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.


70K99 Nonlinear dynamics in mechanics
34C23 Bifurcation theory for ordinary differential equations
65J99 Numerical analysis in abstract spaces
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