Shadowing multiple elbow orbits: An application of dynamical systems to perturbation theory (Q1893071)

Let \(M\) be a smooth compact manifold and \(X_a\) for \(-A < a < A\) a smooth family of vector bundles. Considering an initial value problem solutions won't vary continuously as \(a\) varies, at least not for unbounded time. But if one allows the initial data to vary simultaneously then under certain conditions there are nearby solutions which stay nearby for all times. More precisely the main result of the present paper says: Assume that for \(X_0\) all orbits approach hyperbolic fixed points as \(t \to \pm \infty\) and that the stable and unstable manifolds of these fixed points intersect transversally (meaning that their tangent spaces span the tangent space of the ambiend manifold at each intersection point). Let dist be a metric on \(M\) that gives the topology then there exist constants \(c > 0\) and \(a_0 > 0\) such that for every solution \(p_0 (t)\) of \(X_0\) there is a family \(p_a(t)\) of solutions of \(X_a\) for \(|a|\leq a_0\) such that \[ \text{dist} (p_0(t), p_a(t)) < ca \quad \text{for} \quad -\infty < t < \infty. \]