Geometric tools to determine the hyperbolicity of limit cycles (Q878468)

The authors consider autonomous differential systems \[ \dot x= V(x)\tag{\(*\)} \] in the phase plane assuming \(V\) to be twice continuously differentiable. Let \(\gamma\) be a limit cyclic of \((*)\) with period \(T\), where \(\gamma:= \{x\in R^2: x= p(t)\), \(0\leq t\leq T\}\). The goal of the paper is to find criteria for the hyperbolicity of \(\gamma\). (Note that the given definition of hyperbolicity is wrong, but the obtained results are correct.) For this purpose, the function \[ \nu(x):= {[V(x), W(x)]\wedge V(x)\over V(x)\wedge W(x)} \] is introduced, where \(W\) is a vector field transversal to \(V\), \([\cdot,\cdot]\) denotes the Lie bracket, \(V\wedge W\) means the determinant of the matrix \((V,W)\). The authors prove the relation \[ \int^T_0 \text{div\,}V(p(t))\,dt= \int^T_0\nu(p(t))\,dt. \] Since \(\int^T_0 \text{div\,}V(p(t))\,dt\neq 0\) characterizes the hyperbolicity of the limit cycle \(\gamma\) and the sign of the integral determines the stability of \(\gamma\), the authors study different classes of \(V\) and \(W\) in order to use \(\nu(x)\) to study the location and uniqueness of \(\gamma\). To these classes there belong Liénard systems and perturbed conservative systems.