Étude macroscopique de systèmes différentiels. (French) Zbl 0544.34053

Troesch, Albert

Proc. Lond. Math. Soc., III. Ser. 48, 121-160 (1984).

The paper belongs to the area of nonstandard analysis and consists of three different parts. The first part is devoted to the philosophy and most general set-theoretic principles of nonstandard analysis (axiomatics of the Internal Set Theory via Nelson) and is of very brief informative character. The second part explains the appearance of infinitesimally near objects in nonstandard topology and analysis, especially in the theory of ordinary differential equations containing a small parameter, and discusses the nonstandard theory of asymptotic developments. The third part is the most concrete and thorough one. It deals with the qualitative theory of the van der Pol’s equation \(x''+(x^ 2+1)x'+x=e(t)\) and Liénard’s equation \(ax''+f(x)x'+g(x)=e(t)\) (boundedness of solutions, existence of limit cycles, asymptotic behaviour, minimal sets). Some of the results are new but also the familiar classical results often possess more simple demonstrations.

Reviewer: J.Chrastina

MSC:

34E05	Asymptotic expansions of solutions to ordinary differential equations
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C11	Growth and boundedness of solutions to ordinary differential equations
26E35	Nonstandard analysis
03H05	Nonstandard models in mathematics

Keywords:

nonstandard analysis; small parameter; asymptotic developments; van der Pol’s equation; Liénard’s equation; boundedness of solutions; limit cycles; minimal sets