In a
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
, multistability is the property of having multiple
stable equilibrium points in the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
spanned by the states in the system. By mathematical necessity, there must also be unstable equilibrium points between the stable points. Points that are stable in some dimensions and unstable in others are termed unstable, as is the case with the first three
Lagrangian points
In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of t ...
.
Bistability
Bistability is the special case with two stable equilibrium points. It is the simplest form of multistability, and can occur in systems with only one state variable, as it only takes a one-dimensional space to separate two points.
Initial instability
Near an unstable equilibrium, any system will be sensitive to noise, initial conditions and system parameters,
which can cause it to develop in one of multiple divergent directions.
In economics and social sciences, path dependence gives rise to divergent directions of development.
Some path dependent processes are adequately described by multistability,
by being initially sensitive to input, before reaching a stagnant state –
for example
market share instability, which can develop into a stable
monopoly
A monopoly (from Greek el, μόνος, mónos, single, alone, label=none and el, πωλεῖν, pōleîn, to sell, label=none), as described by Irving Fisher, is a market with the "absence of competition", creating a situation where a speci ...
for one of multiple possible vendors.
Multistable perception
In vision science,
multistable perception
Multistable perception (or bistable perception) is a perceptual phenomenon in which an observer experiences an unpredictable sequence of spontaneous subjective changes. While usually associated with visual perception (a form of optical illusion ...
characterizes the wavering percepts that can be brought about by certain visually ambiguous pattern such as the
Necker cube
The Necker cube is an optical illusion that was first published as a Rhomboid in 1832 by Swiss crystallographer Louis Albert Necker. It is a simple wire-frame, two dimensional drawing of a cube with no visual cues as to its orientation, so ...
,
monocular rivalry or
binocular rivalry Binocular rivalry is a phenomenon of visual perception in which perception alternates between different images presented to each eye.
When one image is presented to one eye and a very different image is presented to the other (also known as dich ...
.
Through lateral inhibition, a pattern in which one image, when stimulated, inhibit the activity of neighboring images.
External links
Definition of multistability from icbm.de
Optical illusions
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