Oh, after almost half a day, finally, I found the exact solution.
The problem was with the Live Script URL.
In the sample code, they set the live Script URL, and I did not save it locally.
<link rel="stylesheet" href="http://code.jquery.com/ui/1.10.3/themes/smoothness/jquery-ui.css" /> <script src="http://code.jquery.com/jquery-1.9.1.js"></script> <script src="http://code.jquery.com/ui/1.10.3/jquery-ui.js"></script>
So it can be on mobile that all js and css are not loading, so for this reason they cannot find
'$( "#pur_cat_text" ).autocomplete'
and giving me an error.
Finally, I save both JS and CSS locally, and now I can run my code
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