Hi,
It's probably a javascript error on that page triggered by a module/plugin which displays something for logged in users with additional javascript conflicting with another piece of javascript.
Because of that, the carousel and accordion systems cannot be initialized properly.
The first step is to look at your browser's javascript error console log on that page in order to get the error message(s).
That will give more information on the issue.
Another thing to do is to look at the javascript files added in the head of the page's HTML and see the ones that are there when you're not logged in/logged in and see if there is any difference, which could point out to the culprit javascript files.
Then, based on these results, it will be possible to either disable the module/plugin generating the error, or fix the bug/conflicting.