Chaos near a resonant inclination-flip

Horseshoes play a central role in dynamical systems and are observed in many chaotic systems. However most points in a neighborhood of the horseshoe escape after finitely many iterations. In this work we construct a new model by re-injecting the points that escape the horseshoe. We show that this model can be realized within an attractor of a flow arising from a three-dimensional vector field, after perturbation of an inclination-flip homoclinic orbit with a resonance. The dynamics of this model, without considering the re-injection, often contains a cuspidal horseshoe with positive entropy, and we show that for a computational example the dynamics with re-injection can have more complexity than the cuspidal horseshoe alone.