The Potters, who were sitting in eighth spot in December, have won once in their last 14 Premier League outings and are only three points above the bottom three.
They have crucial games against QPR, Norwich and Sunderland to come before Tottenham come to the Britannia Stadium on the occasion of their 150th anniversary.
And they finish the season at Southampton, but Pulis believes the club will 'enjoy' the challenge.
He told the Independent on Sunday: "You never see me doing somersaults when things are going well and you won't see me hanging from a tree.
"I accept that in life and football you have good times and bad, and it's how you deal with it.
"Adversity is when you show your strength of character. We're a very close and tight unit here and hopefully they'll enjoy this challenge."
With Pulis under pressure from his own supporters, the 55-year-old, who came back to the club for a second stint in charge in 2006, claims he is not fussed.
"Supporters have the right to say what they want," he said.
"I had booing when I was here the first time and when I came back. When I came to the club first time round (in 2002) we were getting 10,000, now we sell out nearly 28,000 every week."