Computation of the spectrum of spatial Lyapunov exponents for the spatially extended beam-plasma systems and electron-wave devices

Abstract

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. PACS numbers: 05.45.-a, 05.10.-a Keywords: spectrum of Lyapunov exponents, spatially extended systems, beam plasma systems, electron wave media, Pierce diode, transverse field backward wave oscillator, reference system state.

Route to hyperchaos in Rayleigh-Bénard convection

“…The three largest Lyapunov exponents, λ 1 ≥ λ 2 ≥ λ 3 , were computed using the technique described in [26], with recourse to the operator of linearisation of the governing equations (1)-(3) and the Gramm-Shmidt orthonormalisation. Translational invariance of the convective system in the horizontal directions gives rise to two vanishing Lyapunov exponents, which were disregarded in computations by removing the corresponding components of the perturbations (note that for non-steady attractors at least one vanishing Lyapunov exponent remains since eqs.…”

Intermittency route to chaos and broadband high-frequency generation in semiconductor superlattice coupled to external resonator

“…The figure demonstrates the dependence of four largest Lyapunov exponents 1 -4 upon V 0 calculated by method discussed in Refs. [58,59]. For V 0 < 841 mV the largest Lyapunov exponent is 1 = 0, whereas 2 , 3 , 4 < 0.…”

Метод Расчета Спектра Показателей Ляпунова Для Систем С Запаздыванием

“…(см., на-пример, [9,10]). Одним из важных способов изучения динамики является расчет спектра показателей Ляпунова, который снабжает исследовате-лей более детальной информацией о природе наблюдаемых режимов и поведении системы при изменении управляющих параметров [11,12].…”