The Application Of Equation's Scaling Laws In The Bifurcation Problems

From reference [29] if the equations with the property of scaling laws, thesolutions of those equations can be related by a simple rescaling, thus saving muchcomputational effort. This paper studies the bifurcation problems with symmetry byscaling laws and uses the scaling laws of reference [29] and popularizes it in generalcases, which are based on the reference [31]. We still apply the method to constructthe conditions of a special case and show how the scaling law affects the popularizedbifurcation problems. We all know the reduced bifurcation equation withLiapunov-Schmidt method can inherits the nature of symmetry. Through the detaileddiscussing, we derive that the reduced bifurcation equation with Liapunov-Schmidtmethod can inherits this good nature of scaling laws. Finally, we exploit thosediscusses in the problem of seeking its 2Ï€-periodic solutions. The behavior of thebifurcation point is first analyzed by the combination of LS method and recognitionproblem of singularity theory. Then from the scaling laws can know the behaviors ofall the bifurcation points.