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| In [[physics]] and [[mathematics]], the '''Ikeda map''' is a discrete-time [[dynamical system]] given by the [[complex map]]
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| :<math> z_{n+1} = A + B z_n e^{i (|z_n|^2 + C)} </math>
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| The original map was proposed first by Ikeda as a model of light going around
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| across a nonlinear optical resonator ([[Ring laser|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 <ref>K.Ikeda,
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| Multiple-valued Stationary State and its Instability of the Transmitted Light by a Ring Cavity System, Opt. Commun. 30 257-261 (1979); K. Ikeda, H. Daido and O. Akimoto, Optical Turbulence: Chaotic Behavior of Transmitted Light from a Ring Cavity, Phys. Rev. Lett. 45, 709–712 (1980)</ref> <math>z_n</math> stands for the electric field inside the resonator at the n-th step of rotation in the resonator, and <math>A</math> and <math>C</math> are parameters which indicates laser light applied from the outside, and linear phase across the resonator, respectively. In particular
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| the parameter <math>B \leq 1</math> is called dissipation parameter characterizing
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| the loss of resonator, and in the limit of <math>B=1</math> the Ikeda map becomes
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| a conservative map.
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| The original Ikeda map is often used in another modified form in order
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| to take the saturation effect of nonlinear dielectric medium into account:
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| :<math> z_{n+1} = A + B z_n e^{i K/(|z_n|^2 +1)+C} </math>
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| A 2D real example of the above form is:
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| :<math> x_{n+1} = 1 + u (x_n \cos t_n - y_n \sin t_n), \, </math>
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| :<math> y_{n+1} = u (x_n \sin t_n + y_n \cos t_n), \, </math>
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| where ''u'' is a parameter and
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| :<math> t_n = 0.4 - \frac{6}{1+x_n^2+y_n^2}. </math>
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| For some values of ''u'', this system has a [[chaotic attractor]].
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| ==Attractor==
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| This [[media:Ikeda map.ogg|animation]] shows how the attractor of the system changes as the parameter <math>u</math> 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 <math>u</math> is increased.
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| {|
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| | [[image:Ikeda0300.png|thumb|left| <math>u=0.3</math>]]
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| | [[image:Ikeda0500.png|thumb|left| <math>u=0.5</math>]]
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| |-
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| | [[image:Ikeda0700.png|thumb|left| <math>u=0.7</math>]]
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| | [[image:Ikeda0900.png|thumb|left| <math>u=0.9</math>]]
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| |}
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| ==Point trajectories==
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| The plots below show trajectories of 200 random points for various values of <math>u</math>. 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.
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| {|
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| | [[image:ikeda sim u0.1.png|thumb|250px|left| u = 0.1]]
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| | [[image:ikeda sim u0.5.png|thumb|250px|left| u = 0.5]]
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| | [[image:ikeda sim u0.65.png|thumb|250px|left| u = 0.65]]
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| |-
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| | [[image:ikeda sim u0.7.png|thumb|250px|left| u = 0.7]]
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| | [[image:ikeda sim u0.8.png|thumb|250px|left| u = 0.8]]
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| | [[image:ikeda sim u0.85.png|thumb|250px|left| u = 0.85]]
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| |-
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| | [[image:ikeda sim u0.9.png|thumb|250px|left| u = 0.9]]
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| | [[image:ikeda sim u0.908.png|thumb|250px|left| u = 0.908]]
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| | [[image:ikeda sim u0.92.png|thumb|250px|right| u = 0.92]]
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| |-
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| |}
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| ===Octave/MATLAB code for point trajectories===
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| The Octave/MATLAB code to generate these plots is given below:
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| <source lang="matlab">
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| % u = ikeda parameter
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| % option = what to plot
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| % 'trajectory' - plot trajectory of random starting points
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| % 'limit' - plot the last few iterations of random starting points
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| function ikeda(u, option)
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| P = 200;%how many starting points
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| N = 1000;%how many iterations
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| Nlimit = 20; %plot these many last points for 'limit' option
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| x = randn(1,P)*10;%the random starting points
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| y = randn(1,P)*10;
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| for n=1:P,
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| X = compute_ikeda_trajectory(u, x(n), y(n), N);
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| switch option
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| case 'trajectory' %plot the trajectories of a bunch of points
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| plot_ikeda_trajectory(X);hold on;
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| case 'limit'
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| plot_limit(X, Nlimit); hold on;
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| otherwise
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| disp('Not implemented');
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| end
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| end
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| axis tight; axis equal
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| text(-25,-15,['u = ' num2str(u)]);
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| text(-25,-18,['N = ' num2str(N) ' iterations']);
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| end
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| % Plot the last n points of the curve - to see end point or limit cycle
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| function plot_limit(X,n)
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| plot(X(end-n:end,1),X(end-n:end,2),'ko');
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| end
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| % Plot the whole trajectory
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| function plot_ikeda_trajectory(X)
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| plot(X(:,1),X(:,2),'k');
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| %hold on; plot(X(1,1),X(1,2),'bo','markerfacecolor','g'); hold off
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| end
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| %u is the ikeda parameter
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| %x,y is the starting point
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| %N is the number of iterations
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| function [X] = compute_ikeda_trajectory(u, x, y, N)
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| X = zeros(N,2);
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| X(1,:) = [x y];
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| for n = 2:N
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| t = 0.4 - 6/(1 + x^2 + y^2);
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| x1 = 1 + u*(x*cos(t) - y*sin(t)) ;
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| y1 = u*(x*sin(t) + y*cos(t)) ;
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| x = x1;
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| y = y1;
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| X(n,:) = [x y];
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| end
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| end
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| </source>
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| ==References==
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| {{reflist}}
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| {{Chaos theory}}
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| {{DEFAULTSORT:Ikeda Map}}
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| [[Category:Chaotic maps]]
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Hi there, I am Alyson Pomerleau and I believe it sounds fairly great when you say it. My working day job is an invoicing officer but I've currently utilized for another one. Her family members life in Alaska but her spouse wants them to transfer. To perform lacross is 1 of the things she loves most.
My page ... best psychics (publicpledge.com)