Higher order averaging theory for finding periodic solutions via Brouwer degree
Nonlinearity, 27 (3):
563 (2014)Abstract
In this paper we deal with nonlinear differential systems of the form ##IMG## http://ej.iop.org/images/0951-7715/27/3/563/non465661ueqn001.gif equation*x'(t)=\sum_i=0^k\varepsilon^i F_i(t,x)+\varepsilon^k+1 R(t,x,\varepsilon), equation* where ##IMG## http://ej.iop.org/images/0951-7715/27/3/563/non465661ieqn001.gif $F_i:RD\rightarrowR^n$ for i = 0, 1, …, k , and ##IMG## http://ej.iop.org/images/0951-7715/27/3/563/non465661ieqn002.gif $R:RD\times(-\varepsilon_0,\varepsilon_0)\rightarrowR^n$ are continuous functions, and T -periodic in the first variable, D being an open subset of ##IMG## http://ej.iop.org/images/0951-7715/27/3/563/non465661ieqn003.gif $R^n$ , and ε a small parameter. For such differential systems, which do not need to be of class ##IMG## http://ej.iop.org/images/0951-7715/27/3/563/non465661ieqn004.gif $C^1$ , under convenient assumptions we extend the averaging theory for computing their periodic solutions to k -th order in ε . Some applications are also performed.