7 problems

Let $f:M\rightarrow M$ be a $\mathrm{C}^{2}$ diffeomorphism with a compact basic set $\Lambda$ which has associated an almost homoclinic sequence. Let $p\in\Lambda$ be a hyperbolic…

Tangency-destruction conjecture. This strange attractor is destroyed in the tangency $\mathbf{Tan}_{\Gamma_o}$ between $W^u(\Gamma_o)$ and $W^s(\Gamma_o)$; however, the situation m…

Smale–horseshoe boundary conjecture. The Smale–horseshoe region was conjectured to be bounded by a first quadratic tangency between the stable and unstable manifolds of a saddle pe…

Let $C_{l,\epsilon}$ denote the left equilibrium of the FitzHugh-Nagumo system, let $q$ be the relevant saddle equilibrium, and let $W^u(C_{l,\epsilon})$ and $W^s(q)$ be their unst…

Fold-mechanism conjecture. This fold is a direct consequence of a portion of the unstable manifold passing close to the fold point of the critical manifold $M_{eq}$.

Perko's asymptotic conjecture. As $\mu_2\rightarrow-\infty$,

Perko's inequality conjecture. The inequality