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Repeated, Delayed Torque Variations Following X-ray Flux Enhancements in the Magnetar 1E$\;$1048.1$-$5937

We report on two years of flux and spin evolution monitoring of 1E 1048.1$-$5937, a 6.5-s X-ray pulsar identified as a magnetar. Using {\it Swift} XRT data, we observed an X-ray outburst consisting of an increase in the persistent 1--10 keV flux by a factor of 6.3$\pm$0.2, beginning on 2011 December 31 (MJD 55926). Following a delay of $\sim100$ days, the magnetar entered a period of large torque variability, with $\dotν$ reaching a factor of $4.55\pm0.05$ times the nominal value, before decaying in an oscillatory manner over a time scale of months. We show by comparing to previous outbursts from the source that this pattern of behavior may repeat itself with a quasi-period of $\sim1800$ days. We compare this phenomenology to periodic torque variations in radio pulsars, finding some similarities which suggest a magnetospheric origin for the behavior of 1E 1048.1$-$5937.