Harmonic balance (HB) method is well known principle for analyzing periodic oscillations on nonlinear networks and systems. Because the HB method has a truncation error, approximated solutions have been guaranteed by error bounds. However, its numerical computation is very time-consuming compared with solving the HB equation. This paper proposes an algebraic representation of the error bound using Grobner base. The algebraic representation enables to decrease the computational cost of the error bound considerably. Moreover, using singular points of the algebraic representation, we can obtain accurate break points of the error bound by collisions.

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.