Summary

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

2012

Session Number:C3L-D

Session:

Number:755

Chaotic delayed maps and their natural measure

Valentin Flunkert,  Ingo Fischer,  

pp.755-758

Publication Date:

Online ISSN:2188-5079

DOI:10.15248/proc.1.755

PDF download (710.2KB)

Summary:
The natural measure of a chaotic attractor describes many statistical properties of the system. It is well known that the natural measure is related to the unstable periodic orbits in the attractor. Here, we investigate the natural measure of simple delayed maps. We argue that for large delay there are two types of unstable periodic orbits that need to be considered: (i) periodic orbits with periods much smaller than the delay and (ii) periodic orbits with periods close to multiples of the delay time. In a space time representation of the dynamics the latter orbits correspond to unstable pulse-like solutions. Our results suggest that in the limit of large delay times, the natural measure converges in the sense of the spatio-temporal interpretation of the delay system.

References:

[1] T. Erneux, Applied delay differential equations (Springer, ADDRESS, 2009).

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

[3] K. Ikeda and K. Matsumoto, Physica D 29, 223 (1987).

[4] I. Fischer, O. Hess, W. Elsäßer, and E. O. Göbel, Phys. Rev. Lett. 73, 2188 (1994).

[5] L. Larger and J. M. Dudley, Nature 465, 41 (2010).

[6] M. C. Mackey and L. Glass, Science 197, 287 (1977).

[7] S. Lepri, G. Giacomelli, A. Politi, and F. T. Arecchi, Physica D 70, 235 (1993).

[8] J. D. Farmer, Physica D 4, 366 (1982).

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

[10] C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. A 37, 1711 (1988).

[11] Y. C. Lai, Y. Nagai, and C. Grebogi, Phys. Rev. Lett. 79, 649 (1997).

[12] Y.-C. Lai, Phys. Rev. E 56, 6531 (1997).

[13] V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, Phys. Rev. Lett. 105, 254101 (2010).

[14] B. Sandstede and A. Scheel, Journal of Differential Equations 172, 134 (2001).