# Hysteresis

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Hysteresis refers to systems that have memory, where the effects of the current input (or stimulus) to the system are experienced with a certain lag. Such a system may exhibit path dependence, or "rate-independent memory" [1]. Hysteresis phenomena occur in magnetic materials, ferromagnetic materials and ferroelectric materials, as well as in the elastic, electric, and magnetic behavior of materials, in which a lag occurs between the application and the removal of a force or field and its subsequent effect. Electric hysteresis occurs when applying a varying electric field, and elastic hysteresis occurs in response to a varying force. The term "hysteresis" is sometimes used in other fields, such as economics or biology, where it describes a memory, or lagging effect.

In a deterministic system with no dynamics or hysteresis, it is possible to predict the system's output at an instant in time, given only its input at that instant in time. In a system with hysteresis, this is not possible; there is no way to predict the output without knowing the system's current state, and there is no way to know the system's state without looking at the history of the input. This means that it is necessary to know the path that the input followed before it reached its current value.

Many physical systems naturally exhibit hysteresis. A piece of iron that is brought into a magnetic field retains some magnetization, even after the external magnetic field is removed. Once magnetized, the iron will stay magnetized indefinitely. To demagnetize the iron, it would be necessary to apply a magnetic field in the opposite direction. This is the effect that provides the element of memory in a hard disk drive.

A system may be explicitly designed to exhibit hysteresis, especially in control theory, by introducing a positive feedback.[clarification needed] For example, consider a thermostat that controls a furnace. The furnace is either off or on, with nothing in between. The thermostat is a system; the input is the temperature, and the output is the furnace state. If one wishes to maintain a temperature of 20 °C, then one might set the thermostat to turn the furnace on when the temperature drops below 18 °C, and turn it off when the temperature exceeds 22 °C. This thermostat has hysteresis. If the temperature is 21 °C, then it is not possible to predict whether the furnace is on or off without knowing the history of the temperature.

The word hysteresis is often used specifically to represent rate-independent state. This means that if some set of inputs X(t) produce an output Y(t), then the inputs X(αt) produce output Y(αt) for any α > 0. The magnetized iron or the thermostat have this property. Not all systems with state (or, equivalently, with memory) have this property; for example, a linear low-pass filter has state, but its state is rate-dependent.

The term is derived from ὑστέρησις, an ancient Greek word meaning "deficiency" or "lagging behind". It was coined by Sir James Alfred Ewing.