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Detecting unstable periodic orbits (UPOs) from chaotic dynamic systems is a challenging problem. For a large number of complex systems, we can collect some experimental time series data but cannot find theoretical models to describe them. Thus, detecting unstable periodic orbits from experimental data can help us understand the chaotic properties of physical phenomenon without using theoretical models. We, in this paper, first use the dynamical transformation (DT) algorithm to detect unstable periodic orbits from chaotic systems, and find that the original DT algorithm can detect the UPOs from the time series of chaotic discrete map, but it is infeasible for the time series from continuous chaotic flow. In this regard, we then propose an improved DT algorithm that is based on the Poincare section method to detect the UPOs from continuous chaotic flow. In particular, we transform the continuous flow data into discrete map time series in terms of Poincare section, and then detect unstable periodic orbits from the transformed discrete map time series. In addition, we take Rössler and Lorenz chaotic systems as examples to demonstrate the effectiveness of our proposed method.
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Keywords:
- unstable periodic orbits /
- dynamical transformation /
- continuous chaotic dynamical system /
- Poincare section
[1] Grebogi C, Ott E, Yorke J A 1988 Phys. Rev. A 37 1711
[2] Auerbach D, Cvitanovic P, Eckmann J P, Gunaratne G, Procaccia I 1987 Phys. Rev. Lett. 58 2387
[3] Biham O, Wenzel W 1989 Phys. Rev. Lett. 63 819
[4] Pierson D, Moss F 1995 Phys. Rev. Lett. 75 2124
[5] Ott E, Grebogi C, Yorke J A 1990 Rhys. Rev. Lett. 64 1196
[6] Zhang X H, Shen K 1999 Acta Phys. Sin. 48 2186 (in Chinese) [张晓辉, 沈柯 1999 48 2186]
[7] Berry M V 1989 Proc. R. Soc. Lond. A423 219
[8] Luo X S, Chen G R, Wang B H, Fang J Q, Zou Y L, Quan H J 2003 Acta Phys. Sin. 52 790 (in Chinese) [罗晓曙, 陈关荣, 汪秉宏, 方锦清, 邹艳丽, 全宏俊 2003 52 790]
[9] Wu S H, Hao J H, Xu H B 2010 Chin. Phys. B 19 020509
[10] Xie Y, Xu J X, Kang Y M, Hu S J, Dan Y B 2003 Acta Phys. Sin. 52 1112 (in Chinese) [谢勇, 徐建学, 康艳梅, 胡三觉, 段玉斌 2003 52 1112]
[11] So P, Francis J T, Netoff T I, Gluckman B J, Schiff S J 1998 Biophysical Journal 74 2776
[12] Lathrop D P, Kostelich E J 1989 Phys. Rev. A 40 4028
[13] So P, Ott E, Schiff S J, Kaplan D T, Sauer T, Grebogi C 1996 Phys. Rev. Lett. 76 4705
[14] So P, Ott E, Sauer T, Gluckman B J, Grebogi C, Schiff S J 1997 Phys. Rev. E 55 5398
[15] Schmelcher P, Diakonos F K 1997 Phys. Rev. Lett. 23 4733
[16] Davidchack R L, Lai Y C 1999 Phys. Rev. E 60 6172
[17] Davidchack R L, Lai Y C, Klebanoff A, Bollt E M 2001 Rhys. Rev. A 287 99
[18] Pei X, Dolan K, Moss F, Lai Y C 1998 Chaos 8 853
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[1] Grebogi C, Ott E, Yorke J A 1988 Phys. Rev. A 37 1711
[2] Auerbach D, Cvitanovic P, Eckmann J P, Gunaratne G, Procaccia I 1987 Phys. Rev. Lett. 58 2387
[3] Biham O, Wenzel W 1989 Phys. Rev. Lett. 63 819
[4] Pierson D, Moss F 1995 Phys. Rev. Lett. 75 2124
[5] Ott E, Grebogi C, Yorke J A 1990 Rhys. Rev. Lett. 64 1196
[6] Zhang X H, Shen K 1999 Acta Phys. Sin. 48 2186 (in Chinese) [张晓辉, 沈柯 1999 48 2186]
[7] Berry M V 1989 Proc. R. Soc. Lond. A423 219
[8] Luo X S, Chen G R, Wang B H, Fang J Q, Zou Y L, Quan H J 2003 Acta Phys. Sin. 52 790 (in Chinese) [罗晓曙, 陈关荣, 汪秉宏, 方锦清, 邹艳丽, 全宏俊 2003 52 790]
[9] Wu S H, Hao J H, Xu H B 2010 Chin. Phys. B 19 020509
[10] Xie Y, Xu J X, Kang Y M, Hu S J, Dan Y B 2003 Acta Phys. Sin. 52 1112 (in Chinese) [谢勇, 徐建学, 康艳梅, 胡三觉, 段玉斌 2003 52 1112]
[11] So P, Francis J T, Netoff T I, Gluckman B J, Schiff S J 1998 Biophysical Journal 74 2776
[12] Lathrop D P, Kostelich E J 1989 Phys. Rev. A 40 4028
[13] So P, Ott E, Schiff S J, Kaplan D T, Sauer T, Grebogi C 1996 Phys. Rev. Lett. 76 4705
[14] So P, Ott E, Sauer T, Gluckman B J, Grebogi C, Schiff S J 1997 Phys. Rev. E 55 5398
[15] Schmelcher P, Diakonos F K 1997 Phys. Rev. Lett. 23 4733
[16] Davidchack R L, Lai Y C 1999 Phys. Rev. E 60 6172
[17] Davidchack R L, Lai Y C, Klebanoff A, Bollt E M 2001 Rhys. Rev. A 287 99
[18] Pei X, Dolan K, Moss F, Lai Y C 1998 Chaos 8 853
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