In a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
, 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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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 gravitational influence of two massive orbiting bodies. Mathematically, this involves 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
Market share is the percentage of the total revenue or sales in a Market (economics), market that a company's business makes up. For example, if there are 50,000 units sold per year in a given industry, a company whose sales were 5,000 of those ...
instability, which can develop into a stable
monopoly
A monopoly (from Greek language, Greek and ) is a market in which one person or company is the only supplier of a particular good or service. A monopoly is characterized by a lack of economic Competition (economics), competition to produce ...
for one of multiple possible vendors.
External links
Definition of multistability from icbm.de
Optical illusions
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