Characterization and dynamics of certain classes of polynomial vector fields on the torus (Q6627036)

Let \(X=(P,Q,R)\) be a polynomial vector field in \(\mathbb{R}^3\) given by\N\[\N\dot x=P(x,y,z), \quad \dot y=Q(x,y,z), \quad \dot z=R(x,y,z).\N\]\NGiven \(a\in\mathbb(1,\infty)\), consider the torus\N\[\N\mathbb{T}^2=\{(x,y,z)\in\mathbb{R}^3\colon(x^2+y^2-a^2)^2+z^2-1=0\}.\N\]\NLet also \(M\), \(P\subset\mathbb{R}^3\) be planes given respectively by\N\[\Nax+by=0, \quad z-k=0,\N\]\Nwith \(a\), \(b\), \(k\in\mathbb{R}\). A circle \(C\subset\mathbb{T}^2\) is a \emph{meridian} (resp. \emph{parallel}) of \(\mathbb{T}^2\) if it is given by one of the connected components of \(M\cap\mathbb{T}^2\) (resp. \(P\cap\mathbb{T}^2\)).\N\NIn this paper the author provide a study of the qualitative properties of \(X|_{\mathbb{T}^2}\). Among the main results there is a characterization on \(P\), \(Q\) and \(R\), with \(\deg X=\max\{\deg P,\deg Q,\deg R\}\leqslant3\), for \(\mathbb{T}^2\) to be an invariant algebraic surface for \(X\); characterization of the Lie Bracket \([X,Y]\), with \(X\) and \(Y\) of degree two; a characterization of \(X\) with \(P\) and \(Q\) of any degree and \(R=0\); a characterization of \(X\) when it is cubic and Kolmogorov.\N\NThey also provided characterizations and sharp upper bounds on the number of invariant meridians and parallels for these vector fields. In particular, they proved that if \(X\) has degree \(n\), then \(\mathbb{T}^2\) has at most \(2(n-1)\) isolated invariant meridians and this upper bound is sharp.