You should always look at the Javascript console for errors.
jquery.mobile-1.0.min.js:22 Uncaught TypeError: Cannot set property 'touchstart' of undefined
at String.<anonymous> (jquery.mobile-1.0.min.js:22)
at Function.each (jquery-1.10.2.min.js:4)
at jquery.mobile-1.0.min.js:22
at jquery.mobile-1.0.min.js:28
But this error may not be responsible for the menu problem. I tried on Safari, and get the error above on the console. But the menu works.
You are using jQuery Mobile 1.0, which is LONG obsolete. It is unreasonable to expect it to work properly with current browsers.
Current jQuery Mobile is 1.4.5. Unfortunately, it is significantly different, and probably a big job to update.
I don't understand why this site using jQuery Mobile. It doesn't seem to be using any of it's features. This is basically a Bootstrap site. No idea why JQM is tossed in there too.
The page loads a kitchen-sink of scripts, some redundant. You should load only ONE copy of jQuery! First you load 1.11.3. Then you load an older version, 1.10.2. Your page will use the second script, 1.10.2. BOTH are obsolete versions.
You should check ALL of your scripts for freshness. Since your jQuery and jQuery Mobile are so old, I'd guess that others are equally aged.
I guess you are updating a site that is several years old and that gets some updates every year with new content. I'd guess it's been passed-around to various people to maintain, and each one added their own special flourish and another script or two (I see you load two different carousel scripts - are both actually used?), and so the list of scripts loaded grows year by year. But it hasn't been maintained with current scripts, and I'd wager it's not understood why each one is there.
You are free to curse your predecessors for not keeping it up to date with current scripts that work with current browsers. ;)
<script type="text/javascript" src="js/jquery-1.10.2.min.js"></script>
<script type="text/javascript" src="js/jquery.ui.map.min.js"></script>
<script type="text/javascript" src="js/bootstrap.min.js"></script>
<script type="text/javascript" src="js/jquery.flexslider-min.js"></script>
<script type="text/javascript" src="js/jquery.carousel.js"></script>
<script type="text/javascript" src="js/jquery.scrollTo-1.4.3.1-min.js"></script>
<script type="text/javascript" src="js/jquery.parallax-1.1.3.js"></script>
<script type="text/javascript" src="js/jquery.localscroll-1.2.7-min.js"></script>
<script type="text/javascript" src="js/jquery.nav.js"></script>
<script type="text/javascript" src="js/jquery.countdown.min.js"></script>
<script type="text/javascript" src="js/jquery.validationengine-en.min.js"></script>
<script type="text/javascript" src="js/jquery.validationengine.min.js"></script>
<script type="text/javascript" src="js/tweetie.min.js"></script>
<script type="text/javascript" src="js/custom.js"></script>
<script src="js/lightbox.js"></script>
