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Centers and limit cycles for a family of Abel equations.
- Source :
-
Journal of Mathematical Analysis & Applications . Sep2017, Vol. 453 Issue 1, p485-501. 17p. - Publication Year :
- 2017
-
Abstract
- Given trigonometric monomials A 1 , A 2 , A 3 , A 4 , such that A 1 , A 3 have the same signs as sin t , and A 2 , A 4 the same signs as cos t , and natural numbers n , m > 1 , we study the family of Abel equations x ′ = ( a 1 A 1 ( t ) + a 2 A 2 ( t ) ) x m + ( a 3 A 3 ( t ) + a 4 A 4 ( t ) ) x n , a 1 , a 2 , a 3 , a 4 ∈ R . The center variety is the set of values a 1 , a 2 , a 3 , a 4 such that the Abel equation has a center (every bounded solution is periodic). We prove that the codimension of the center variety is one or two. Moreover, it is one if and only if A 1 = A 3 and A 2 = A 4 and it is two if and only if the family has non-trivial limit cycles (different from x ( t ) ≡ 0 ) for some values of the parameters. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0022247X
- Volume :
- 453
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Journal of Mathematical Analysis & Applications
- Publication Type :
- Academic Journal
- Accession number :
- 122773204
- Full Text :
- https://doi.org/10.1016/j.jmaa.2017.04.017