On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems
The paper deals with a two-dimensional nonlinear quasi-periodic system \[ \begin{cases} \dot x = \Omega y + h_1(x,y,t) + f_1(x,y,t), \\ \dot y = x^{2n+1} + h_2(x,y,t) + f_2(x,y,t), \end{cases}\tag{1} \] where \(\Omega > 0,\) \(h = (h_1, h_2)^T\) consists of higher-order terms with \[ h(x,y,t) = \sum_{j \geq 2 \;or \;i+j \geq 2n+2} h_{ij}(t)x^i y^j \] and \(f = (f_1, f_2)^T\) is a small perturbation term of lower-order terms with \[ f(x,y,t) = \sum_{j \leq 1 \;and \;i+j \leq 2n+1} f_{ij}(t)x^i y^j. \] The functions \(f_{ij}(t)\) and \(h_{ij}(t)\) are quasi-periodic with a frequency \(\omega\) satisfying a Diophantine condition. It is assumed that the unperturbed system (\(f = 0\)) has a hyperbolic-type degenerate equilibrium point \((0,0).\) Using the KAM iteration, the author proves that if \(f\) is sufficiently small then there exists a quasi-periodic transformation such that system (1) can be reduced to a suitable normal form with an equilibrium point at \((0,0)\). Hence, system (1) has a small quasi-periodic solution.
hyperbolic-type degenerate equilibrium point
scientific article; zbMATH DE number 6157319