During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator $B_w^n:{\lambda }_p\left(A\right)\to {\lambda }_p\left(A\right)$ defined on the Köthe sequence space ${\lambda }_p\left(A\right)$ exhibits distributional $ϵ$-chaos for any $0<ϵ<\mathrm{diam}{\lambda }_p\left(A\right)$ and any $n\in \mathbb{N}$ is obtained. Under this assumption, the principal measure of $B_w^n$ is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional $ϵ$-chaos for any $0<ϵ<\mathrm{diam}{\lambda }_p\left(A\right)$.

This paper is a continuation of [1], where a explicit description of the scrambled sets of weakly unimodal functions of type 2∞ was given. Its aim is to show that, for an appropriate non-trivial subset of the above family of functions, this description can be made in a much more effective and informative way.

Si dimostra l'esistenza di infinite soluzioni «multi-bump» - e conseguentemente il comportamento caotico - per una classe di sistemi Hamiltoniani del secondo ordine della forma $-\ddot{q}+q=\left(g_1\left(\omega t\right)+g_2\left(t/\omega \right)\right)V^{\prime }\left(q\right)$ per $\omega$ sufficientemente piccolo. Qui $q\in {\mathbb{R}}^n$ , $g_1$ e $g_2$ sono funzioni strettamente positive e periodiche e $V$ è un potenziale superquadratico (ad esempio $V\left(q\right)={\left|q\right|}^4$ ).