Ikeda map

The trajectories of 2000 random points in an Ikeda map with u = 0.918.

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

$${\displaystyle z_{n+1}=A+Bz_{n}e^{i(|z_{n}|^{2}+C)}}$$

The original map was proposed first by Kensuke 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 ^{[1] [2]} ${\displaystyle z_{n}}$ stands for the electric field inside the resonator at the n-th step of rotation in the resonator, and ${\displaystyle A}$ and ${\displaystyle C}$ are parameters which indicate laser light applied from the outside, and linear phase across the resonator, respectively. In particular the parameter ${\displaystyle B\leq 1}$ is called dissipation parameter characterizing the loss of resonator, and in the limit of ${\displaystyle B=1}$ 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:

$${\displaystyle z_{n+1}=A+Bz_{n}e^{iK/(|z_{n}|^{2}+1)+C}}$$

A 2D real example of the above form is:

$${\displaystyle x_{n+1}=1+u(x_{n}\cos t_{n}-y_{n}\sin t_{n}),\,}$$

$${\displaystyle y_{n+1}=u(x_{n}\sin t_{n}+y_{n}\cos t_{n}),}$$

where u is a parameter and

$${\displaystyle t_{n}=0.4-{\frac {6}{1+x_{n}^{2}+y_{n}^{2}}}.}$$

For ${\displaystyle u\geq 0.6}$, this system has a chaotic attractor.

Attractor

This animation shows how the attractor of the system changes as the parameter ${\displaystyle u}$ is varied from 0.0 to 1.0 in steps of 0.01. The Ikeda dynamical system is simulated for 500 steps, starting from 20000 randomly placed starting points. The last 20 points of each trajectory are plotted to depict the attractor. Note the bifurcation of attractor points as ${\displaystyle u}$ is increased.

${\displaystyle u=0.3}$	${\displaystyle u=0.5}$
${\displaystyle u=0.7}$	${\displaystyle u=0.9}$

Point trajectories

The plots below show trajectories of 200 random points for various values of ${\displaystyle u}$. 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.

u = 0.1	u = 0.5	u = 0.65
u = 0.7	u = 0.8	u = 0.85
u = 0.9	u = 0.908	u = 0.92

Octave/MATLAB code for point trajectories

The Ikeda map is composed by a rotation (by a radius-dependent angle), a rescaling, and a shift. This "stretch and fold" process gives rise to the strange attractor.

The Octave/MATLAB code to generate these plots is given below:

% u = ikeda parameter% option = what to plot%  'trajectory' - plot trajectory of random starting points%  'limit' - plot the last few iterations of random starting pointsfunctionikeda(u, option)P=200;% how many starting pointsN=1000;% how many iterationsNlimit=20;% plot these many last points for 'limit' optionx=randn(1,P)*10;% the random starting pointsy=randn(1,P)*10;forn=1:P,X=compute_ikeda_trajectory(u,x(n),y(n),N);switchoptioncase'trajectory'% plot the trajectories of a bunch of pointsplot_ikeda_trajectory(X);holdon;case'limit'plot_limit(X,Nlimit);holdon;otherwisedisp('Not implemented');endendaxistight;axisequaltext(-25,-15,['u = 'num2str(u)]);text(-25,-18,['N = 'num2str(N)' iterations']);end% Plot the last n points of the curve - to see end point or limit cyclefunctionplot_limit(X, n)plot(X(end-n:end,1),X(end-n:end,2),'ko');end% Plot the whole trajectoryfunctionplot_ikeda_trajectory(X)plot(X(:,1),X(:,2),'k');% hold on; plot(X(1,1), X(1,2), 'bo', 'markerfacecolor', 'g'); hold offend% u is the ikeda parameter% x,y is the starting point% N is the number of iterationsfunction[X]=compute_ikeda_trajectory(u, x, y, N)X=zeros(N,2);X(1,:)=[xy];forn=2:Nt=0.4-6/(1+x^2+y^2);x1=1+u*(x*cos(t)-y*sin(t));y1=u*(x*sin(t)+y*cos(t));x=x1;y=y1;X(n,:)=[xy];endend

Python code for point trajectories

importmathimportmatplotlib.pyplotaspltimportnumpyasnpdefmain(u:float,points=200,iterations=1000,nlim=20,limit=False,title=True):"""    Args:        u:float            ikeda parameter        points:int            number of starting points        iterations:int            number of iterations        nlim:int            plot these many last points for 'limit' option. Will plot all points if set to zero        limit:bool            plot the last few iterations of random starting points if True. Else Plot trajectories.        title:[str, NoneType]            display the name of the plot if the value is affirmative    """x=10*np.random.randn(points,1)y=10*np.random.randn(points,1)forninrange(points):X=compute_ikeda_trajectory(u,x[n][0],y[n][0],iterations)iflimit:plot_limit(X,nlim)tx,ty=2.5,-1.8else:plot_ikeda_trajectory(X)tx,ty=-30,-26plt.title(f"Ikeda Map ({u=:.2g}, {iterations=})")iftitleelseNonereturnpltdefcompute_ikeda_trajectory(u:float,x:float,y:float,N:int):"""Calculate a full trajectory    Args:        u - is the ikeda parameter        x, y - coordinates of the starting point        N - the number of iterations    Returns:        An array.    """X=np.zeros((N,2))forninrange(N):X[n]=np.array((x,y))t=0.4-6/(1+x**2+y**2)x1=1+u*(x*math.cos(t)-y*math.sin(t))y1=u*(x*math.sin(t)+y*math.cos(t))x=x1y=y1returnXdefplot_limit(X,n:int)->None:"""    Plot the last n points of the curve - to see end point or limit cycle    Args:        X: np.array            trajectory of an associated starting-point        n: int            number of "last" points to plot    """plt.plot(X[-n:,0],X[-n:,1],'ko')defplot_ikeda_trajectory(X)->None:"""    Plot the whole trajectory    Args:        X: np.array            trajectory of an associated starting-point    """plt.plot(X[:,0],X[:,1],"k")if__name__=="__main__":main(0.9,limit=True,nlim=0).show()

References

1. Ikeda, Kensuke (1979). "Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system". Optics Communications. 30 (2). Elsevier BV: 257–261. Bibcode:1979OptCo..30..257I. CiteSeerX   10.1.1.158.7964 . doi:10.1016/0030-4018(79)90090-7. ISSN   0030-4018.
2. Ikeda, K.; Daido, H.; Akimoto, O. (1980-09-01). "Optical Turbulence: Chaotic Behavior of Transmitted Light from a Ring Cavity". Physical Review Letters. 45 (9). American Physical Society (APS): 709–712. Bibcode:1980PhRvL..45..709I. doi:10.1103/physrevlett.45.709. ISSN   0031-9007.