Year: 2026
Type: article
Abstract: For a system in $$\mathbb {R}^n$$ R n consisting of an unstable star node and homogeneous nonlinearity, i.e. $$\dot{x}=\mu x+Q(x)$$ x ˙ = μ x + Q ( x ) with $$\mu >0$$ μ > 0 and Q ( x ) a homogeneous polynomial satisfying $$xQ(x)<0$$ x Q ( x ) < 0 for all $$x\ne 0$$ x ≠ 0 , Field in 1989 proved the existence of a unique topological invariant sphere. Here restricted to $$n=2$$ n = 2 , we prove that the topological invariant circle is of star shape, and obtain the classification of all their phase portraits in the Poincaré disc when Q is of degree 5 and the invariant circle is the circle $${\mathbb {S}}^1=\{(x,y,z) \in \mathbb {R}^3|\ x^2+y^2+z^2=1\}$$ S 1 = { ( x , y , z ) ∈ R 3 | x 2 + y 2 + z 2 = 1 } . Moreover, without the restriction $$xQ(x)<0$$ x Q ( x ) < 0 we characterize the relation between equilibria (both finite and infinite) and the invariant straight lines passing through the origin. These results extend and improve the previous results of other authors.
Source: Aequationes Mathematicae
Authors: Jaume Llibre, Xiang Zhang
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
Awards: G252719983, G3936534585
OpenAlex ID: W7161631746