Discussion of the bifurcation equations
The cases considered in Chapters 3 and 4 show that the bifurcation equations are either a set of n c nonlinear ordinary differential equations of first order ((3.19) and (3.67)), a set of n c difference equations (3.47), or a set of n c nonlinear algebraic equations (3.58). The number n c is given by the number of eigenvalues with zero real part for differential equations, or of modulus one for difference equations, or by the multiplicity of the critical eigenvalue for the linear operator G u for static systems in Section 3.2.1, respectively. The number of nonlinear terms in the bifurcation equations can become quite large, as the examples in Chapters 4 and 5 show. Hence, the question arises whether a simplification of the nonlinear part in the equations is possible. By simplification, we mean a reduction of the number of terms without changing the qualitative behavior of the flow in the phase space defined by the bifurcation system. We shall see that such a simplification is possible and will be provided by normal form theory. In general, a strong reduction of the number of terms can be achieved. However, we remark that the reduction to a normal form is not unique. This non-uniqueness need not bother those who apply normal form theory because in the cases of low codimension, that is, low number of essential parameters, the normal form into which the problem should be transformed is known. This will be shown in the next sections when we discuss the classified cases.
KeywordsNormal Form Hopf Bifurcation Bifurcation Diagram Rectangular Plate Parameter Plane
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