  The spectrum of Lyapunov exponents is powerful tool for the analysis of the
complex system dynamics. In the general framework of nonlinear dynamical
systems a number of the numerical technics have been developed to obtain the
spectrum of Lyapunov exponents for the complex temporal behavior of the systems
with a few degree of freedom. Unfortunately, these methods can not apply
directly to analysis of complex spatio-temporal dynamics in plasma devices
which are characterized by the infinite phase space, since they are the
spatially extended active media. In the present paper we propose the method for
the calculation of the spectrum of the spatial Lyapunov exponents (SLEs) for
the spatially extended beam-plasma systems. The calculation technique is
applied to the analysis of chaotic spatio-temporal oscillations in three
different beam-plasma model: (1) simple plasma Pierce diode, (2) coupled Pierce
diodes, and (3) electron-wave system with backward electromagnetic wave. We
find an excellent agreement between the system dynamics and the behavior of the
spectrum of the spatial Lyapunov exponents. Along with the proposed method, the
possible problems of SLEs calculation are also discussed. It is shown that for
the wide class of the spatially extended systems the set of quantities included
in the system state for SLEs calculation can be reduced using the appropriate
feature of the plasma systems.
