LE_李雅普诺夫_洛伦兹系统李雅普诺夫指数_李雅普诺夫LE_againstyi9_

%功能：计算李亚普诺夫指数 function [Texp,Lexp]=lyapunov(n,rhs_ext_fcn,fcn_integrator,tstart,stept,tend,ystart,ioutp); % % 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 % n1=n; n2=n1*(n1+1); % Number of steps nit = round((tend-tstart)/stept); % Memory allocation y=zeros(n2,1); cum=zeros(n1,1); y0=y; gsc=cum; znorm=cum; % Initial values y(1:n)=ystart(:); for i=1:n1 y((n1+1)*i)=1.0; end; t=tstart; % Main loop for ITERLYAP=1:nit % Solutuion of extended ODE system [T,Y] = feval(fcn_integrator,rhs_ext_fcn,[t t+stept],y); t=t+stept; y=Y(size(Y,1),:); for i=1:n1 for j=1:n1 y0(n1*i+j)=y(n1*j+i); end; end; % % construct new orthonormal basis by gram-schmidt % znorm(1)=0.0; for j=1:n1 znorm(1)=znorm(1)+y0(n1*j+1)^2; end; znorm(1)=sqrt(znorm(1)); for j=1:n1 y0(n1*j+1)=y0(n1*j+1)/znorm(1); end; for j=2:n1 for k=1:(j-1) gsc(k)=0.0; for l=1:n1 gsc(k)=gsc(k)+y0(n1*l+j)*y0(n1*l+k); end; end; for k=1:n1 for l=1:(j-1) y0(n1*k+j)=y0(n1*k+j)-gsc(l)*y0(n1*k+l); end; end; znorm(j)=0.0; for k=1:n1 znorm(j)=znorm(j)+y0(n1*k+j)^2; end; znorm(j)=sqrt(znorm(j)); for k=1:n1 y0(n1*k+j)=y0(n1*k+j)/znorm(j); end; end; % % update running vector magnitudes % for k=1:n1 cum(k)=cum(k)+log(znorm(k)); end; % % normalize exponent % for k=1:n1 lp(k)=cum(k)/(t-tstart); end; % Output modification if ITERLYAP==1 Lexp=lp; Texp=t; else Lexp=[Lexp; lp]; Texp=[Texp; t]; end; if (mod(ITERLYAP,ioutp)==0) fprintf('t=%6.4f',t); for k=1:n1 fprintf(' %10.6f',lp(k)); end; fprintf('\n'); end; for i=1:n1 for j=1:n1 y(n1*j+i)=y0(n1*i+j); end; end; end;