Limit cycles of a Liénard cubic system with quadratic friction function
Consider the planar autonomous system \[ {dx\over dt}= y,\;{dy\over dt}= x[1-(1+ L)x+Lx^2]- \varepsilon \sum^3_{j=1} a_j x^{j-1}y,\tag{\(*\)} \] where \(-1<L<0\), \(\varepsilon>0\). System \((*)\) has three equilibria: \(E_1=(1/L,0)\), \(E_2= (1,0)\), \(S=(0,0)\), where \(S\) is a saddle, \(E_1\) and \(E_2\) are anti-saddles. Using the notation \((i_{E_1}, i_{E_2}, i_S)\) which denotes that \((*)\) has \(i_{E_1}(i_{E_2})\) limit cycles surrounding the equilibrium \(E_1(E_2)\), and \(i_S\) limit cycles surrounding all three equilibria, the authors describe a method to construct systems \((*)\) with the following distribution of limit cycles: \((2,0,0)\), \((0,2,0)\), \((1,1,1)\), \((1,1,0)\), \((1,0,1)\), \((0,1,1)\), \((0,0,2)\), \((0,1,2)\), \((1,0,2)\).
scientific article; zbMATH DE number 5659344