These are only ideas, as I have never experienced the program you described with any of my cable modem/router combinations (and none of them include a Linksys).
(1) very unlikely, in my opinion, that a firmware update, or lack thereof, to the cable modem or the router, would be causing this problem
(2) it sounds like something is running (perhaps hidden in background?) on your computer that is attempting to control the cable modem/router, so that when the computer is turned off and then back on, the cable modem "loses sync" and then attempts to re-establish a new connection, as though it was being connected to your Internet Service Provider for the very first time.
(3) since this may be something (even possibly something nefarious) running on your computer, I would try, if it were my setup, to do a System Restore of my computer to a date prior to September 2. If you have System Restore turned on in your computer, which usually sets a Restore Point every time you do a Windows Update or other system software change, you still should be able to find a Restore Date before September 2.
So I would try #3, with the computer DISCONNECTED from the router when you do the System Restore (and have both the modem and router unplugged from power0, then after the Restore completes, shut the computer down, then power up the modem and wait for its lights to settle down, then power up the router and wait for its lights to settle down, then connect the computer to the router and power up the computer.