Xiaolong He, Rafael de la Llave
Construction of Quasi-periodic Solutions
of State-dependent Delay Differential
Equations by the Parameterization Method I: finitely differentiable, hyperbolic case
(261K, PDF)
ABSTRACT. In this paper, we use the parameterization method
to construct
quasi-periodic solutions of state-dependent delay differential equations.
For example
$$
\left\{
egin{aligned}
\dot{x}(t)&=f( heta,x(t),\epsilon x(t- au(x(t))))\
\dot heta(t)&=\omega.
\end{aligned}
ight.
$$
Under the assumption of exponential dichotomies
for the $\epsilon=0$ case, we use a contraction
mapping argument to prove the existence and smoothness of the
quasi-periodic solution. Furthermore, the result is given in an
$a-posteriori$ format. The method is very general and applies also
to equations with several delays, distributed delays etc.