hysteresis (hĭsˌtərēˈsĭs) [key], phenomenon in which the response of a physical system to an external influence depends not only on the present magnitude of that influence but also on the previous history of the system. Expressed mathematically, the response to the external influence is a doubled-valued function; one value applies when the influence is increasing, the other applies when the influence is decreasing. Magnetic hysteresis occurs when a permeable material like soft iron is magnetized by being subjected to an external magnetic field. The induced magnetization tends to lag behind the magnetizing force. If a field is applied to an initially unmagnetized sample and is then removed, the sample retains a residual magnetization (it has become a permanent magnet). The graph of the magnetic induction B versus the magnetic field H is called a hysteresis loop. The area of the loop is proportional to the energy dissipated as heat when the system goes through a cycle; this represents a considerable energy loss in alternating-current machinery. Thermal hysteresis occurs when the value of a given property of a body depends not only on the body's temperature but also on whether the temperature is rising or falling. An example is the dielectric constant versus temperature for certain crystals. Another kind of hysteresis is a common feature of control or cybernetic systems. A familiar example is a thermostat controlling a source of heat and set at some temperature T0. When the room temperature falls through T0 to some lower temperature T1, the heating power is switched on. When the room temperature rises through T0 to some higher temperature T2, the power is switched off. Thus, for temperatures lower than T1, the heat is always on; for temperatures higher than T2, the heat is always off; but for temperatures between T1 and T2, the heat may be on or off (double-valued response), depending on which of the two temperatures T1 and T2 occurred most recently in the system's history. Unlike the previous examples, this hysteresis effect is not naturally occurring; it is designed into the control system to prevent the damage to the system that would arise from switching on and off too frequently.