moreover, there isn't any err.log in system after running.

Your code does not compile (at least not for me):

$ perl
use strict;
use warnings;

open (STDERR ">>err.log") or die "error!$!\n";
__END__
Missing comma after first argument to open function at - line 4, near 
+"">>err.log") "
Execution of - aborted due to compilation errors.
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