Bifurcations Of Holonomic First-order Differential Equations With Complete Integral

This paper is devoted to studying the bifurcations of completely integrable holonomicsystems of first-order di?erential equation germs. By using the theory of Legendrian un-foldings, S. Izumiya has classified the generic completely integrable holonomic systemsof n-variable first-order di?erential equation germs in the case of 1≤n≤3, and alsogiven the classification of this equation germs which have R+-simple and stable integraldiagrams when n≥1. The next purpose is natural to classify one-parameter familiesof equation germs with complete integral and to bifurcate the phase portraits (completesolutions and singular solutions). In this paper, we investigate the classification of bifur-cations of completely integrable holonomic first-order di?erential equation germs whichsatisfy that the corresponding one-parameter integral diagrams are R+-simple and stable.Since the S.P-K-simple function germs is generic in the case of S.P-K-cod≤5. Therefore,we can also give a generic classification of bifurcations of completely integrable n-variablefirst-order di?erential equation germs when n≤2. To complete the proof of ClassificationTheorem, we need the theory of the one-parameter complete Legendrian unfolding. Sincea one-parameter complete Legendrian unfolding is a kind of Legendrian immersion germ,there exists the generating family in Arnold-Zakalyukin theory. Our equivalence relationis given by the group of point transformations following S.Lie’s view. In the process ofclassifying, we will utilize the notion of t-P-K-equivalence between the smooth functiongerms with distinguished parameters. As a natural generalization of t-P-K-equivalence,we give some properties of I-P-K-equivalence in the last.Chapter 1 is the Introduction. We first introduce the history of Singularity The-ory and its applications, rising from other branches of mathematics and interdisciplinary,where we highlights the application in di?erential equations. Then the research of bifur-cations relative to di?erential equations is shown. Also, the basic frame of this thesis isgiven. Chapter 2 is the Preliminaries. Firstly, the basic notions of Singularity Theory in-volved in this paper are given. Secondly, we introduce the relevant knowledge of holonomicfirst-order di?erential equation germs with complete integral. At last, some notions andequivalent relations with respect to 1-parameter family of holonomic first-order di?erentialequation germs with complete integral are defined.For the proof of Classification Theorems, we investigate the theory of the one-parameter complete Legendrian unfolding and the corresponding generating family inChapter 3.In Chapters 4 and 5, we classify the bifurcations of completely integral holonomicn-variable first-order di?erential equation germs when n≤2 and the bifurcations of thisequation germs which have R+-simple and stable one-parameter integral diagrams in thecase of n≥1.In Chapter 6, we obtain some properties of I-P-K-equivalence between the smoothfunction germs with distinguished parameters.