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The strange attractor of a chaotic system is composed of numerous periodic orbits densely covered. The periodic orbit is the simplest invariant set except for the fixed point in the nonlinear dynamic system, it not only reflects all the characteristics of the chaotic motion, but also is closely related to the amplitude generation and change of chaotic system. Therefore, it is of great significance to obtain the periodic orbits in order to analyze the dynamical behaviors of the complex system. In this paper, we study the periodic orbits of the diffusionless Lorenz equations which are derived in the limit of high Rayleigh and Prandtl numbers. A new approach to establishing one-dimensional symbolic dynamics is proposed, and the periodic orbits based on a topological structure are systematically calculated. We use the variational method to locate the cycles, which is proposed to explore the periodic orbits in high-dimensional chaotic systems. The method not only preserves the robustness characteristics of most of other methods, such as the Newton descent method and multipoint shooting method, but it also has the characteristics of fast convergence when the search process is close to the real cycle in practice. In order to apply the method, a rough loop guess must be made first based on the entire topology for the cycle to be searched, and then the variational algorithm will bring the initial loop guess to evolving toward the real periodic orbit in the system. In the calculations, the Newton descent method is used to achieve stability. Two cycles can be used as basic building blocks for initialization, searching for more complex cycles with multiple circuits around the two fixed points requires more delicate initial conditions; otherwise, it will probably lead to nonconvergence. We can initialize the loop guess for longer cycles constructed by cutting and gluing the short, known cycles. For this system, such a method yields quite a good systematic initial guess for longer cycles. Even if we deform the orbit manually into a closed loop, the variational method still shows its powerfulness for good convergence. The topological classification based on the entire orbital structure is shown to be effective. Furthermore, the deformation of periodic orbits with the change of parameters is discussed, which provides a route to the periods of cycles. The present research may provide a method of performing systematic calculation and classification of periodic orbits in other similar chaotic systems.
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Keywords:
- diffusionless Lorenz system /
- periodic orbit /
- variational method /
- symbolic dynamics
[1] Lorenz E N 1963 J. Atmos. Sci. 20 130
[2] Rössler O E 1976 Phys. Lett. A 57 397
[3] Chen G R, Ueta T 1999 Int. J. Bifurcation Chaos 9 1465
[4] Lü J H, Chen G R 2002 Int. J. Bifurcation Chaos 12 1789
[5] Schrier G V D, Maas L R M 2000 Physica D 141 19
[6] Dwivedi A, Mittal A K, Dwivedi S 2012 Iet Commun. 6 2016
[7] Pehlivan I, Uyaro Y 2007 Iet Commun. 1 1015
[8] Xu Y, Gu R, Zhang H, Li D 2012 Int. J. Bifurcation Chaos 22 1250088
[9] He S, Sun K, Banerjee S 2016 Eur. Phys. J. Plus 131 254
[10] Huang D 2003 Phys. Lett. A 309 248
[11] Wei Z, Yang Q 2009 Comput. Math. Appl. 58 1979
[12] Wang Z, Li Y X, Xi X J, Wang X F 2014 Adv. Mater. Res. 905 651
[13] Strogatz S H 2000 Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (New York: Perseus Books Publishing) p301
[14] Artuso R, Aurell E, Cvitanović P 1990 Nonlinearity 3 325
[15] Artuso R, Aurell E, Cvitanović P 1990 Nonlinearity 3 361
[16] Cvitanovi P, Artuso R, Mainieri R, Tanner G, Vattay G, Whelan N, Wirzba A 2012 Chaos: Classical and Quantum (Copenhagen: Niels Bohr Institute) p395
[17] Hao B L, Zheng W M 1998 Applied Symbolic Dynamics and Chaos (Singapore: World Scientific) p13
[18] Lan Y, Cvitanović P 2004 Phys. Rev. E 69 016217
[19] Press W H, Teukolsky S A, Veterling W T, Flannery B P 1992 Numerical Recipes in Fortran 77 The Art of Scientific Computing (New York: Cambridge) p34
[20] Dong C, Lan Y 2014 Commun. Nonlinear Sci. Numer. Simul. 19 2140
[21] Dong C 2018 Mod. Phys. Lett. B 32 1850155
[22] Dong C 2018 Int. J. Mod. Phys. B 32 1850227
[23] Dong C 2018 Chin. Phys. B 27 080501
[24] Dong C 2018 Europhys. Lett. 123 20005
[25] Dong C, Wang P, Du M, Uzer T, Lan Y 2016 Mod. Phys. Lett. B 30 1650183
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[1] Lorenz E N 1963 J. Atmos. Sci. 20 130
[2] Rössler O E 1976 Phys. Lett. A 57 397
[3] Chen G R, Ueta T 1999 Int. J. Bifurcation Chaos 9 1465
[4] Lü J H, Chen G R 2002 Int. J. Bifurcation Chaos 12 1789
[5] Schrier G V D, Maas L R M 2000 Physica D 141 19
[6] Dwivedi A, Mittal A K, Dwivedi S 2012 Iet Commun. 6 2016
[7] Pehlivan I, Uyaro Y 2007 Iet Commun. 1 1015
[8] Xu Y, Gu R, Zhang H, Li D 2012 Int. J. Bifurcation Chaos 22 1250088
[9] He S, Sun K, Banerjee S 2016 Eur. Phys. J. Plus 131 254
[10] Huang D 2003 Phys. Lett. A 309 248
[11] Wei Z, Yang Q 2009 Comput. Math. Appl. 58 1979
[12] Wang Z, Li Y X, Xi X J, Wang X F 2014 Adv. Mater. Res. 905 651
[13] Strogatz S H 2000 Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (New York: Perseus Books Publishing) p301
[14] Artuso R, Aurell E, Cvitanović P 1990 Nonlinearity 3 325
[15] Artuso R, Aurell E, Cvitanović P 1990 Nonlinearity 3 361
[16] Cvitanovi P, Artuso R, Mainieri R, Tanner G, Vattay G, Whelan N, Wirzba A 2012 Chaos: Classical and Quantum (Copenhagen: Niels Bohr Institute) p395
[17] Hao B L, Zheng W M 1998 Applied Symbolic Dynamics and Chaos (Singapore: World Scientific) p13
[18] Lan Y, Cvitanović P 2004 Phys. Rev. E 69 016217
[19] Press W H, Teukolsky S A, Veterling W T, Flannery B P 1992 Numerical Recipes in Fortran 77 The Art of Scientific Computing (New York: Cambridge) p34
[20] Dong C, Lan Y 2014 Commun. Nonlinear Sci. Numer. Simul. 19 2140
[21] Dong C 2018 Mod. Phys. Lett. B 32 1850155
[22] Dong C 2018 Int. J. Mod. Phys. B 32 1850227
[23] Dong C 2018 Chin. Phys. B 27 080501
[24] Dong C 2018 Europhys. Lett. 123 20005
[25] Dong C, Wang P, Du M, Uzer T, Lan Y 2016 Mod. Phys. Lett. B 30 1650183
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