Science China Mathematics

, Volume 61, Issue 3, pp 439–452 | Cite as

Affine-periodic solutions by averaging methods



This paper concerns the existence of affine-periodic solutions for perturbed affine-periodic systems. This kind of affine-periodic solutions has the form of x(t + T) ≡ Qx(t) with some nonsingular matrix Q, which may be quasi-periodic when Q is an orthogonal matrix. It can be even unbounded but \(\frac{{x(t)}}{{|x(t)|}}\) is quasi-periodic, like a helical line, for example x(t) = eat(cosωt, sinωt), when Q is not an orthogonal matrix. The averaging method of higher order for finding affine-periodic solutions is given by topological degree.


affine-periodic solutions averaging method topological degree 


34C27 34C25 


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This work was supported by National Basic Research Program of China (Grant No. 2013CB834100) and National Natural Science Foundation of China (Grant Nos. 11571065, 11171132 and 11201173).


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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.College of Mathematics, Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of EducationJilin UniversityChangchunChina
  2. 2.School of Mathematics and StatisticsNortheast Normal UniversityChangchunChina
  3. 3.Center for Mathematics and Interdisciplinary SciencesNortheast Normal UniversityChangchunChina

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