Necessary and sufficient conditions for the boundedness of solutions to two-dimensional quadratic systems in a positively invariant half-plane
The author studies the planar system \[ \frac{dx}{dt}= x^2+xy+y, \quad \frac{dy}{dt}= a_2x^2+ b_2xy+c_2y^2+ \alpha_2x+\beta_2y \tag{1} \] under the conditions \[ c_2\neq 0, \qquad c_2\neq-1, \qquad c_2\neq b_2-a_2. \] He shows that the half-plane \(\Gamma:= \{x>-1\), \(y\in\mathbb R\}\) is positively invariant under the flow of (1) and that (1) can be reduced to a Liénard system. By this way the author is able to derive conditions on the coefficients of (1) such that {\parindent=7mm \begin{itemize}\item[(i)] All trajectories of (1) staring in \(\Gamma\) are bounded; \item[(ii)] System (1) has a limit cycle in \(\Gamma\); \item[(iii)] System (1) has two limit cycle sin \(\Gamma\). \end{itemize}}
scientific article; zbMATH DE number 5816854