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  • Last update: 5 years ago
  • Total downloads: 127
  • Operating system: Linux, Mac OS X, Win All, BSD, Solaris
  • Publisher: Vasiliy Govorukhin
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scripts default iconCalculation Lyapunov Exponents for ODE (Scripts) Publisher's description

Lyapunov exponent calcullation for ODE-system

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)];
R-z,-1,-x; y, x,-BETA];

% 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.

System Requirements:

MATLAB 6.0 (R12)
Program Release Status: New Release
Program Install Support: Install and Uninstall

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