Summary

Proceedings of the 2012 International Symposium on Nonlinear Theory and its Applications

2012

Session Number:A4L-D

Session:

Number:276

Finite-time Lyapunov exponents in nonlinear dynamical systems with time-delayed feedback

Kazutaka Kanno,  Atsushi Uchida,  

pp.276-279

Publication Date:

Online ISSN:2188-5079

DOI:10.15248/proc.1.276

PDF download (1019.4KB)

Nonlinear dynamical systems with time-delayed feedback show very rich dynamics due to the high-dimensionality of the systems induced by time-delayed feedback. Dynamical systems generate chaotic and regular motions, which induce local instability. These local motions can be quantified with finite-time Lyapunov exponents (FTLEs). However, methods for the calculation of FTLEs in time-delayed dynamical systems have not been well established yet because of the difficulty of calculating Lyapunov exponents in time-delayed high-dimensional systems. We present a method for calculating FTLEs in time-delayed dynamical systems, and apply it to the Mackey-Glass model and the Lang-Kobayashi equations for a semiconductor laser with optical feedback. We investigate the distributions of FTLEs for different parameter values. It is found that both the variance of the distribution of FTLEs and the maximum Lyapunov exponent decrease with increase of the delay time in chaotic regions.

[1] J. D. Farmer, Physica D, Vol. 4, pp. 366 (1982).

[2] R. Vicente, J. Dauden, P. Colet, and R. Toral, IEEE J. Quantum Electron., Vol. 41, pp. 541 (2005).

[3] A. Uchida K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, Nat. Photonics Vol. 2, pp. 728 (2008).

[4] I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, Phys. Rev. Lett., Vol. 103, pp. 024102 (2009).

[5] L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, Nat. Commun., Vol. 2, pp. 468 (2011).

[6] L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, Opt. Express, Vol. 20, pp. 3241 (2011).

[7] J. D. Farmer and J. J. Sidorowich, Phys. Rev. Lett., Vol. 59, pp. 845 (1987).

[8] T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, Phys. Rev. E, Vol. 85, pp. 016211 (2012).

[9] B. Eckhardt and D. Yao, Physica D, Vol. 65, pp. 100 (1993).

[10] A. Prasad and R. Ramaswamy, Phys. Rev. E, Vol. 60, pp. 2761 (1999).

[11] A. Uchida, K. Yoshimura, P. Davis, S. Yoshimori, and R. Roy, Phys. Rev. E, Vol. 78, pp. 036203 (2008).

[12] R. Lang and K. Kobayashi, IEEE J. Quantum Electron., Vol. 16, pp. 347 (1980).