On the persistence of lower-dimensional tori in reversible systems with high dimensional degenerate equilibrium under small perturbations (Q2081933)

The authors consider reversible dynamical systems with low-dimensional tori. They provide conditions for the persistence of such tori in the case of high-dimensional degenerate equilibrium under small perturbations. They call a dynamical system on \(\mathbb{T}^n \times \mathbb{R}^m \times \mathbb{R}^p \times \mathbb{R}^p\) (with \(n \leq m\)) reversible if it has an involutive symmetry \(G\) such that \(G: (x,y,u,v) \rightarrow (-x,y,-u,v)\) for \(x\in\mathbb{T}^n, y\in\mathbb{R}^m, u\in\mathbb{R}^p\) and \(v\in\mathbb{R}^p\). The dynamical system they consider has the general form \(\dot{x} = \omega(y) + P^1(x,y,u,v)\), \(\dot{y} = P^2(x,y,u,v)\), \(\dot{u} = A(y)v +P^3(x,y,u,v)\), and \(\dot{v} = B(y)u + P^4(x,y,u,v)\). This system is called reversible if \(\mathcal{D}G \cdot F = -F \circ G\), where \(\mathcal{D}G\) is the differential of \(G\) and \(F = (\omega + P^1, P^2, Av+P^3, Bu + P^4)^T\) is the vector field of the dynamical system. The aim of the authors is to generalize the results by \textit{X. Wang} et al. [J. Math. Anal. Appl. 387, No. 2, 776--790 (2012; Zbl 1260.37029); Ergodic Theory Dyn. Syst. 35, No. 7, 2311--2333 (2015; Zbl 1351.37231)] on the persistence of low-dimensional tori for certain reversible systems. The main theorem in this paper applies to a class of nearly integrable reversible systems with a high-dimensional degenerate equilibrium. In this case, under certain technical assumptions (including a Diophantine condition on the frequency \(\omega\)), the authors establish the existence of an invariant \(n\)-dimensional torus that persists under small perturbations.