15-105 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) Nov 9, 15

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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.

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