nonlinear analysis of a dynamic system at selected control parameter c
Contents
- plot time series y1
- Calculation of autocorelation function
- estimation of power spectrum using matlab function
- attractor reconstruction using time delay
- Estimation of Embeding imension via calculation of correlation dimension
- vs radius of the sphere
- ploting correlatin integral versus radius in loglog scale
- ploting correlation dimension d2 versus radius in loglog scale
- estimation of correlation dimension out of original attrcator
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close all; clear c=28; %seting of control parameter c value Fs=100; % sampling frequency [Hz] T=[0:1/Fs:600-(1/Fs)]; % integration interval and sampling time y0=[2.9 -1.3 25.9]; % initial conditions Lorenz %y0=[2.0 2.0 2.0]; % initial conditions Rossler depend on DYNAMIC SYSTEM % integration of a dyanamic system [t,y] = ode45(@dynsys,T,y0,[],c); ndata = 50000; % number of data to be analysed y1=y(end-ndata+1:end,1); % last ndata for x variable y2=y(end-ndata+1:end,2); % last ndata for y variable y3=y(end-ndata+1:end,3); % last ndata for z variable y3D = [y1 y2 y3]; % three dimansional vector of ndata save dynsys_y.txt y3D -ascii -tabs % save all three components to file dynsys_y.txt save dynsys_yCDim.txt y3D -ascii -tabs % save all three components to file dynsys_yCDim.txt
plot time series y1
plot(y1);
% plot(y1(1:2000)); % plot components from 1 to 2000

======================================================================== Tisean Path: Note that some TISEAN programmes add heading lines to the output files and that the comment mark is # instead of %. This requires that we change the # sign to the % sign in order to import the file into Matlab.
% Path: to TISEAN bin files) tiseanPath = 'C:\Tisean\Tisean_3.0.0\bin\'; % ========================================================================
Calculation of autocorelation function
aoutocorelation function "tiesean_corr" (corr) calculated from the first columns c1 of file for D1000 points ... tau from 1Ts to 1000 Ts stored into file dynsys_ycorr.txt
system([tiseanPath,'tisean_corr -c1 -D1000 -o dynsys_ycorr.txt dynsys_y.txt']); dynsys_ycorr = load('dynsys_ycorr.txt'); % figure plot(dynsys_ycorr(:,1),dynsys_ycorr(:,2),'k');
TISEAN 3.0.0 (C) R. Hegger, H. Kantz, T. Schreiber (1998-2007) C:\Tisean\Tisean_3.0.0\bin\tisean_corr: Estimates the autocorrelations of a data set Using dynsys_y.txt as datafile, reading column 1 Use 50000 lines. Opened dynsys_ycorr.txt for writing

estimation of power spectrum using matlab function
using Hamming window window lenght ndata/100 Fs sampling frequency
h1 = spectrum.welch('Hamming',ndata/10); spek1 = msspectrum(h1,y(:,1),'Fs',Fs); % sample frequency ... Fs = 100 ps1 = spek1.Data; f1 = spek1.Frequencies; % frequwncy in Hz figure semilogy(f1,ps1,'k'); %plot(f1,ps1,'k'); % plot powerspectrum in linear scale

attractor reconstruction using time delay
delay function on input file dynsys_y.txt -M1 read one component -c1 read first column -m3 embeding dimension 3 -F3 format of embeding (xi,xi+1d,xi+2d) -d150 using time delay 150 -o store results into dynsys_delay.txt
system([tiseanPath,'delay -M1 -c1 -m3 -F3 -d10 dynsys_y.txt -o dynsys_delay.txt']); dynsys_ydelay = load('dynsys_delay.txt'); % reading dynsys_dela.txt figure; % ploting two dimensional projection plot(dynsys_ydelay(:,1),dynsys_ydelay(:,2)); % plot(dynsys_ydelay(1:2000,1),dynsys_ydelay(1:2000,2)); % plot 2D reconstruction from 1 to 2000 components % plot3(dynsys_ydelay(1:2000,1),dynsys_ydelay(1:2000,2),dynsys_ydelay(1:2000,3)); plot three 3D reconstructio
TISEAN 3.0.0 (C) R. Hegger, H. Kantz, T. Schreiber (1998-2007) C:\Tisean\Tisean_3.0.0\bin\delay: Produces delay vectors Using dynsys_y.txt as datafile! Using columns: 1 get_multi_series: first data item(s) used: 13.533056 Use 50000 lines. Opened dynsys_delay.txt for writing

