An effective method for finding periodic solutions
of a class of non--autonomous periodic nonlinear
differential equations
Abstract
In the study of nonlinear ODE problems it is of interest to find
periodic solutions of a given equation. We consider a
non-autonomous system in the complex plane with the right hand
side f(u,t) being a complex analytic function of the space
variable u with the coefficients c_j(t) being
nonconstant T-periodic functions in the time
variable t and having non-zero Fourier coefficients for
non-negative indices only. This class is large enough to contain
all polynomial equations, e.g. the Bernoulli and Riccati equations
with periodic coefficients as above. We prove that the formal Fourier
obtained by
expanding u and c_j into series, substituting,
differentiating and comparing both sides,
is convergent. Despite the fact
that the proof of convergence is geometric, the method gives the solution
effectively - as a Fourier expansion.
