and the neon lamp oscillator is limited to a top frequency of about 20 kHz. The breakdown and extinction voltages of neon lamps may vary between similar parts; manufacturers usually specify only wide ranges for these parameters. So if a precise frequency is desired the circuit must be adjusted by trial and error. The thresholds also change with temperature, so the frequency of neon lamp oscillators is not particularly stable. Forced oscillations and chaotic behavior Like other relaxation oscillators, the neon bulb oscillator has poor frequency stability, but it can be synchronized (entrained) to an external periodic voltage applied in series with the neon bulb. Even if the external frequency is different from the natural frequency of the oscillator, the peaks of the applied signal can exceed the breakdown threshold of the bulb, discharging the capacitor prematurely, so that the period of the oscillator becomes locked to the applied signal. Interesting behavior can result from varying the amplitude and frequency of the external voltage. For instance, the oscillator may produce an oscillating voltage whose frequency is a submultiple of the external frequency. This phenomenon is known as "submultiplication" or "demultiplication", and was first observed in 1927 by Balthasar van der Pol and his collaborator Jan van der Mark. In some cases the ratio of the external frequency to the frequency of the oscillation observed in the circuit