The existence of invariant tori of reversible systems with Liouvillean frequency and its applications

Abstract

This paper focuses on the quasi-periodically forced reversible system: $\left(\begin{array}{c}\dot{x}=\omega ,\\ \dot{u}=\text{i}A\left(\xi \right)u+\text{i}f(x,u,\overline{u},\xi ),\\ \dot{\overline{u}}=-\text{i}A\left(\xi \right)\overline{u}-\text{i}f(x,u,\overline{u},\xi ),\end{array}\right)$ where the frequency vector $\omega =(1,\alpha )$ with $\alpha$ being an irrational number, and $(x,u,\overline{u})\in T^2\times {ℂ}^l\times {ℂ}^l$. The matrix $A\left(\xi \right)$ is an $l\times l$ constant matrix that depends only on the parameter $\xi$, and $f$ represents a small perturbation that also depends on $\xi$ as a parameter. Based on the CD bridge method and the improved KAM iteration with parameters, it is proved that for most of the parameter $\xi$, the reversible system can be reduced to the form: $\left(\begin{array}{c}\dot{x}=\omega ,\\ \dot{u}=\text{i}A\left(\xi \right)u+\text{i}{V}_{\ast }(x,\xi )u+\text{i}{f}_{\ast }(x,u,\overline{u},\xi ),\\ \dot{\overline{u}}=-\text{i}A\left(\xi \right)\overline{u}-\text{i}{V}_{\ast }(x,\xi )\overline{u}-\text{i}{f}_{\ast }(x,u,\overline{u},\xi ),\end{array}\right)$ where $V_{\ast }(x,\xi )$ is an analytic function, and $f_{\ast }(x,u,\overline{u},\xi )$ represents the higher-order term. This indicates the existence of lower-dimensional invariant tori with Liouvillean frequency in the reversible system investigated. The result is further applied to reversible harmonic oscillators, and the existence of response solutions is proved.