The Duffing map (also called as 'Holmes map') is a
discrete-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
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. Examples include the mathematical models that describe the swinging of a ...
. It is an example of a dynamical system that exhibits
chaotic behavior
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have ...
. The Duffing
map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
takes a point (''x
n'', ''y
n'') in the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
and maps it to a new point given by
:
:
The map depends on the two
constants ''a'' and ''b''. These are usually set to ''a'' = 2.75 and ''b'' = 0.2 to produce chaotic behaviour. It is a discrete version of the
Duffing equation
The Duffing equation (or Duffing oscillator), named after Georg Duffing (1861–1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by
:\ddot + \delta \dot + \ ...
.
References
External links
Duffing oscillator on Scholarpedia
Chaotic maps
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