An intrinsic multiple-scale harmonic balance method for nonlinear vibration and bifurcation problems.

*(English)*Zbl 0757.70017The 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.

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.

Reviewer: Á.Bosznay (Budapest)

##### MSC:

70K99 | Nonlinear dynamics in mechanics |

34C23 | Bifurcation theory for ordinary differential equations |

65J99 | Numerical analysis in abstract spaces |