为揭示混沌的生成机理,探讨了强迫Lorenz模型混沌行为的力学机制。将强迫Lorenz模型转化成柯尔莫哥洛夫系统,把系统的力矩分解成惯性力矩,耗散力矩和外力矩。分析各种力矩耦合模式对系统动力学行为的影响,从中揭示系统生成混沌的力学机理。研究表明只有耗散因素,驱动因素和内能都存在的全力矩模式下,并且驱动因素和耗散因素相匹配时强迫Lorenz系统才能产生混沌。构造卡西米尔函数进行全局稳定性分析,估计强迫Lorenz模型吸引子的边界。
In order to reveal the generation mechanism of chaos, the mechanical mechanism of forced Lorenz model is discussed. The forced Lorenz model is transformed into Kolmogorov system, and the torques of the system are decomposed into inertial torques, dissipative torques and external torques. The effects of various torque coupling modes on the System dynamics behavior are analyzed to reveal the mechanism of chaos generation. It is shown that only when the dissipation factor, the driving factor and the internal energy exist in the full moment mode, and the driving factor and the dissipation factor match, the forced Lorenz system can produce chaos. The global stability of the forced Lorenz model is analyzed by constructing the Casimir function, and the boundary of the attractor of the forced Lorenz model is estimated.
[1] EDWARD L. Deterministic non-periodic flow[J]. Journal of the Atmospheric Sciences, 1963, 20: 130-141.
[2] PALMER T N, BUIZZA R, MOLTENI F, et al. Chaos and forecasting-singular vectors and the predictability of weather and climate[J]. Philosophical Transactions: Physical Sciences and Engineering, 1994, 348(1688): 459-475.
[3] BHATTHACHARJEE J K, MCKANE A J. Lorenz model for the rotating Rayleigh-Bénard probiem[J]. J Phs A: MathGen, 1998, 21: 555-558.
[4] 张银,许兰喜. 旋转的Rayleigh-Bénard问题的Lorenz模型及数值模拟[J]. 北京化工大学学报(自然科学版), 2010, 37(3): 140-143.
ZHANG Y, XU L X. Lorenz model for the rotating Rayleigh-Bénard problem and its numerical simulation[J] Journal of Beijing University of Chemical Technology (Natural Science Edition), 2010, 37(3): 140-143.
[5] XU L X. On the nonlinear stability of sane hydrodynam ic problems [D]. Bayreuth: Deparment of Mathematics and Physics Bayreuth University, 1997: 3-17.
[6] WANG H Y. The dynamical behavior and the numerical simulation of a five-mode system of the navier-stokes equations for a two-dimensional incompressible fluid on a torus[J]. Journal of Applied Mathematics and Physics, 2016, 14(7):7-9.
[7] 王贺元,崔进. Navier-Stokes方程九模类Lorenz方程组的动力学行为及数值仿真[J]. 工程数学学报, 2015, 32(6): 893-897.
WANG H Y, CUI Jin. The dynamic behavior and numerical simulation of nine mode lorenz equation system for navier-stokes equation [J] Journal of Engineering Mathematics, 2015, 32(6): 893-897.
[8] VALTER F, CLAUDLO T. A seven modes truncation of the plane incompressible Navicr-Stokes equations[J]. Journal of Statistical Physics, 1981, 25(3): 397-417.
[9] BOLDRIGHINI C, FRANCESHINI V. A five-dimensional truncation of the plane incompressible Navier-Stokes equations[J]. Communications in Mathematical Physics, 1979,64:159-170.
[10] CHEN G R,UETA T. Yet another chaotic attractor[J]. International Journal of Bifurcation and Chaos, 1999, 9(7): 1465-1469.
[11] LÜ J H, CHEN G R. A new chaotic attractor[J]. International Journal of Bifurcation and Chaos, 2002, 12(3): 659-661.
[12] 黎爱兵, 张立凤, 项杰. 外强迫对Lorenz系统初值可预报性的影响[J]. 物理学报, 2012, 61(11): 9.
