Global planar dynamics with star nodes: beyond Hilbert's $16^{th}$ problem

TL;DR

The paper develops a global theory for planar polynomial vector fields with a nonzero multiple of the identity as the linear part and homogeneous nonlinearities, using the Poincaré compactification to analyze equilibria at infinity and their impact on the global flow. It derives bounds on the number of finite equilibria, characterizes invariant radii and flow-invariant sectors, and provides a complete description of dynamics with star nodes, including conditions for the existence of polycycles and limit cycles. The authors illustrate their results with detailed degree-2 and degree-3 examples, connecting their findings to classical classifications and yielding explicit phase portraits. This work extends the understanding of global planar dynamics beyond Hilbert's 16th problem by focusing on a structured class of nonlinearities and offering a constructive, geometry-driven framework for phase portraits.

Abstract

This is a full study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a homogeneous polynomial. It extends previous work by other authors that was mainly concerned with the existence and number of limit cycles. The general results are also applied to two classes of examples where the nonlinearities have degrees 2 and 3, for which we provide a set of phase portraits.

Figures (16)

• Figure 1: Dynamics of Example \ref{['ex:grau3']} on the Poincaré disk.
• Figure 2: Dynamics in invariant half-cones in Theorem \ref{['teo:NewGlobalDynamics']} for $\lambda>0$. In case \ref{['teoGlobalPP']} the unstable manifold of the finite equilibrium splits the half-cone in two basins of attraction. In case \ref{['teoGlobalPH']} there is a robust heteroclinic connection between two finite equilibria. In the remaining cases the origin is a global repellor for the interior of the half-cone.
• Figure 3: Dynamics in invariant half-cones in Theorem \ref{['teo:NewGlobalDynamics']} for $\lambda<0$. In case \ref{['teoGlobalPP']} the stable manifold of the finite equilibrium splits the half-cone in two basins of attraction. In case \ref{['teoGlobalHP']} there is a robust heteroclinic connection between two finite equilibria. In the remaining cases the origin is a global attractor for the interior of the half-cone.
• Figure 4: Dynamics in invariant half-cones in Theorem \ref{['teo:globalDynamics']} when there is an angle $H_0^\pm$ consisting of two sectors. The dashed line is the trajectory separating the sectors, it splits the half-cone in two basins of attraction or repulsion. In (a) and (b) $\lambda<0$, in (c) and (d) $\lambda>0$. In cases (b) and (d) there is a parabolic sector at infinity connecting the two infinite equilibria as in Example \ref{['ex:grau3']}.
• Figure 5: Phase portraits on the Poincaré disk for normal form \ref{['Date1']} in Proposition \ref{['prop:10vf-2']} with $\lambda>0$: (A) $q_1-q_2<1$; (B) $q_1-q_2>1$; (C) $q_1-q_2=1$. The dashed line in the case (C) where $q_1-q_2=1$ indicates the possibility of a parabolic sector at the infinite equilibrium $\theta=-\pi/4$. Whether this sector does or does not exist has to be decided by other methods, see Appendix \ref{['app:correspondence']}.

Theorems & Definitions (44)

• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Corollary 3.4
• Theorem 3.5