Outstanding problems in magnetospheric substorms include what causes substorm onsets and whether substorms have some specific periodicity. This paper provides observations and interpretations that relate to these problems. We study a specific type of geosynchronous particle injections during magnetic storms and the possible relationship between the solar wind and injections. Three storm events are analyzed: the March--April 2001, October 2000, and September 1998 storms. During the main phase or initial recovery phase of the storms, recurrent particle injections with a period of 2--3 hours were observed by geosynchronous satellites. The particle fluxes exhibited a well-defined sawtooth profile with gradual decreases (flux dropouts) followed by rapid increases (injections). The sawtooth injections coincided well with AE increases and are interpreted as the signature of periodic substorms. In one case, sawtooth substorm injections were correlated with continuous solar wind pressure oscillations. In two other cases, sawtooth substorm injections occurred after a solar wind pressure impulse arrived at the magnetosphere. The observations show that substorms during storm times can be indeed periodic with a period of 2--3 hours and that solar wind variations may trigger the periodic substorms. We suggest that magnetospheric substorms have an intrinsic cycle time of 2--3 hours. If solar wind pressure oscillations with periods comparable to the substorm cycle time are imposed on the magnetosphere, some magnetospheric resonant state may be excited, and periodic substorms can be triggered. Otherwise, if the solar wind oscillates too fast, not enough energy is accumulated in the magnetotail for substorms to occur; if the period of solar wind pressure oscillations is too long, magnetospheric energy may be released through other processes including internally triggered substorms.