Efficient Applications of Invariants to Harmonic Balance Equation Using Grobner Base

Masakazu YAGI  Takashi HISAKADO  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E90-A   No.10   pp.2178-2186
Publication Date: 2007/10/01
Online ISSN: 1745-1337
DOI: 10.1093/ietfec/e90-a.10.2178
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Nonlinear Theory and its Applications)
Category: Nonlinear Phenomena and Analysis
Keyword: 
Grobner base,  invariant,  harmonic balance method,  symmetry,  

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Summary: 
This paper presents efficient applications of invariants to harmonic balance (HB) methods using Grobner base. The Grobner base is a powerful tool based on ideal theory. Using the Grobner base, we can obtain the solutions of the HB equation. However, its computation is very time-consuming when the equation has equivalent different solutions based on symmetries of the system. We show that invariants enable to transpose the equivalent different solutions to a unique solution. The bifurcation diagram of the invariant is simpler than the original bifurcation diagram, and its computation is considerably decreased. Further, we can obtain the relation among the amplitudes of each frequency component using the invariants. We propose a method for finding the circuit parameters using the amplitude relation.