On closed orbits for a class of planar systems of high degree
Consider the system of differential equations \[ dx/dt= 1-x^py^q,\quad dy/dt= \alpha(x^py^q- y),\quad x\geq 0,\;q\geq 0,\tag{\(*\)} \] where \(p,q\geq 1\) are integers and \(\alpha> 0\). Under one of the following conditions: (1) \(q= 2\); (2) \(q= 3\) and \(p= 1, 2\) or 5; (3) \(q\geq 4\) and \(1\leq p\leq M\), the system \((*)\) has at least one unstable limit cycle near the equilibrium point \(C(1,1)\) when \(p/(q- 1)< \alpha\ll\alpha_2\). Moreover, if \(q\geq 4\) and \(p>M\), \((*)\) has at least one unstable limit cycle near \(C\) when \(\alpha_1\ll \alpha< p/(q- 1)\). For \(p\geq 2\) and \(q\geq 2\), \((*)\) has no closed orbit when \(0<\alpha<\delta\) or \(\Delta\) when \(\Delta<\alpha<\infty\). Here \(\alpha_{1,2}= p/(\sqrt q\pm 1)^2\) if \(q>1\) and \(\alpha_1= \alpha_2= p/4\) if \(q= 1\); \(M\), \(\delta\) and \(\Delta\) are some constants dependent on \(p\) and \(q\).
scientific article; zbMATH DE number 1104083
scientific article