Systems with hysteresis are nonlinear, and can be mathematically challenging to model. However, phenomena such as the magnetic hysteresis loops are mainly rate-independent, which makes a durable memory possible. Hysteresis can be a dynamic lag between an input and an output that disappears if the input is varied more slowly this is known as rate-dependent hysteresis. It is incorporated in many artificial systems: for example, in thermostats and Schmitt triggers, it prevents unwanted frequent switching. Hysteresis can be found in physics, chemistry, engineering, biology, and economics. In natural systems it is often associated with irreversible thermodynamic change such as phase transitions and with internal friction and dissipation is a common side effect. Hysteresis occurs in ferromagnetic and ferroelectric materials, as well as in the deformation of rubber bands and shape-memory alloys and many other natural phenomena. This history dependence is the basis of memory in a hard disk drive and the remanence that retains a record of the Earth's magnetic field magnitude in the past. Plots of a single component of the moment often form a loop or hysteresis curve, where there are different values of one variable depending on the direction of change of another variable. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past.

Hysteresis is the dependence of the state of a system on its history. Electric displacement field D of a ferroelectric material as the electric field E is first decreased, then increased.

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