Ikeda Map

In physics and mathematics, the Ikeda map is a discrete-time dynamical system given by the complex map

The original map was proposed first by Ikeda as a model of light going around across a nonlinear optical resonator (ring cavity containing a nonlinear dielectric medium) in a more general form. It is reduced to the above simplified "normal" form by Ikeda, Daido and Akimoto stands for the electric field inside the resonator at the n-th step of rotation in the resonator, and and are parameters which indicates laser light applied from the outside, and linear phase across the resonator, respectively. In particular the parameter is called dissipation parameter characterizing the loss of resonator, and in the limit of the Ikeda map becomes a conservative map.

The original Ikeda map is often used in another modified form in order to take the saturation effect of nonlinear dielectric medium into account:

A 2D real example of the above form is:

where u is a parameter and

For some values of u, this system has a chaotic attractor.