Characteristic equation for autonomous planar half-linear differential systems
The authors investigate the planar half-linear differential system \[ x'=a(t)x+b(t)\phi_{p^\ast}(y),\, y'=c(t)\phi_p(x)+d(t)y,\leqno(S) \] where \(a\), \(b\), \(c\), \(d\in C(\mathbb R)\), \(p\), \(p^\ast>1\) with \(\frac{1}{p}+\frac{1}{p^\ast}=1\), \(\phi_q(s)=| s| ^{q-2}s\) if \(s\not=0\), and \(\phi_q(s)=0\) if \(s=0\), for \(q>1\), and the corresponding autonomous system \((S_0)\) with \(a\), \(b\), \(c\), \(d\) real constants. First they introduce and study the characteristic equation of \((S_0)\). Then, by using the generalized Prüfer transformation, they prove the existence and uniqueness of solutions to \((S)\) with the initial conditions \((x(t_0),y(t_0))=(x_0,y_0)\). A detailed asymptotic behavior of solutions for \((S_0)\) is also established by roots of the characteristic equation, and a classification of the origin \((0,0)\) for \((S_0)\) is finally addressed.
scientific article; zbMATH DE number 6806136
scientific article