On stronger forms of Devaney chaos

We define and study stronger forms of Devaney chaos and name it as $\mathcal{F}-$Devaney chaos, where $\mathcal{F}$ is a family of subsets of $\mathbb{N}$. Examples of maps which is ${\mathcal{F}}_t-$Devaney chaotic but not ${\mathcal{F}}_{cf}-$Devaney chaotic, ${\mathcal{F}}_s-$Devaney chaotic but neither ${\mathcal{F}}_t-$Devaney chaotic nor ${\mathcal{F}}_{cf}-$Devaney chaotic are discussed. Further, we show that for the maps on infinite metric space without isolated points, $\mathcal{F}-$sensitivity is a redundant condition in the definition $\mathcal{F}-$Devaney chaos. Here $\mathcal{F}={\mathcal{F}}_s,{\mathcal{F}}_t,{\mathcal{F}}_{ts}$ or ${\mathcal{F}}_{cf}$. We also obtain conditions under which Devaney chaos implies ${\mathcal{F}}_s-$Devaney chaos or ${\mathcal{F}}_t-$Devaney chaos. Next, we define the concept of $(\mathcal{F},\mathcal{G})-P-$chaos and obtain conditions under which $({\mathcal{F}}_1,{\mathcal{G}}_1)-P-$chaos implies $\mathcal{F}-$Devaney chaos for different families ${\mathcal{F}}_1$ and ${\mathcal{G}}_1$.