My understanding is that 3135 occurs when a connection is lost. This doesn't tell you why the connection was lost, though. It may have been terminated by the server because the server failed to recieve a response to a probe for a certain amount of time, and assumed that the connection was dead. Or (I'm not sure about this) the exact reverse of that: the client failed to recieve a probe response from the server for a certain amount of time, so it assumed the connection was lost. The "certain amount of time" is cotrolled by SQLNET.EXPIRE_TIME=[minutes] in sqlnet.ora.
As for 12571, my (again vague) understanding is that there was a sudden failure to send a packet during communication with the server, and that this is typically caused by some software or hardware interfering with the connection (either by design, or by error). For instance, if you pull out your ethernet cable and then try to execute a query, you'll probably get this. Or if a firewall or anti-malware application decides to block the traffic.