Substantiation of the Generalized Method for Quasi-harmonic Linearization

Authors: Gribov A.F., Shakhtarin B.I. Published: 14.02.2014
Published in issue: #1(52)/2014  

Category: Mathematics and Mechanics  
Keywords: phase system, harmonic balance method, estimate of accuracy

The strict mathematical substantiation of the method for quasi-harmonic linearization is given, which has been proposed as a modification of the harmonic balance method for analysis of phase systems. Many authors (R. Bass, E.S. Pyatnitskii, E.N. Rosenwasser, et al.) were engaged in mathematical substantiation of the classical method for harmonic linearization. The phase system specificity - the presence of a secular term - has required the statement of the problem on substantiation of the procedure for obtaining a solution using the generalized method for quasi-harmonic linearization. The existence of both an O-cycle and an l-bypass φ-cycle is possible for phase systems. Conditions of existence of an l-bypass φ-cycle can be used for searching chaotic systems with the denumerable many of different l-turn φ-cycles, l = 1,2,3,... If a solution to the obtained system of algebraic equations exists with all l for the same values of parameters then the initial system has a denumerable number of periodic movements. Since the denumerable set of p-cycles can be only a saddle one, the system is chaotic. The nonlinear function entering the phase-system equation appears to be periodic and is expanded into a Fourier series. Next, the terms containing identical harmonics are equated. The conditions are determined, with which the solutions that are found using the quasi-harmonic linearization method differ very little from the exact solution.


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