On the effective reducibility of a class of Quasi-periodic nonlinear systems near the equilibrium

ABSTRACT

In this paper, we consider the effective reducibility of the following quasi-periodic nonlinear system $\dot{x}=(A+\epsilon Q(t,\epsilon \left)\right)x+\epsilon g(t,\epsilon )+h(x,t,\epsilon ),|\epsilon |\le {\epsilon }_0,$ where A is a constant matrix with the different and nonzero eigenvalues, $h=O\left(x^2\right)(x\to 0)$, and $h(x,t)$, $Q\left(t\right)$ and $g\left(t\right)$ are analytic quasi-periodic on $D_{\rho }$ with respect to t. Under non-resonance conditions, without any non-degeneracy condition, by a quasi-periodic transformation, the system can be reducible to a quasi-periodic system $\dot{y}=\left(A^{\ast }\right(\epsilon )+\epsilon R^{\ast }(t,\epsilon \left)\right)y+\epsilon g^{\ast }(t,\epsilon )+O\left(y^2\right),|\epsilon |\le {\epsilon }_0,$ where $R^{\ast }$ and $g^{\ast }$ are exponentially small in ϵ.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors are supported by the Natural Science Foundations for Colleges and Universities in Jiangsu Province grant 18KJB110029, National Natural Science Foundation of China grant 11501234 and 11526177 (Tianyuan Foundation), and the Scientific Research Foundation of Xuzhou Institute of Technology grant XKY2016214, XKY2016215 and XKY2017113.