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Applied Physics 2023
保积的Lozi映射中与抛物周期点和双曲周期点相关的轨道特征
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Abstract:
本文研究了与Lozi保积映射的抛物和双曲周期点相关的轨道特征。证明了其全轨道相对于y=-x都是对称的。首先证明了当a=2时Lozi映射的轨道绕原点旋转，并且指出其轨道可能是发散的，也可能是稳定的周期轨。然后证明了a=-2时Lozi映射的轨道全平面发散，并且发散轨道最终都是属于平行于二四象限的对角平分线的平行直线族。最后证明了当｜a｜＞2时Lozi映射的轨道沿着双曲线在第二或第四象限发散。
In this paper, the orbital characteristics associated with a parabolic and hyperbolic periodic point of the Lozi area-preserving map are studied. It is proved that the full trajectory is symmetric about y=-x . We first prove that the trajectory of the Lozi map rotates around the origin, and point out that the trajectory may be divergent or stable periodic when a=2 . Then it is proved that the trajectory of the Lozi map diverges in all planes, and the divergent trajectory eventually belongs to the family of lines parallel to the diagonal bisectors of the two and four quadrants when a=-2 . Finally, it is proved that the trajectory of the Lozi map diverges along the hyperbola in the second or fourth quadrant when ｜a｜＞2 .
| [1] | Lozi, R. (1978) Un Attracteur étrange (?) du type attracteur de Hénon. Journal de Physique Colloque, 39, 5-9.
https://doi.org/10.1051/jphyscol:1978505 |
| [2] | Misiurewicz, M. and ？timac, S. (2016) Symbolic Dynamics for Lozi Maps. Nonlinearity, 29, 3031-3046.
https://doi.org/10.1088/0951-7715/29/10/3031 |
| [3] | Gupta, C., Holland, M. and Nicol, M. (2011) Extreme Value Theory for Sinai Dispersing Billiards, Lozi Maps and Lorenz Like Maps. Ergodic Theory and Dynamical Systems, 31, 1363-1390.
https://doi.org/10.1017/S014338571000057X |
| [4] | Botella-Soler, V., Castelojm, J.M., Oteo, J.A., et al. (2011) Bifurcations in the Lozi Map. Journal of Physics A: Mathematical and Theoretical, 44, Article ID: 305101. https://doi.org/10.1088/1751-8113/44/30/305101 |
| [5] | Rybalova, E., Semenova, N., Strelkovag, G., et al. (2017) Transition from Complete Synchronization to Spatio-Temporal Chaos in Coupled Chaotic Systems with Nonhyperbolic and Hyperbolic Attractors. European Physical Journal Special Topics, 226, 1857-1866. https://doi.org/10.1140/epjst/e2017-70023-1 |
| [6] | Lopesino, C., Balibrea-Iniesta, F., Wiggins, S., et al. (2015) The Chaotic Saddle in the Lozi Map, Autonomous and Nonautonomous Versions. International Journal of Bifur-cation & Chaos, 25, Article ID: 1550184.
https://doi.org/10.1142/S0218127415501849 |
| [7] | Cao, Y. and Liu, Z. (1998) Strange Attractors in the Ori-entation-Preserving Lozi Map. Chaos Solitons and Fractals, 9, 1857-1864. https://doi.org/10.1016/S0960-0779(97)00180-X |
| [8] | Devaney, R.L. (1984) A Piecewise Linear Model for the Zones of Instability of an Area-Preserving Map. Physica D: Nonlinear Phenomena, 10, 387-393. https://doi.org/10.1016/0167-2789(84)90187-8 |
| [9] | Aharonov, D., Devaney, R.L., Elias, U. (1997) The Dy-namics of a Piecewise Linear Map and Its Smooth Approximation. International Journal of Bifurcation and Chaos, 7, 351-372. https://doi.org/10.1142/S0218127497000236 |
| [10] | Li, H., Li, K., Chen, M., et al. (2020) Coexisting Infinite Orbits in an Area-Preserving Lozi Map. Entropy, 22, Article 1119. https://doi.org/10.3390/e22101119 |
| [11] | Li H, Bao H, Zhu L, et al. (2020) Extreme Multistability in Simple Area-Preserving Map. IEEE Access, 8, 175972-175980. https://doi.org/10.1109/ACCESS.2020.3026676 |
| [12] | Rybalova, E. and Strelkova, G. (2022) Response of Sol-itary States to Noise-Modulated Parameters in Nonlocally Coupled Networks of Lozi Maps. Chaos, 32, Article ID: 021101. https://doi.org/10.1063/5.0082431 |
| [13] | Zheng, W.M. and Liu, J. (1994) Symbolic Analysis of At-tractor Geometry for the Lozi Map. Physical Review E, 50, 3241-3244. https://doi.org/10.1103/PhysRevE.50.3241 |
| [14] | Duarte, P. (1994) Plenty of Elliptic Islands for the Standard Family of Area Preserving Maps. Annales De Linstitut Henri Poincare, 11, 359-409. https://doi.org/10.1016/s0294-1449(16)30180-9 |
| [15] | Reichl, L.E. (1992) The Transition to Cha-os/Conservative Classical Systems and Quantum Manifestations. Springer-Verlag, New York. |
| [16] | Rybalova, E.V., Strelkovag, I. and Anishchenko, V.S. (2020) Impact of Sparse Inter-Layer Coupling on the Dynamics of a Heter-ogeneous Multilayer Network of Chaotic Maps. Chaos, Solitons & Fractals, 142, Article ID: 110477.
https://doi.org/10.1016/j.chaos.2020.110477 |
| [17] | Sander, E. and Meissj, D. (2020) Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps. Physica D: Nonlinear Phenomena, 411, Article ID: 132569. https://doi.org/10.1016/j.physd.2020.132569 |
| [18] | Lagarias, J.C. and Rains, E. (2005) Dynamics of a Family of Piecewise-Linear Area-Preserving Plane Maps I, Rational Rotation Numbers. Journal of Difference Equations and Applications, 11, 1089-1108.
https://doi.org/10.1080/10236190500273069 |
| [19] | Lagarias, J.C. and Rains, E. (2005) Dynamics of a Family of Piecewise-Linear Area-Preserving Plane Maps II, Invariant Circles. Journal of Difference Equations and Applications, 11, 1137-1163.
https://doi.org/10.1080/10236190500273127 |
| [20] | Gu, E.G. (2018) On the Existence of Chaos in a Discon-tinuous Area-Preserving Map Arising in Financial Markets. International Journal of Bifurcation and Chaos, 28, Article ID: 1850177. https://doi.org/10.1142/S0218127418501778 |
| [21] | Gu, E.G., He, Z.H., Ni, J., et al. (2023) Invariant Regions in Piecewise Linear Area-Preserving Map. Chaos Soliton and Fractals, 169, Article ID: 113308. https://doi.org/10.1016/j.chaos.2023.113308 |
