Abstract
A method is proposed for calculating the normal form coefficients of the degenerate Hopf bifurcation system and the steady periodic solutions of a nonlinear vibration system. The results obtained by this method are the same as those obtained by the classical one. The present method is much simpler and can easily be implemented: that is, given the coefficients of the governing equations, the response is obtained directly by substitution.
Highlights
The idea of the normal transformation came from Poincare (1889). It has been taken up by many authors (Birkhoff, 1966; Arnold, 1978; Moser, 1973; 100s and Joseph, 1980). It consists of carrying out a near identity change of variable allowing one to transform one system of nonlinear ordinary differential equations to a simpler one
Mathematicians are interested in the kinds of normal form different differential equations can have
Many kinds of normal forms have been given at present, such as the Hopf, pitchfork, saddle-node bifurcation for the generate and degenerate cases
Summary
The idea of the normal transformation came from Poincare (1889) It has been taken up by many authors (Birkhoff, 1966; Arnold, 1978; Moser, 1973; 100s and Joseph, 1980). Engineers and scientists are more interested in how to get the normal form for a given system of equations, that is, what is the relationship of the coefficients between the original equations and the normal form equations To this end, we give a simple method to calculate the coefficients of the normal form of a Hopf bifurcation system for both the nondegenerate and the degenerate case. We give a simple method to calculate the coefficients of the normal form of a Hopf bifurcation system for both the nondegenerate and the degenerate case We use this method to get the steady periodic solution of nonlinear free vibrations.
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