Studying the concept of basins of attraction, I reach several articles regarding the riddled basins.
The system under my study present some features that might resemble to this property. However, one of the conditions to present riddled basins according to [1] is that the system presents negative Lyapunov exponents transverse to the invariant set of the basin in which the attractor lies.
Could anyone explain me or direct me to an article in which this method is described?
Thank you in advance
[1]S. Camargo, R. L. Viana & C. Anteneodo, Intermingled basins in coupled Lorenz systems. Physical Review E, Vol. 85(3), 036207. (2012).
This can be done using Benetdin algorithm
that uses the Gramm-Schmidt orthogonalization
When You calculate Jacobian matrix along trajectory You can calculate eigenvectors.
Yes, of course. And on each step You have to make eigenvectors orthogonal to each other.
Thank You very much for your answers !!!
Dr. Politanskyi, Im havent been able to find anything on the internet about the Benetdin Algorithm. Could you please provide me a reference on this.
Prof. Koshel, if I calculate the Jacobian Matrix along the trajectory, am I calculating the Lyapunov exponent which is transverse to the direction of the trajectory?
Is there any difference between the transverse Lyapunov exponent and the regular Lyapunov exponent calculated by means of the natural logarithm of infinitesimal close trajectories?
in 2d case the Jacobian Matrix have two eigenvalueы and two eigenvectors. Usually the positive larger one is that You mean as the regular Lyapunov exponent. You must study what is eigenvalue and eigenvector. I do not know the good book for linear algebra and geomentry in English. I will try to find tomorrow. Also You must study what mean "And on each step You have to make eigenvectors orthogonal to each other." or "the Gramm-Schmidt orthogonalization" All things are explained in elementary books. After that You would understand what is the Lyapunov exponent which is transverse to the direction of the trajectory.
If we give You recipe it will be useless.
Thank You very much Prof. Koshel.
Let me study this matter.
Best Regards
May that book will be useful
Cheney, Ward; Kincaid, David (2009). Linear Algebra: Theory and Applications. Sudbury, Ma: Jones and Bartlett. pp. 544, 558. ISBN 978-0-7637-5020-6.
Hasegawa, et al., First-principles calculations of electron states of a silicon nanowire with 100,000 atoms on the K computer. 2011
Bau III, David; Trefethen, Lloyd N. (1997), Numerical linear algebra, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-361-9.
Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins, ISBN 978-0-8018-5414-9.
Greub, Werner (1975), Linear Algebra (4th ed.), Springer.
Soliverez, C. E.; Gagliano, E. (1985), "Orthonormalization on the plane: a geometric approach" (PDF), Mex. J. Phys. 31 (4): 743–758.
Or see here for first view https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
I have many books on linear algebra in English, but I do not know which ones are good.
I have an a code in C++ for Transverse Lyapunov Exponen calculation
Thank You again for your comments.
Im checking the bibliography that you provided and searching fot the Benetdin algorithm.