As once was pointed out by an Apple Engineer: never ever do calendar calculations by yourself. There are tons of small and large pitfalls that most people are unaware of and thus are not taking into account.
If you need an accurate depiction of the (current) time in another Timezone there is no other way then use a library that has all the information regarding Timezones and its peculiarities.
If you do not care about being exact every day of the year then this might be overkill but you have to decide for yourself.
For Javascript there exists the https://github.com/mde/timezone-js library, which itself uses the Olsen Database (https://en.wikipedia.org/wiki/Tz_database).
You have to download the Timezone-Infos and provide them along your source code. Then you can do all that TZ-related calculations/
// 'now' in the Timezone of Chicago (Central Time, should be equal to that of Dallas)
var dt_chicago = new timezoneJS.Date('America/Chicago');
// 'now' in London
var dt_london = new timezoneJS.Date('Europe/London');
// 'now' in Brisbane - you should look up which timezones
// there are in Australia and which you want to display
var dt_brisbane = new timezoneJS.Date('Australia/Brisbane');
I just found out that there is another library http://momentjs.com/timezone/ but I have not used it yet so I may not recommend or advise against using it.