When an internet modem keep disconnecting and reconnecting it's usually caused by bad connections either inside the home or outside or the modem could be going bad.
The problem with an internet modem that keeps disconnecting and reconnecting can also be caused by something on the ISP end.
For example I have CenturyLink DSL at my home and for the last 6 months my DSL through Centurylink has been going down and the internet light keeps going out.
I have no other ISP around to sign up too so I'm stuck with Centurylink.
If you have DSL check the connections at your telephone lines and telephone box inside and check the wires in the box outside.
Sometimes water can get into the box and cause corroded connections which can cause your internet to go in and out.
But if you're on centurylink internet then it's likely the problem with the ISP as Centurylink has been having internet outages all over the USA.