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 "poi ...
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 ...
. It is an example of a dynamical system that exhibits
chaotic behavior. The Duffing
map
A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...
takes a point (''x
n'', ''y
n'') in the
plane
Plane most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
* Plane (mathematics), generalizations of a geometrical plane
Plane or planes may also refer to:
Biology
* Plane ...
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 Harmonic oscillator, damped and driven oscillators. The equation is given by
\dd ...
.
External links
Duffing oscillator on Scholarpedia
Chaotic maps
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