Numerical algorithm for constructing Lyapunov functions of polynomial differential system (Q1032546)

The authors consider polynomial planar differential systems \[ {dx\over dt}= P(x,y),\quad {dy\over dt}= Q(x,y)\tag{1} \] assuming that \(x= y= 0\) is an asymptotically stable equilibrium. Their goal is to present an algorithm for constructing a Lyapunov function \(V(x,y)\) in the form \[ V(x,y)= R_0(x)+ R_1(x)y+ R_2(x) y^2, \] where the unknown functions \(R_i\), \(i= 0,1,2\), are assumed to have the representation \(R_i(x)= \sum^\infty_{j=0} r_{ij} x^j\), such that \(V\) can be used to estimate the stability region of the origin. Writing the derivative of \(V\) along system (1) in the form \[ {dV\over dt}\biggl|_{(1)}= \sum^m_{k=0} S_k(x) y^k, \] the basic idea of the algorithm consists in requiring either \(S_k= 0\) for \(k= 1,\dots, m\), that means \(dV/dt\) depends only on the variable \(x\), or \(S_k(x)\equiv 0\) for \(k\) odd and \(S_k(x)= d_k(x)\leq 0\) for \(k\) even. The present examples are generalized Liénard systems.