In some cases, it may seem that cookies do not end after the browser is closed, because in fact not all browser instances are closed.
There are a number of situations where a browser session may be running in the background after it seems to have been closed.
Browser error in IE, may leave a hidden session open. In Google Chrome, some extensions use "hidden pages" that persist after the browser closes. On a Mac, inexperienced users might think that they closed the browser by pressing the X button, but they neglected clicking Safari> Quit Safar or Command-Q.
This is especially true for using cookies to log in. This has become a problem for us with Sharepoint 2013 and ADFS. Hope this helps!
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