There is a documented problem with Mavericks and the Cisco AnyConnect VPN Client, where the VPN connection frequently becomes unstable causing it to disconnect/reconnect continuously.

Cisco has released a temporary workaround to address this issue, which requires downloading an unreleased version of the AnyConnect VPN Client. This requires a SMARTnet-enabled login to download, so Cisco device owners that don’t have an active support contract will need to purchase one to get this and future official releases. More information is available at: https://supportforums.cisco.com/thread/2247235