Hi Gracefull,
You stumbled upon one of many known issues in Mavericks. A list is here:
http://www.displaylink.com/support/ticket.php?id=354
I have sort of good news, we are implementing a wide rage of workarounds to take care of this and some slowness and corruption and other gifts from Mavericks.
The situation is likely not going to be perfect, for example at the moment I have all my screens up but the layout is still lost at every connection. Not sure we can work around that.
As usual I can't give any dates for a release unfortunately. We are working on this as I'm writing, if you look at the date you can tell it's important to us.
Carlo