Description |
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)
dy/dt = r*x - y - x*z
dz/dt = x*y - b*z
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: lyapunov.m - algorithm
lorenz_ext - Lorenz system extended rhs
run_lyap - example of calling and result visualization
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. |