Don't forget to chomp the incoming list. Also, if all you do is read the results immediately and use them directly, backticks are simpler.

chomp( my @Files = `ls -1t $LOGSTR` );
die "No files?" if ! @Files;
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I agree with the idea of using ls to get a sorted list of files. It's a lot easier than a manual stat and sort (though there's probably a module to do it for you).