function [Texp,Lexp]=lyapunov(n,tstart,stept,tend,ystart,ioutp);
global DS;
global P;
global calculation_progress first_call;
global driver_window;
global TRJ_bufer Time_bufer bufer_i;
%
% Lyapunov exponent calcullation for ODE-system.
%
% The alogrithm employed in this m-file for determining Lyapunov
% exponents was proposed in
%
% A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano,
% "Determining Lyapunov Exponents from a Time Series," Physica D,
% Vol. 16, pp. 285-317, 1985.
%
% For integrating ODE system can be used any MATLAB ODE-suite methods.
% This function is a part of MATDS program - toolbox for dynamical system investigation
% See: http://www.math.rsu.ru/mexmat/kvm/matds/
%
% Input parameters:
% n - number of equation
% rhs_ext_fcn - handle of function with right hand side of extended ODE-system.
% This function must include RHS of ODE-system coupled with
% variational equation (n items of linearized systems, see Example).
% fcn_integrator - handle of ODE integrator function, for example: @ode45
% tstart - start values of independent value (time t)
% stept - step on t-variable for Gram-Schmidt renormalization procedure.
% tend - finish value of time
% ystart - start point of trajectory of ODE system.
% ioutp - step of print to MATLAB main window. ioutp==0 - no print,
% if ioutp>0 then each ioutp-th point will be print.
%
% Output parameters:
% Texp - time values
% Lexp - Lyapunov exponents to each time value.
%
% Users have to write their own ODE functions for their specified
% systems and use handle of this function as rhs_ext_fcn - parameter.
%
% Example. Lorenz system:
% dx/dt = sigma*(y - x) = f1
% dy/dt = r*x - y - x*z = f2
% dz/dt = x*y - b*z = f3
%
% The Jacobian of system:
% | -sigma sigma 0 |
% J = | r-z -1 -x |
% | y x -b |
%
% Then, the variational equation has a form:
%
% F = J*Y
% where Y is a square matrix with the same dimension as J.
% Corresponding m-file:
% function f=lorenz_ext(t,X)
% SIGMA = 10; R = 28; BETA = 8/3;
% x=X(1); y=X(2); z=X(3);
%
% Y= [X(4), X(7), X(10);
% X(5), X(8), X(11);
% X(6), X(9), X(12)];
% f=zeros(9,1);
% f(1)=SIGMA*(y-x); f(2)=-x*z+R*x-y; f(3)=x*y-BETA*z;
%
% Jac=[-SIGMA,SIGMA,0; R-z,-1,-x; y, x,-BETA];
%
% f(4:12)=Jac*Y;
%
% Run Lyapunov exponent calculation:
%
% [T,Res]=lyapunov(3,@lorenz_ext,@ode45,0,0.5,200,[0 1 0],10);
%
% See files: lorenz_ext, run_lyap.
%
% --------------------------------------------------------------------
% Copyright (C) 2004, Govorukhin V.N.
% This file is intended for use with MATLAB and was produced for MATDS-program
% http://www.math.rsu.ru/mexmat/kvm/matds/
% lyapunov.m is free software. lyapunov.m is distributed in the hope that it
% will be useful, but WITHOUT ANY WARRANTY.
%
%
% n=number of nonlinear odes
% n2=n*(n+1)=total number of odes
%
options = odeset('RelTol',DS(1).rel_error,'AbsTol',DS(1).abs_error,'MaxStep',DS(1).max_step,...
