Stability Implications of Bendixson’s Criterion

This note presents a proof that the omega limit set of a solution to a planar system satisfying the Bendixson criterion is either empty or is a single equilibrium. The proof involves elementary techniques which should be accessible to senior undergraduates and graduate students

A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\mathbb{R}^d$ and those of the parabolic equations $\dot u=\Delta u +f(x,u,\nabla u)$ on a bounded domain $\Omega$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\geq 3$ and in the corresponding cases $\Omega=(0,1)$, $\Omega=\mathbb{T}^1$ and dim($\Omega$)$\geq 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations

Entropy-Expansiveness and Domination for Surface Diffeomorphisms

Let f : M → M be a Cr-diffeomorphism, r ≥ 1, deffined on a closed manifold M. We prove that if M is a surface and K ⊂ M is a compact invariant set such that TKM = E ⊕ F is a dominated splitting then f/K is entropy expansive. Moreover C¹ generically in any dimension, isolated homoclinic classes H(p), p hyperbolic, are entropy expansive. Conversely, if there exists a C1 neighborhood U of a surface diffeomorphism f and a homoclinic class H(p), p hyperbolic, such that for every g ∈ U the continuation H(pg) of H(p) is entropy-expansive then there is a dominated splitting for f/H(p).Let f : M → M be a Cr-diffeomorphism, r ≥ 1, defined on a closed manifold M. We prove that if M is a surface and K ⊂ M is a compact invariant set such that TK M = E ⊕ F is a dominated splitting then f/K is entropy expansive. Moreover C1 generically in any dimension, isolated homoclinic classes H(p), p hyperbolic, are entropy expansive. Conversely, if there exists a C1 neighborhood U of a surface diffeomorphism f and a homoclinic class H(p), p hyperbolic, such that for every g ∈U the contin¬uation H(pg) of H(p) is entropy-expansive then there is a dominated splitting for f/H(p)

Robustly transitive sets and heterodimensional cycles

It is known that all non-hyperbolic robustly transitive sets Λφ have a dominated splitting and, generically, contain periodic points of different indices. We show that, for a C1-dense open subset of diffeomorphisms φ, the indices of periodic points in a robust transitive set Λφ form an interval in ℕ. We also prove that the homoclinic classes of two periodic points in Λφ are robustly equal. Finally, we describe what sort of homoclinic tangencies may appear in Λφ by studying its dominated splittings