Asymptotics of the Hausdorff dimension of the basic set which is generated in the vanishing of an equilibrium state of the saddle-saddle type (Q578682)

A one parameter family of flows \(v_{\epsilon}\) is considered in \(R^ 3\) which has an equilibrium in the origin with eigenvalues 0, \(\lambda <0\), \(\gamma >0\). It is assumed further that the restriction of the flow \(v_ 0\) to the center manifold is equivalent to \(z'=z^ 2+\beta z^ 3\), and that it has \(p\geq 2\) transversal homoclinic orbits corresponding to the origin. The assumptions imply that for small \(\epsilon >0\) the flow \(v_{\epsilon}\) has a nontrivial basic set \(\Lambda_{\epsilon}\) in a small neighbourhood of the set formed by the homoclinic orbits of \(v_ 0\). The main result of this concise paper is the proof of the theorem that establishes the asymptotic behaviour of the Hausdorff dimensions of the stable and unstable sets \((W_ s\) and \(W_ u)\) of \(\Lambda_{\epsilon}\). According to this \[ \dim W_ s(\Lambda_{\epsilon}) = 2 + \sqrt{\epsilon} \log (p/(\pi \gamma)) + o(\sqrt{\epsilon})\quad, \] \[ \dim W_ u(\Lambda^{\epsilon})=2 + \sqrt{\epsilon} \log (p/(-\pi \lambda)) + o(\sqrt{\epsilon}). \]