LI A B, ZHANG L F, XIANG J. Influence of external forcing on the predictability of Lorenz systems [J] Journal of Physics, 2012, 61(11): 9.
[13] 尹社会, 张勇, 张付臣, 等. 基于Lorenz系统的强迫Lorenz混沌系统的动力学研究[J]. 东北师大学报:自然科学版, 2014,46(1): 6.
YIN S H, ZHANG Y, ZHANG F C, et al. Dynamics study of forced lorenz chaotic system based on Lorenz system[J]. Journal of Northeast Normal University: Natural Science Edition, 2014, 46(1): 6.
[14] 李保生, 丁瑞强, 李建平,等. 强迫Lorenz系统的可预报性研究[J]. 物理学报, 2017, 66(6): 8.
LI B S, DING R Q, LI J P, et al. A study on the predictability of forced Lorenz systems[J]. Journal of Physics, 2017, 66(6): 8.
[15] 刘绍刚, 李艳平. Lorenz系统的非线性动力学行为及仿真[J]. 计算机工程与设计. 2019, 40(3): 807-811.
LIU S G, LI Y P. Nonlinear dynamic behavior and simulation of Lorenz system[J]. Computer Engineering and Design, 2019, 40(3): 807-811.
[16] 宋宇. 含季节强迫项的Lorenz-84大气模型的动力学研究[D]. 北京: 北京交通大学, 2012.
SONG Y. A dynamic study of Lorenz-84 atmospheric model with seasonal forcing term[D]. Beijing: Beijing Jiaotong University, 2012.
[17] 孙叶平, 李德雪, 张勇, 等. 受扰动Lorenz混沌系统的动力学分析[J]. 计算机工程与应用, 2015, 51(11): 55-56.
SUN Y P, LI D X, ZHANG Y, et al. Dynamic analysis of for forced Lorenz chaotic systems[J]. Computer Engineering and Applications, 2015, 51(11): 55-56.
[18] 李一伟, 范广洲. Lorenz模型中外源强迫强弱对初值可预报性的影响研究[J]. 高原气象, 2020, 39(1): 9.
LI Y W, FAN G Z. Research on the influence of externally forced intensity on the predictability of initial values in the Lorenz model[J]. Plateau Meteorology, 2020, 39(1): 9.
[19] PELINO V, PASINI A. Energy cycle for the Lorenz attractor[J]. Chaos, Solitons and Fractals: Applications in Science and Engineering: An Interdisciplinary Journal of Nonlinear Science, 2014,64:67-77.
[20] QI G, LIANG X. Mechanical analysis of Qi four-wing chaotic system[J]. Nonlinear Dyn, 2016, 86(2): 1095-1106.
[21] 王贺元. Couette-Taylor流的力学机理与能量转换[J]. 数学物理学报,2020,40(1): 243-256.
WANG H Y. The mechanical mechanism and energy conversion of Couette-Taylor flow[J] Acta Mathematica Scientia, 2020, 40(1): 243-256.
[22] ANTONELLO P, VINICIO P. A unified view of Kolmogorov and Lorenz systems[J]. Physics Letters A, 2000, 275(5): 435-446.
[23] ANTONELLO P, VINICIO P, SERGIO P. Torsion and attractors in the Kolmogorov hydrodynamical system[J]. Physics Letters A, 1998, 241(1): 77-83.
[24] VINICIO P, ANTONELLO P. Dissipation in Lie-Poisson systems and the Lorenz-84 model[J]. Physics Letters A, 2001, 291(6): 389-396.
[25] LIANG X Y, QI G Y. Mechanical analysis of Chen chaotic system[J]. Chaos, Solitons and Fractals: the Interdisciplinary Journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, 2017, 98: 173-177.
[26] DOERING C R,GIBBON J D. On the shape and dimension of the Lorenz zttractor[J]. Dyn Stab Syst, 1995, 10(3): 255.
