Bifurcation and Chaos in the Second oscillatory Window of the Classical Pierce diode

The second oscillatory window of the classical Pierce diode is explored by a particle-in-cell simulation, and for the first time, results that support the existence of another chaotic region are presented. The classical Pierce model consists of a one-dimensional plasma-filled diode into which a monoenergetic electron beam is injected. This system presents a rich dynamical behavior as a function of a single control parameter α (the electron transit angle) and has four different operating regimes: stable, oscillatory, chaotic and unstable oscillatory with virtual cathode formation. The second oscillatory region, revealed by linear approximation analysis, presents a similar dynamical behavior to the first one, studied by many works and located below α = 3π. By gradually diminishing α from 5π, we report here, through numerical experiments, the existence of a sequence of subharmonic bifurcations leading to chaos, periodic windows, and a crisis from which just unstable oscillations with virtual cathode can be observed.