Abstract We use the variational method to extract the short periodic orbits of the Qi system within a certain topological length. The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by the means of phase portraits, Lyapunov exponents, and Poincaré maps. Based on several periodic orbits with different sizes and shapes, they are encoded systematically with two letters or four letters for two different sets of parameters. The periodic orbits outside the attractor with complex topology are discovered by accident. In addition, the bifurcations of cycles and the bifurcations of equilibria in the Qi system are explored by different methods respectively. In this process, the rule of orbital period changing with parameters is also investigated. The calculation and classification method of periodic orbits in this study can be widely used in other similar low-dimensional dissipative systems.
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 12205257, 11647085, and 11647086), the Shanxi Province Science Foundation for Youths (Grant No. 201901D211252), Fundamental Research Program of Shanxi Province (Grant No. 202203021221095), and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi of China (Grant Nos. 2019L0505, 2019L0554, and 2019L0572).
Corresponding Authors:
Chengwei Dong, Hantao Li
E-mail: dongchengwei@tsinghua.org.cn;lihantao@nuc.edu.cn
Cite this article:
Lian Jia(贾莲), Chengwei Dong(董成伟), Hantao Li(李瀚涛), and Xiaohong Sui(眭晓红) Unstable periodic orbits analysis in the Qi system 2023 Chin. Phys. B 32 040502
[1] Lorenz E N 1963 J. Atmos. Sci.20 130 [2] Chen G and Ueta T 1999 Int. J. Bifurcat. Chaos9 1465 [3] Lü J and Chen G 2002 Int. J. Bifurcat. Chaos12 659 [4] Čelikovský S and Chen G 2002 Int. J. Bifurcat. Chaos12 1789 [5] Čelikovský S and Chen G 2005 Chaos Soliton. Fract.26 1271 [6] Zhang D, Shi J Q, Sun Y, Yang X H and Ye Lei 2019 Acta Phys. Sin.68 240502 (in Chinese) [7] Yang Q, Chen G and Huang K 2007 Int. J. Bifurcat. Chaos17 3929 [8] Le X, Zhang S and Zeng Y 2018 Chin. J. Phys.56 2381 [9] Dadras S, Momeni H R and Qi G 2010 Nonlinear Dyn.62 391 [10] Zolfaghari-Nejad M, Charmi M and Hassanpoor H 2022 Complexity2022 4488971 [11] Mobayen S, Volos C K, Kaar S and Avuolu N 2018 Nonlinear Dyn.91 939 [12] Ding P F, Feng X Y and Wu C M 2020 Chin. Phys. B29 108202 [13] Qi G, Chen G, Du S, Chen Z and Yuan Z 2005 Physica A352 295 [14] Vaidyanathan S 2012 Adaptive controller and synchronizer design for the Qi-Chen chaotic system (Berlin: Springer Berlin Heidelberg) [15] Luo R 2008 Phys. Lett. A372 648 [16] Song L, Yang J and Xu S 2010 Nonlinear Anal.72 2326 [17] Li J F, Li N, Liu Y P and Gan Y 2008 Acta Phys. Sin.58 779 (in Chinese) [18] Qi G and Zhang J 2017 Chaos Soliton. Fract.99 7 [19] Wu W, Chen Z and Yuan Z 2009 Chaos Soliton. Fract.41 2756 [20] Harrington H A and Gorder R A V 2017 Nonlinear Dyn.88 715 [21] Esen O, Choudhury A G and Guha P 2016 Int. J. Bifurcat. Chaos26 1650215 [22] Wang J, Chen Z, Che G and Yuan Z 2008 Int. J. Bifurcat. Chaos18 3309 [23] Wang J Z, Chen Z Q and Yuan Z Z 2006 Chin. Phys.15 1216 [24] Jia H Y, Chen Z Q and Yuan Z Z 2009 Acta Phys. Sin.58 4469 (in Chinese) [25] Chen Z, Yang Y, Qi G and Yuan Z 2007 Phys. Lett. A360 696 [26] Wu W and Chen Z 2010 Nonlinear Dyn.60 615 [27] Artuso R, Aurell E and Cvitanovic P 1990 Nonlinearity3 325 [28] Artuso R, Aurell E and Cvitanovic P 1990 Nonlinearity3 361 [29] Lan Y and Cvitanović P 2004 Phys. Rev. E69 16217 [30] Dong C 2022 Fractal and Fractional6 190 [31] Lan Y and Li Y C 2008 Nonlinearity21 2801 [32] Wang D, Wang P and Lan Y 2018 Phys. Rev. E98 042204 [33] Azimi S, Ashtari O and Schneider T M 2022 Phys. Rev. E105 014217 [34] Boghosian B M, Brown A, Lätt J, Tang H, Fazendeiro L M and Coveney P V 2011 Philos. Trans. Roy. Soc. A369 2345 [35] Hao B L and Zheng W M 1998 Applied Symbolic Dynamics and Chaos (Singapore: World Scientific) pp. 6-10 [36] Cvitanović P, Artuso R, Mainieri R, Tanner G and Vattay G 2012 Chaos: Classical and Quantum (Copenhagen: Niels Bohr Institute) [37] Press W H, Teukolsky S A, Veterling W T and Flannery B P 1992 Numerical Recipes in C (Cambridge: Cambridge University Press) [38] Galias Z and Tucker W 2011 Int. J. Bifurcat. Chaos21 551 [39] Strogatz S H 2000 Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Cambridge: Perseus Books Publishing) pp. 312-313 [40] Wiggins S, Wiggins S and Golubitsky M 2003 Introduction to applied nonlinear dynamical systems and chaos (New York: Springer) [41] Guckenheimer J and Holms P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector (New York: Springer)
Altmetric calculates a score based on the online attention an article receives. Each coloured thread in the circle represents a different type of online attention. The number in the centre is the Altmetric score. Social media and mainstream news media are the main sources that calculate the score. Reference managers such as Mendeley are also tracked but do not contribute to the score. Older articles often score higher because they have had more time to get noticed. To account for this, Altmetric has included the context data for other articles of a similar age.
