×

zbMATH — the first resource for mathematics

An intrinsic multiple-scale harmonic balance method for nonlinear vibration and bifurcation problems. (English) Zbl 0757.70017
The intrinsic harmonic balance is a combination of a perturbation scheme (introduced by the first author) and the classical harmonic balance method. This paper introduces a modification of the above combination such that a particular approximation for a specific solution can be obtained at a comparatively lower order of perturbation. The modification is the application of a multiple time scaling. By this new method it is not necessary to assign the role of the perturbation parameter to one of the basic varibles. Further: perturbation analysis produces approximate differential equations without additional effort, and these equations yield the stability properties of the solutions; this method doesn’t require solution of partial differential equations at each perturbation step.
The paper shows the application of the method to Hopf bifurcation in a specific nonlinear system, then to the analysis of general dynamical bifurcation problems. According to the authors the method is applicable to more degenerate bifurcations, and also to non-autonomous systems. Convergence and existence questions are not dealt with in this paper.

MSC:
70K99 Nonlinear dynamics in mechanics
34C23 Bifurcation theory for ordinary differential equations
65J99 Numerical analysis in abstract spaces
PDF BibTeX XML Cite
Full Text: DOI