Small-amplitude limit cycles of some Liénard-type systems
Consider the Liénard systems \[ \dot x= y-{\sum^n_{i=0} a_i x^i\over 1+\sum^m_{i=1} b_i x^i},\quad \dot y= -g(x),\tag{1} \] \[ \dot x= y- {\sum^n_{j=0} a_j x^{2j}\over \sum^m_{j=0} b_j x^{2j}},\quad \dot y= -x(x^2- 1).\tag{2} \] The authors prove the following results: (i) Let \(g\) be a smooth function satisfying \(g(0)= 0\), \(g'(0)> 0\). Then the cyclicity of the equilibrium point \((0,a_0)\) of (1) is \([{n+m-1\over 2}]\). (ii) Let \(\sum^m_{j=0} b_j\neq 0\). Then system (2) has at most \(2[{n+m-1\over 2}]\) small amplitude limit cycles near the equilibrium \((0,a_0)\), where this number can be reached.
scientific article; zbMATH DE number 5649974