From local to global investigations of linear differential equations of the \(n\)-th order. (English) Zbl 0548.34009

Neuman, František

Jahrb. Überblicke Math. 1984, Math. Surv. 17, 55-80 (1984).

[For the entire collection see Zbl 0535.00005.]
In this survey paper main definitions, methods and results of a new approach to global properties of linear homogeneous ordinary differential equations of the n-th order, \(n\geq 2\), are given. On several occasions the author compares the global results with previous local investigations (e.g. in the case of global canonical forms), or gives an easy explanation based on the new approach to some rather complicated constructions. Algebraic, topological and geometrical tools together with the methods of the theory of dynamical systems and functional equations make it possible to deal with problems concerning global properties of solutions by contrast to previous local or isolated results. For example, the structure of the set of all global transformations of linear differential equations is described by algebraic means (theory of categories: Brandt and Ehresmann groupoids), the construction of global canonical forms is given by methods of differential geometry (including E. Cartan’s moving-frame-of-reference method). The theory in question also includes effective methods for solving several special problems, e.g., on the global equivalence of two given equations, or from the area of questions on distribution of zeros of solutions, such as disconjugacy, oscillatory behavior of solutions, etc.

MSC:

34A30	Linear ordinary differential equations and systems
34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

functional equations; global properties of solutions; global transformations; algebraic means; global equivalence

Citations:

Zbl 0535.00005