Ikeda map

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

Contents

• Attractor
• Point trajectories
• OctaveMATLAB code for point trajectories
• References

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 z n stands for the electric field inside the resonator at the n-th step of rotation in the resonator, and A and C are parameters which indicates laser light applied from the outside, and linear phase across the resonator, respectively. In particular the parameter B ≤ 1 is called dissipation parameter characterizing the loss of resonator, and in the limit of B = 1 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 u ≥ 0.6 , this system has a chaotic attractor.