A corollary of the Poincaré-Bendixson theorem and periodic canards
Consider the planar system \[ \dot x= f(x,a),\tag{\(*\)} \] where \(f\) is smooth and \(a\) is a parameter varying in the interval \([a_-, a_+]\). Let \(\Gamma\) be a Jordan curve that bounds an open domain \(D\). Assume: \((\text{A}_1)\). For \(a\in [a_-, a_+]\) system \((*)\) has a unique equilibrium \(e(a)\) in \(\overline D\). \((\text{A}_2)\). \(\text{det\,}J(e(a))> 0\) for \(a\in [a_-, a_+]\). \((\text{A}_3)\). \(\text{tr\,}J(e(a_-))\,\text{tr\,}J(e(a_+))< 0\). Here, \(J\) is the Jacobian of \(f\) and tr denotes the trace. The authors prove as a Corollary of the Poincaré-Bendixson theorem: Proposition. If \((*)\) has no limit cycle in \(\overline D\) for \(a= a_-\) and \(a= a_+\), then there exists \(a^*\in (a_-, a_+)\) such that \((*)\) for \(a= a^*\) has a limit cycle located in \(\overline D\) and which touches \(\Gamma\). The authors apply this Proposition to singularly perturbed systems of the type \[ \dot x= y,\quad\varepsilon\dot y= -x+ F(y+ a), \] \[ \dot x= x(p- f(y)),\quad\dot y= y(-q+ x(r+ g(y)- ah(y))) \] in order to prove the existence of a periodic canard. The results of this paper are relevant to the use of topological degree in the analysis of canard solutions.
scientific article; zbMATH DE number 5638340