'OutputFcn',@odeoutp,'Refine',0,'InitialStep',0.001);
n_exp = DS(1).n_lyapunov;
n1=n; n2=n1*(n_exp+1);
neq=n2;
% Number of steps
nit = round((tend-tstart)/stept)+1;
% Memory allocation
y=zeros(n2,1);
cum=zeros(n2,1);
y0=y;
gsc=cum;
znorm=cum;
% Initial values
y(1:n)=ystart(:);
for i=1:n_exp y((n1+1)*i)=1.0; end;
t=tstart;
Fig_Lyap = figure;
set(Fig_Lyap,'Name','Lyapunov exponents','NumberTitle','off');
set(Fig_Lyap,'CloseRequestFcn','');
hold on;
box on;
timeplot = tstart+(tend-tstart)/10;
axis([tstart timeplot -1 1]);
title('Dynamics of Lyapunov exponents');
xlabel('t');
ylabel('Lyapunov exponents');
Fig_Lyap_Axes = findobj(Fig_Lyap,'Type','axes');
for i=1:n_exp
PlotLyap{i}=plot(tstart,0);
end;
uu=findobj(Fig_Lyap,'Type','line');
for i=1:size(uu,1)
set(uu(i),'EraseMode','none') ;
set(uu(i),'XData',[],'YData',[]);
set(uu(i),'Color',[0 0 rand]);
end
ITERLYAP = 0;
% Main loop
calculation_progress = 1;
while t
tt = t + stept;
ITERLYAP = ITERLYAP+1;
if tt>tend, tt = tend; end;
% Solutuion of extended ODE system
% [T,Y] = feval(fcn_integrator,rhs_ext_fcn,[t t+stept],y);
while calculation_progress == 1
[T,Y] = integrator(DS(1).method_int,@ode_lin,[t tt],y,options,P,n,neq,n_exp);
first_call = 0;
if calculation_progress == 99, break; end;
if ( T(size(T,1))
y=Y(size(Y,1),:);
y(1,1:n)=TRJ_bufer(bufer_i,1:n);
t = Time_bufer(bufer_i);
calculation_progress = 1;
else
calculation_progress = 0;
end;
end;
if (calculation_progress == 99)
break;
else
calculation_progress = 1;
end;
t=tt;
y=Y(size(Y,1),:);
first_call = 0;
%
% construct new orthonormal basis by gram-schmidt
%
znorm(1)=0.0;
for j=1:n1 znorm(1)=znorm(1)+y(n1+j)^2; end;
znorm(1)=sqrt(znorm(1));
for j=1:n1 y(n1+j)=y(n1+j)/znorm(1); end;
for j=2:n_exp
for k=1:(j-1)
gsc(k)=0.0;
for l=1:n1 gsc(k)=gsc(k)+y(n1*j+l)*y(n1*k+l); end;
end;
for k=1:n1
for l=1:(j-1)
y(n1*j+k)=y(n1*j+k)-gsc(l)*y(n1*l+k);
end;
end;
znorm(j)=0.0;
for k=1:n1 znorm(j)=znorm(j)+y(n1*j+k)^2; end;
znorm(j)=sqrt(znorm(j));
for k=1:n1 y(n1*j+k)=y(n1*j+k)/znorm(j); end;
end;
%
% update running vector magnitudes
%
for k=1:n_exp cum(k)=cum(k)+log(znorm(k)); end;
%
% normalize exponent
%
rescale = 0;
u1 =1.e10;
u2 =-1.e10;
for k=1:n_exp
lp(k)=cum(k)/(t-tstart);
% Plot
Xd=get(PlotLyap{k},'Xdata');
Yd=get(PlotLyap{k},'Ydata');
if timeplot
u1=min(u1,min(Yd));
u2=max(u2,max(Yd));
end;
Xd=[Xd t]; Yd=[Yd lp(k)];
set(PlotLyap{k},'Xdata',Xd,'Ydata',Yd);
end;
if timeplot
timeplot = timeplot+(tend-tstart)/20;
figure(Fig_Lyap);
axis([tstart timeplot u1 u2]); end;
drawnow;
% Output modification
if ITERLYAP==1
Lexp=lp;
Texp=t;
else
Lexp=[Lexp; lp];
Texp=[Texp; t];
end;
if (mod(ITERLYAP,ioutp)==0)
for k=1:n_exp
txtstring{k}=['\lambda_' int2str(k) '=' num2str(lp(k))];
end
legend(Fig_Lyap_Axes,txtstring,3);
end;
end;
ss=warndlg('Attention! Plot of lyapunov exponents will be closed!','Press OK to continue!');
uiwait(ss);
delete(Fig_Lyap);
fprintf('\n \n Results of Lyapunov exponents calculation: \n t=%6.4f',t);
for k=1:n_exp fprintf(' L%d=%f; ',k,lp(k)); end;
fprintf('\n');
if ~isempty(driver_window)
if ishandle(driver_window)
delete(driver_window);
driver_window = [];
end;
end;
calculation_progress = 0;
update_ds;
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