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It is known that Q Q the appearance of the new bifurcation at the aperiodical branches can lead to the birth of new oscillatory branches being already discussed in [24,40,60,61]. An example of the dispersion curves of such a nonuniform state with e "0. Two PDs for c"1 and e "0. In this point the aperiodical branch has always C"0 and X"0. At this point the O -branch disappears. With a further growth of d an oscillatory branch Q 62 A.
For simplicity, we shall denote this branch also as O. Thus in the case considered all oscillatory branches have discontinuities Q between the points c and k. The in uence of the neutralization parameter on the dispersion branches is demonstrated in Fig.
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As seen from Fig. Boundaries of oscillatory instability Now we are interested in the boundaries of existence of oscillatory modes. The equations for these lines are Q Q Q obtained from Eq. Note that in Eq. Q Q In fact, the lines N are extension of the lines M ; however, for an easier understanding of the Q Q considered picture we shall use both names.
In order to elucidate this picture we choose e"! Let us compare Fig. The only di!
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The intersection points of these lines are denoted by d. Q Q Q Our calculations by Eqs. With an increasing departure of c from 1 the values of s, at which m"1 for the "rst time, increases: for example, at c"0. Let us consider, for example, the zone with s"7 Fig.
We choose e "! Here the line B is the left boundary of the "rst oscillatory unstable region. This becomes clear if we look at the dispersion branches in Fig. The stable solutions occur only in the region located between the points Q and M. As an example we consider small values of the parameter s according to Fig. Therefore in Q each zone 2ps d 2p s 2 the line C at s'0 is the right boundary of the "rst oscillatory Q unstable region see Fig.
There is no oscillatory mode O in the region bounded by the lines C and K. Both of these lines Q Q Q exist only at e 'eQ. Such regions were plotted in Section 7. The study of the aperiodical branches makes it possible to correct to a certain extent the boundary lines for these regions. It should be noted that the growth rate is positive rather than zero Q for these e -values.
These regions are grayish in Figs. For example, take the known Pierce equilibria c"1, A. How the transformation looks like in di!
We proved that extrema were mapped onto itself and that each monotonic potential region had its counterpart apart transformation. Hence, the topology of the PDs remained unchanged. Also typical examples of PDs were exhibited. For given d and c, we then constructed an g, e diagram which shows how the collector potential and the type of equilibrium change as e is varied. Later in the nonlinear stage of the electron dynamics an essential reformation of the potential distribution occurs in the TIC, resulting in an ion redistribution so that the electron beam moves already through a nonuniform and incompletely neutralized ion background.
An analysis has shown that at certain times during the ion redistribution, the conditions of a Pierce-type instability arise again [84,94]. Thus, the investigation of nonneutral diodes is of fundamental interest for a study of the nonlinear oscillations in a TIC. A deeper analysis of these oscillations requests, therefore, a nonuniform ion background. In Refs. Moreover, a strong re ection of the electrons from the virtual cathode is characteristic of some of these solutions. This means that the transition from one state to another is typically accompanied by a strong decrease or increase of the current passing through the diode.
Thus, a collisionless beam plasma diode can be used as a switch element switching on and o!
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If an initial state of the diode is chosen to be near the bifurcation point, such a switch can be controlled by applying a voltage pulse of low intensity. Because the growth rate of an aperiodic instability is always comparable to the transit time of electrons, the switching speed of a device that is based on the collisionless beam plasma diode can be rather high.
So, for example, for d"1 mm and an electron beam energy of 0. With d decreases and beam energy increases the switching time can be still smaller. Another signi"cant advantage of such switches is the high current density which can attain tens of amperes per square centimeter . The possibility of switches being based on the e! Therefore, in such devices after the plasma transition has taken place, the new state will be conserved for in"nitely long time. The regimes of switches constructed on the base of such diodes can be controlled by a slight variation of the density of the ion ow.
An optimization of the generation regime of the microwave generators can be realized by varying the spatial pro"le of the ion ow. Thus the investigation of the beam plasma diodes opens new opportunities for the creation of microwave generators, high e!
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The most important result of this paper is the following: for all devices mentioned the search for equilibrium states, the investigation of their stability as well as the study of the initial stage of an electron instability could be carried out analytically. The main task of future investigations in the scienti"c branches mentioned above will be an extensive use of accurate numerical methods for studying the electron processes in the regime with electron re ection both of the linear stage and of the nonlinear stage.
In addition there is also a need for an analysis of beam plasma diode states with a nonuniform density distribution of ions in space. Acknowledgements We would like to thank P. Akimov for discussions and Mrs. Monika Glaser and F.
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SchmoK gner for their help in the preparation of the manuscript. References  R. McIntyre, J. Baksht et al. Kuznetsov, A. Ender, Sov. Babanin, I. Kolyshkin, V.
Mustafaev, V. Sitnov, A. Coutsias, D. Sullivan, Phys. A 27 Granatstein, I. Hertz, Phys. Farouki, M. Dalvie, L. Pavarino, J. Poukey, J. Quintenz, C.