Persistence of equilibria as periodic solutions of forced systems (Q657453)

The authors prove a nice characterization of persistence in dimension two. Let \(X:U\subset{\mathbb R}^2\to{\mathbb R}^2\) be a Lipschitz continuous vector field defined on an open and bounded neighborhood of the origin, and assume that the autonomous system \[ \dot x=X(x) \tag{1} \] has \(x=0\) as its unique equilibrium. Consider the perturbed system \[ \dot x=X(x)+p(t,x,\varepsilon), \tag{2} \] where \(p:{\mathbb R}\times U\times[0,1]\to{\mathbb R}^2\) is continuous, \(T\)-periodic in its first variable and such that \(p(t,x,0)\equiv0\). The equilibrium \(x=0\) is said to persist as a \(T\)-periodic solution if, given any \(p\) as above, there exists a \(T\)-periodic solution \(\varphi_\varepsilon(t)\) of the perturbed system~(2), for small \(\varepsilon>0\), and \[ \lim_{\varepsilon\to0}\|\varphi_\varepsilon(t)\|=0, \] uniformly in \(t\in{\mathbb R}\). The authors prove that \(x=0\) persists as a \(T\)-periodic solution if and only if system~(1) is not \(T\)-isochronous around the origin and deg\((X,U,0)\neq0\).