Bistability

Bistability is a property of a dynamical system in which two distinct stable steady states exist under the same external conditions. Between them lies an unstable steady state that separates the basins of attraction. As a result, the system can reside in either stable state, with its actual state determined by initial conditions, perturbations, or history. Small perturbations near a stable state decay, while larger perturbations can switch the system to the other state.

Mathematically, for a system dx/dt = F(x, p), fixed points satisfy F(x) = 0. Bistability requires at least two stable fixed points x1, x2 (Jacobian matrices with eigenvalues of negative real parts) and at least one unstable fixed point xu between them (with a positive eigenvalue). Varying parameters p can move the fixed points and produce hysteresis, in which the state tracks different branches depending on the direction of parameter change.

Bistability commonly arises from nonlinear feedback and cooperativity, such as positive feedback loops and threshold-like responses. It is observed in chemical reaction networks with autocatalysis, genetic regulatory circuits, nonlinear optical and electronic resonators, and magnetic systems.

Examples include the genetic toggle switch, where two mutually repressing genes yield stable high/low expression states; in electronics, the Schmitt trigger and bistable multivibrator; and in chemistry, models like the Schlögl reaction showing two stable concentrations.

Applications involve memory storage, decision-making in cells, and robust switching in the presence of noise. Bistability underpins many natural and engineered systems by providing a simple mechanism for switches and memory.