On stationary solutions of some canonical systems of differential equations View Full Text


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Article Info

DATE

1981-11

AUTHORS

S. G. Zhuravlev

ABSTRACT

In our article (Zhuravlev, 1979) a formal method of constructing conditionally periodic solutions of canonical systems of differential equations with a ‘quick-rotating phase’ in the case of sharp commensurability was presented. The existence of stationary (or periodic) solutions of an averaged system of differential equations corresponding to the initial system of differential equations is necessary for an effective application of the method for different problems. Evidently, the stationary solutions do not always exist but in numerous papers on stationary solutions (oscillations or motions), the conditions of existence of such solutions are very often not considered at all. Usually a simple assumption is used that the stationary solutions do exist. Otherwise it is well known that Poincaré's theory of periodic solutions (Poincaré, 1892) let one set up conditions of existence of periodic solutions in different systems of differential equations. Particularly, in papers,MaлИh (1949, 1956), see alsoБИexmah (1971), the necessary and sufficient conditions of the existence of periodic solutions of (non-canonical) systems of differential equations which are close to arbitrary non-linear systems are given. For canonical autonomous systems of differential equations the conditions of existence of periodic solutions and a method of calculation are presented in the paperMepmah (1952). In our paper another approach is given and the conditions of existence of stationary solutions of canonical systems of differential equations with a quick-rotating phase are proved. For this purpose Delaunay-Zeipel's transformation and Poincaré's small parameter method are used. More... »

PAGES

297-315

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bf01228967

DOI

http://dx.doi.org/10.1007/bf01228967

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1051749244


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