Summary

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

2012

Session Number:B3L-C

Session:

Number:443

Visualization analysis on stretch-and-fold mechanism of chaotic attractors

Yutaka Shimada,  Takuya Kobayashi,  Tohru Ikeguchi,  Kazuyuki Aihara,  

pp.443-446

Publication Date:

Online ISSN:2188-5079

DOI:10.15248/proc.1.443

PDF download (2MB)

We propose a simple visualization method for detecting the stretch-and-fold mechanism in chaotic dynamical systems. In the proposed method, we first place a rectangle that is centered at a point on the trajectory of a chaotic attractor and then uniformly arrange points on the rectangle. Next, applying the dynamics and observing the temporal evolution of the center point and the uniformly arranged points on the rectangle, we track the temporal evolution of the distances between the center point and the points on the rectangle. Finally, we express them using multiple colors and draw the colors representing the distances at the initial position on the rectangle. Then, tracking the temporal changes of these colors generated by the temporal evolution of the distances, we can observe the appearance of stripe patterns on the rectangle over time in the case of chaotic dynamics. We show that our method can detect the stretch-and-fold mechanism in chaotic dynamics by the stripe pattern. In addition, the stripe patterns are evaluated quantitatively.

[1] R. Gilmore, Topological analysis of chaotic dynamical systems. Reviews of Modern Physics, 70, 1455-1529, 1998.

[2] R. Abraham and C. D. Shaw, Dynamics:The geometry of behaviour. part 2: Chaotic behavior. Aerial Press Santa Cruz CA, 1983.

[3] S. H. Strogatz, Nonlinear dynamics and chaos with applications to physics, biology, chemistry, and engineering. Westview Press, 2000.

[4] H. Morioka, T. Ikeguchi, and K. Aihara, Analyses on stretching and folding by the Lorenz plot and the Poincaré section. In Proceedings of the 2006 IEICE General Conference, A-2-24, 2005.

[5] M. Suefuji and T. Ikeguchi, Analyses on stretch-and-fold mechanism by the hilbert transform and the Poincaré section. In Proceedings of the 2011 IEICE Society Conference, A-2-8, 2011.

[6] M. Suefuji, K. Fujiwara, and T. Ikeguchi, Visualization of the stretch-and-fold mechanism in chaotic dynamics. In Proceedings of the 2011 IEICE General Conference, A-2-39, 2012.

[a] M. Suefuji, K. Fujiwara, and T. Ikeguchi, A Visualization Method for the Stretch-and-fold Mechanism in Chaotic Dynamics. to appear in this proceedings, 2012.

[7] Y. Shimada, T. Yamada, and T. Ikeguchi, Detecting Stretch-and-fold Mechanism in Chaotic Dynamics. to appear in International Journal of Bifurcation and Chaos, 2012.

[b] http://www.youtube.com/watch?v=h4kgDnDNi1U

[8] W. F. Langford, Numerical Studies of Torus Bifurcations. International Series of Numerical Mathematics, 70, 285-294, 1984.

[9] O. E. Rössler, An equation for continuous chaos. Physics Letters A, 57, 397-398, 1976.

[10] E. N. Lorenz, Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130-141, 1963.

[11] T. Matsumoto, L. O. Chua and M. Komuro, The double scroll. IEEE Transactions on Circuits and Systems, CAS32, 797-818, 1985.

[12] T. Matsumoto, M. Komuro,, H. Kokubu, and R. Tokunaga, Bifurcations:sights, sounds, and mathematics (Springer-Verlag Tokyo).

[13] H. F. Bremen, F. E.Udwadia, and W. Proskurowski, An efficient QR based method for the computation of Lyapunov exponents. Physica D, 101, 1-16, 1997.