Estimation of Embeding imension via calculation of correlation dimension
vs radius of the sphere
d2 function -c1 reading first columns from file dynsys_y.tyt -M1,8 using one component do up to 8 time delay embeding -d150 using tiem delay 150 sampling time Ts -r0.0030 minimal radius/length scale -t500 tailor correction windov approx twice of corr(t)=0 -o write results into file dynsys_y.txt.d2; dynsys_y.txt.c2 Estimation of embeding according to embedding ocena embedding dimenzije preko korelacijske dimenzije
system([tiseanPath,'d2 -c1 -M1,8 -d10 -r0.0030 -t500 dynsys_y.txt -o']); % smothing d2 by av-d2 reading dynsys_y.txt.d2 % av-d2 function % -m1 minimal embeding dimension % -M8 maximal % -a2 number of +-averaging points % -o output file name ab_Cdim1.txt system([tiseanPath,'av-d2 -m1 -M8 -a2 -oav_Cdim1.txt dynsys_y.txt.d2']); % smothing d2 [de1,de2,de3,de4,de5,de6,de7,de8] = read_av_d2('av_Cdim1.txt'); % reding average d2 [c1,c2,c3,c4,c5,c6,c7,c8] = read_d2('dynsys_y.txt.c2'); % reading correlation integral
TISEAN 3.0.0 (C) R. Hegger, H. Kantz, T. Schreiber (1998-2007) C:\Tisean\Tisean_3.0.0\bin\d2: Estimates the correlation sum, -dimension and -entropy Using dynsys_y.txt as datafile! Using columns: 1 Use 50000 lines. TISEAN 3.0.0 (C) R. Hegger, H. Kantz, T. Schreiber (1998-2007) C:\Tisean\Tisean_3.0.0\bin\av-d2: Smoothes the output of the d2 program Using dynsys_y.txt.d2 as datafile! Opened av_Cdim1.txt for writing
ploting correlatin integral versus radius in loglog scale
figure loglog((c1(:,1)),(c1(:,2)),'-') hold on loglog((c2(:,1)),(c2(:,2)),'-') hold on loglog((c3(:,1)),(c3(:,2)),'-') hold on loglog((c4(:,1)),(c4(:,2)),'-') hold on loglog((c5(:,1)),(c5(:,2)),'-') hold on loglog((c6(:,1)),(c6(:,2)),'-') hold on loglog((c7(:,1)),(c7(:,2)),'-') hold on loglog((c8(:,1)),(c8(:,2)),'-')

ploting correlation dimension d2 versus radius in loglog scale
figure semilogx((de1(:,1)),de1(:,2),'-g') hold on semilogx((de2(:,1)),de2(:,2),'-k') hold on semilogx((de3(:,1)),de3(:,2),'-') hold on semilogx((de4(:,1)),de4(:,2),'-') hold on semilogx((de5(:,1)),de5(:,2),'-') hold on semilogx((de6(:,1)),de6(:,2),'-') hold on semilogx((de7(:,1)),de7(:,2),'-') hold on semilogx((de8(:,1)),de8(:,2),'-')

estimation of correlation dimension out of original attrcator
d2 function -M3,1 Using three components do embeding with no time lags
system([tiseanPath,'d2 -M3,1 -r0.0030 -t500 dynsys_yCDim.txt -o']); system([tiseanPath,'av-d2 -m1 -M3 -a4 -odynsys_Cdim2.txt dynsys_yCDim.txt.d2']); [b1,b2,b3] = read_av_d2E('dynsys_Cdim2.txt'); [d1,d2,d3] = read_d2E('dynsys_yCDim.txt.c2'); figure loglog((d3(:,1)),(d3(:,2)),'-') figure semilogx((b3(:,1)),b3(:,2),'-')
TISEAN 3.0.0 (C) R. Hegger, H. Kantz, T. Schreiber (1998-2007) C:\Tisean\Tisean_3.0.0\bin\d2: Estimates the correlation sum, -dimension and -entropy Using dynsys_yCDim.txt as datafile! Using columns: 1 2 3 Use 50000 lines. TISEAN 3.0.0 (C) R. Hegger, H. Kantz, T. Schreiber (1998-2007) C:\Tisean\Tisean_3.0.0\bin\av-d2: Smoothes the output of the d2 program Using dynsys_yCDim.txt.d2 as datafile! Opened dynsys_Cdim2.txt for writing

