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matlab编写的Lyapunov指数计算程序

已有 2406 次阅读 2009-12-29 08:37 |个人分类:其它|系统分类:科普集锦|关键词:李雅普诺夫指数

一、计算连续方程Lyapunov指数的程序

其中给出了两个例子：

计算Rossler吸引子的Lyapunov指数

计算超混沌Rossler吸引子的Lyapunov指数

www.pudn/downloads39/sourcecode/math/detail132231.html

二、recnstitution重构相空间，在非线性系统分析中有重要的作用

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: 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];

maxstep% 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.

% --------------------------------------------------------------------

% This file is intended for use with MATLAB and was produced for MATDS-program

% 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');

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