Isochronous properties in fractal analysis of some planar vector fields (Q616308)

Let \(V\) be the planar autonomous \(C^\infty\) vector field. Set \(p\) be a focus with \(DV (p)\neq 0\) and \(\psi(t)\) be an isolated closed trajectory. Assume that corresponding Poincaré recurrence maps \(P\) are not degenerated, i.e. there exists \(k\in\mathbb{N}\) such that \(\partial^k_x(P(x)- x)|_{x=0}\neq 0\). Then the following statements hold. {\parindent=5mm \begin{itemize}\item[1.] (Belitskii type) Doing convenient changes of variables, the vector field near such invariant sets can be \(C^\infty\) locally changed into \[ \dot R= {g_kR^k\over 1+ bR^{k- 1}},\quad\dot\beta= 1+ d_mR^m+\cdots+ d_{k- 1}R^{k- 1}, \] or a polynomial one \[ \dot R= g_k R^k+ dR^{2k- 1},\quad\dot\beta= 1+ d_m R^m+\cdots+ d_{k- 1} R^{k-1}, \] where \((R,\beta)\) are (generalized) polar coordinates and \(g_k\neq 0\). \item[2.] (Existence) The limitation curve of the pull back of the cross-section exists pointwise, which depends on the initial point but independent of the choosing of the cross-section. Moreover, the limitation curve \({\mathcal L}\) is not a spiral, if and only if the focus or limit cycle is isochronous. \item[3.] (Minkowski box dimension) When the invariant set is anisochronous, then there are two cases. \end{itemize}} {\parindent=7mm \begin{itemize}\item[(a)] In the focus type, constants \(m\) and \(k- 1\) are both even. And we have that \(\dim_B({\mathcal L},\text{rad})= {2(k-m- 1)\over k- m}\) for \(m< k- 1\) and \(\dim_B({\mathcal L},\text{rad})= 1\) for \(m= k- 1\). \item[(b)] In the limit cycle type, we have that \(\dim_B({\mathcal L},\text{rad})= {2(k- m)- 1\over k- m}\). \end{itemize}}