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计算仿真发现，函数f（x，y，z）=sin（k（x2+y2+z2）），f（x，y，z）=k（1-（x2+y2+z2））e（-（x+y+z+u）2），f（x，y，z）=k（（x2+y2+z2）/3）（1-（x2+y2+z2）/3）分别与另外两个随机产生的二次多项式函数均可组合成一个三维离散动力系统，当系数k，u在一定范围内取值时，系统出现混沌吸引子的概率可以大于90%. 通过绘制分岔图、Lyapunov指数图等对上述系统的混沌特性进行了分析. 分析发现，出现混沌概率高的原因是这3 个函数的截面都是中间凸起或中间凹陷的曲面，在这样的截面条件下系统容易出现混沌. 这普遍适用于三维函数，利用这些三维离散动力系统绘制出的大量吸引子图形具有使用价值和研究价值.The calculation and simulation results show that f(x,y,z)=sin(k(x2+y2+z2)), f(x,y,z)=k(1-(x2+y2+z2))e(-(x+y+z+u)2), f(x,y,z)=k((x2+y2+z2)/3)(1-(x2+y2+z2)/3) can easily constructe a three-dimensional (3D) discrete dynamic system by combining other two polynomial functions generated randomly. Through calculating Lyapunov exponent and drawing the bifurcation diagram, the characteristics of chaos of the function are confirmed, and according to the bifurcation diagram of parameters and the Lyapunov exponent curve more chaotic mapping functions are found. Analysis shows that the cross-section geometric shape can determine the chaotic characteristics of 3D function, and the cross-sections are all the median convex or middle concave surfaces, which can constructe chaotic dynamic systems easily. In the future, the mathematical description model and some basic theorems are to be further investigated and their results will be used to solve practical problems such as turbulence.
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Keywords:
- chaos /
- dynamic system /
- three-dimensional function
[1] Madhok V, Riofrío C A, Ghose S, Deutsch I H 2014 Phys. Rev. Lett. 112 014102
[2] Wang X Y, Li F P 2009 Nonlinear Analysis: Theor. 70 830
[3] Secelean N A 2014 J. Math. Appl. 410 847
[4] Shi Y M, Chen G 2004 Chaos Slitons Fract. 22 555
[5] Reza M S 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 3857
[6] Feng J J, Zhang Q C, Wang W, Hao S Y 2013 Chin. Phys. B 22 090503
[7] Yu W B, Yang L Z 2013 Acta Phys. Sin. 62 020503 (in Chinese) [于万波, 杨灵芝 2013 62 020503]
[8] Yu W B, Zhou Y 2013 Acta Phys. Sin. 62 220501 (in Chinese) [于万波, 周洋 2013 62 220501]
[9] Yu W B, Yang L Z 2014 Computer. 39 5 (in Chinese) [于万波, 杨灵芝 2014 计算机工程 39 5]
[10] Yu W B, Yang X S, Wei X P 2011 Appl. Res. Comput. 28 3837 (in Chinese) [于万波, 杨雪松, 魏小鹏 2011 计算机应用研究 28 3837]
[11] Yu W B, Wei X P 2006 Acta Phys. Sin. 55 3969 (in Chinese) [于万波, 魏小鹏 2006 55 3969]
[12] Li X P, Zhang H J, Zhang X 2011 Acta Phys. Sin. 60 080901 (in Chinese) [李孝攀, 张皓晶, 张雄 2011 60 080901]
[13] Wang Y, Wu X 2012 Chin. Phys. B 21 050504
[14] Blázquez-Salcedo J L, Kunz J, Navarro-Lérida F, Radu E 2014 Phys. Rev. Lett. 112 011101
[15] Šuvakov M, Dmitrašinović V 2013 Phys. Rev. Lett. 110 114301
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[1] Madhok V, Riofrío C A, Ghose S, Deutsch I H 2014 Phys. Rev. Lett. 112 014102
[2] Wang X Y, Li F P 2009 Nonlinear Analysis: Theor. 70 830
[3] Secelean N A 2014 J. Math. Appl. 410 847
[4] Shi Y M, Chen G 2004 Chaos Slitons Fract. 22 555
[5] Reza M S 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 3857
[6] Feng J J, Zhang Q C, Wang W, Hao S Y 2013 Chin. Phys. B 22 090503
[7] Yu W B, Yang L Z 2013 Acta Phys. Sin. 62 020503 (in Chinese) [于万波, 杨灵芝 2013 62 020503]
[8] Yu W B, Zhou Y 2013 Acta Phys. Sin. 62 220501 (in Chinese) [于万波, 周洋 2013 62 220501]
[9] Yu W B, Yang L Z 2014 Computer. 39 5 (in Chinese) [于万波, 杨灵芝 2014 计算机工程 39 5]
[10] Yu W B, Yang X S, Wei X P 2011 Appl. Res. Comput. 28 3837 (in Chinese) [于万波, 杨雪松, 魏小鹏 2011 计算机应用研究 28 3837]
[11] Yu W B, Wei X P 2006 Acta Phys. Sin. 55 3969 (in Chinese) [于万波, 魏小鹏 2006 55 3969]
[12] Li X P, Zhang H J, Zhang X 2011 Acta Phys. Sin. 60 080901 (in Chinese) [李孝攀, 张皓晶, 张雄 2011 60 080901]
[13] Wang Y, Wu X 2012 Chin. Phys. B 21 050504
[14] Blázquez-Salcedo J L, Kunz J, Navarro-Lérida F, Radu E 2014 Phys. Rev. Lett. 112 011101
[15] Šuvakov M, Dmitrašinović V 2013 Phys. Rev. Lett. 110 114301
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