Emergence of Chaotic Dynamics from Singularities

Résumé

This book is a journey through singularities,unfolding, bifurcations and strange attractors. The first stage isa walk through essential results in the context ofdiffeomorphisms. After that, we see how these results can beapplied to flows of vector fields via Poincaré return maps. Inthis way, the persistence of strange attractors in families ofdiffeomorphisms which unfold a homoclinic tangency lead to thepersistence of suspended strange attractors in families of vectorfields. We explain how these attractors are formed from homoclinicbifurcations, mostly around a Shilnikov homoclinic orbit or arounda bifocal homoclinic orbit, and from heteroclinic bifurcations as,for instance, in neighborhoods of Bykov cycles. All these globalconfigurations will be introduced later. These structures aredifficult to detect in a phase space, but we prove that they arisein generic unfoldings of certain singularities. As alreadymentioned, proceeding in this way, we provide results that allowto conclude the persistence of strange attractors from thepresence of certain singularities in a given family.