Non-existence, uniqueness of limit cycles and center problem in a system that includes predator-prey systems and generalized Liénard equations
The authors study the number of limit cycles surrounding the origin of the system: \[ dx/dt=b(x)(F(x)- \varphi(y)), \qquad dy/dx=a(y)g(x), \] where \(F(x)\), \(g(x)\) and \(b(x)\) are \(C'\)-functions on the interval \((x_1,x_2)\), \(x_1\in (-\infty,0)\), \(x_2\in (0,+\infty)\), on this interval \(xg(x)>0\) (zero only for \(x=0\)), \(b(x)>0\), \(F(0)=0\); \(b(y)\) and \(a(y)\) are \(C'\)-functions, \(y\varphi(y)\geq 0\) (zero only for \(y=0\)) for all \(y\in (-\infty,+\infty)\). Sufficient conditions are obtained for the non-existence, uniqueness and hyperbolicity of periodic orbits and also for the existence of centers. The main tool used is a generalization of the Filippov transformation by \textit{A. F. Filippov} [Mat. Sb., n. Ser. 30(72), 171-180 (1952; Zbl 0046.31705)]. The authors apply the results on non-existence of periodic orbits and on the existence of centers to some predator-prey systems. The result on uniqueness is applied to some generalized Liénard equations.
scientific article; zbMATH DE number 1011015