In chaos theory, the Ikeda map is a discrete-time dynamical system that produces a strange attractor. It was introduced in 1979 by the physicist Kensuke Ikeda as a model for the behavior of light within a nonlinear optical resonator. The map demonstrates how a simple set of rules can lead to complex, chaotic behavior through a process of repeated rotation, scaling, and translationâÂÂa "stretch and fold" operation common in chaotic systems.
The map is defined by an iterative function on the complex plane. For a given complex number , the next value is calculated as:Here, represents the electric field in the resonator at step . The parameters and relate to the external laser light and the phase of the system, while (where ) is a dissipation parameter representing energy loss in the resonator.
A commonly studied real-valued version of the map is given by the two-dimensional equations:where is a parameter andFor values of the parameter , this system exhibits chaotic behavior, generating the characteristic fractal attractor shown in the article's images.
This shows how the attractor of the system changes as the parameter is varied from 0.0 to 1.0 in steps of 0.01. The Ikeda dynamical system is simulated for 500 steps, starting from 20,000 randomly placed starting points. The last 20 points of each trajectory are plotted to depict the attractor. Note the bifurcation of attractor points as is increased.
The plots below show trajectories of 200 random points for various values of . The inset plot on the left shows an estimate of the attractor while the inset on the right shows a zoomed in view of the main trajectory plot.
The Octave/MATLAB code to generate these plots is given below: