Projections of polynomial vector fields and the Poincaré sphere
Let \(p:\mathbb{R}^n\to\mathbb{R}^n\) be a homogeneous polynomial vector function, and let \(\rho:\mathbb{R}^n \to\mathbb{R}\) be a homogeneous polynomial of positive degree \(d\). The authors introduce the projection of \(p\) with respect to \(\rho\) by the formula \[ p^{\rho}(x)=- {\textstyle\frac{1}{d}} L_p(\rho)(x)x+\rho(x)p(x), \] where \(L_p(\rho)\) is the Lie derivative of \(\rho\) with respect to \(p\). They develop a theory based on the following fact: specially defined projections map trajectories of \(x'=p(x)\) to trajectories of \(x'=p^{\rho}(x)\) (with a natural change of time). The theory is applied to projections of trajectories to the Poincaré sphere; conditions for the existence of unbounded solutions are obtained. A second-order equation \[ z''=g(z,z'), \] where \(g\) is a polynomial, is treated in detail.
scientific article; zbMATH DE number 1067762
scientific article