查看“︁拉比诺维奇-法布里康特方程”︁的源代码

'''拉比诺维奇-法布里康特方程'''(Rabinovich-Fabrikant equations)是 1979年苏联物理学家拉比诺维奇和法布里康特提出模拟非平衡介质自激波动的非线性常微分方程组<ref> Rabinovich, Mikhail I.; Fabrikant, A. L. (1979). "Stochastic Self-Modulation of Waves in Nonequilibrium Media". Sov. Phys. JETP 50: 311. Bibcode:1979JETP...50..311R.</ref>：
[[File:Rabinovich-Fabrikant equations.gif|thumb|200px|拉比诺维奇-法布里康特方程 3D动画 γ=0.803...0.917]]
[[File:Rabinovich Fabrikant plot2.gif|thumb|200px|拉比诺维奇-法布里康特图2 t<20 动画γ=0.803...0.917]]
[[File:Rabinovich Fabrikant plot3.gif|thumb|200px|拉比诺维奇-法布里康特图3 t>20 动画γ=0.803...0.917]]
[[File:Rabinovich-Fabrikant attractor xy plot.png|thumb|拉比诺维奇-法布里康特吸引子 xy 相图alpha = 1.1, gamma = .86666666666666666667]]
: <math>\dot{x} = y (z - 1 + x^2) + \gamma x \, </math>
: <math>\dot{y} = x (3z + 1 - x^2) + \gamma y \, </math>
: <math>\dot{z} = -2z (\alpha + xy), \, </math>

其中 ''α'', ''γ'' 是控制系统的参数.

Danca and Chen<ref name="DancaChen">Danca</ref>指出由于拉比诺维奇-法布里康特方程包含平方项，因此比较难以分析，即便选择的参数相同，但由于求解微分方程组的步骤的不同也会导致不同的吸引子。

==数值解==
利用[[Maple]]中以[[龙格－库塔法]][[rkf45]]为核心的软件包odeplot和plot、seq可以得出拉比诺维奇-法布里康特方程的数值解的3D动画图，以便观察拉比诺维奇-法布里康特系统随参数γ和时间t的变化：

参数值：α=1.1，γ=0.803..0.917，t=0...130

初始条件：x(0)=-1,y(0)=0,z(0)=0.5

在t<20时，系统表现为自激振动，当t>20,系统进入混沌态。


===平衡点===
[[File:Rabinovich-Fabrikant equilibrium point existence regions.svg|thumb|平衡点区域图 <math>\tilde{\mathbf{x}}_{1,2,3,4}</math>.]]


拉比诺维奇-法布里康特系统具有5个双曲线平衡点，一个在原点，4个依赖于系统参数α '' '' ''和γ ' ： <ref> name="DancaChen"</ref>。


: <math>\tilde{\mathbf{x}}_0 = (0,0,0)</math>
: <math>\tilde{\mathbf{x}}_{1,2} = \left( \pm q_-, - \frac{\alpha}{q_-}, 1- \left(1-\frac{\gamma}{\alpha}\right)q_-^2 \right)</math>
: <math>\tilde{\mathbf{x}}_{3,4} = \left( \pm q_+, - \frac{\alpha}{q_+}, 1- \left(1-\frac{\gamma}{\alpha}\right)q_+^2 \right)</math>

其中：

: <math>q_{\pm} = \sqrt{ \frac{ 1 \pm \sqrt{ 1- \gamma \alpha \left( 1- \frac{3 \gamma}{4\alpha} \right) } }{2 \left(1- \frac{3\gamma}{4\alpha}\right) }}</math>

这些平衡点只存在于参数 ''α'' and ''γ'' > 0 的一些区域。

当α=1.1,γ=0.87
代人上式可得：
# [0,0,0]
#  [.46748585798513339859, -2.3530123557983251267, .95430463972208895291]
#  [-.46748585798513339859, -2.3530123557983251267, .95430463972208895291]
#  [1.3347123182858183570, -.82414763460993508052, .62751354209609286530]
#  [-1.3347123182858183570, -.82414763460993508052, .62751354209609286530]
{{Wikibooks|Maple/例子#Rabinovich Fabrikant 平衡点|Rabinovich Fabrikant 平衡点}}

===γ = 0.87, α = 1.1===
当 ''&gamma;'' = 0.87 and ''&alpha;'' = 1.1，初始条件为(−1, 0, 0.5).<ref name="SprottJC">Sprott</ref> The [[关联维数]] 为 2.19 ± 0.01.<ref name="Grassberger">Grassberger</ref>  [[李雅普诺夫指数]], ''&lambda;'' 约为 0.1981, 0, −0.6581  [[卡普兰 - 约克量纲]], ''D''<sub>KY</sub> ≈ 2.3010<ref name="SprottJC">Sprott</ref>。

===γ = 0.1===

[[File:Rabinovich Fabricant xy plot-0.05.png|thumb|left|200px|Rabinovich Fabricant xy plot alpha=-0.05]]
[[File:Rabinovich Fabricant xy plot 0.05.png|thumb|right|200px|Rabinovich Fabricant xy plot alpha=0.05]]
[[File:Rabinovich Fabricant xy plot0.png|thumb|left|200px|Rabinovich Fabricant xy plot alpha=0]]
[[File:Rabinovich Fabricant xy plot 0.15.png|right|200px|Rabinovich Fabricant xy plot alpha=0.15]]
[[File:Rabinovich Fabricant xy plot 0.25.png|thumb|left|200px|Rabinovich Fabricant xy plot alpha=0.25]]

Danca and Romera<ref name="DancaRomera">Danca</ref>指出当参数 ''γ'' = 0.1、 ''α'' = 0.98时，系统进入[[混沌]]状态，当 ''α'' = 0.14时，系统进入[[极限环]]。



利用Maple中以龙格－库塔法rkf45为核心的软件包odeplot和plot、seq可以得出拉比诺维奇-法布里康特方程的数值解的3D动画图，以便观察拉比诺维奇-法布里康特系统随参数α的变化：

参数值：γ=0.14，t=0...130,

初始条件：x(0)=-1,y(0)=0,z(0)=0.5
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==参考文献==
{{reflist}}

{{cite journal
 | first1  = P. 
 | last1   = Grassberger
 | first2  = I.
 | last2   = Procaccia
 | title   = Measuring the strangeness of strange attractors
 | journal = Physica D
 | year    = 1983
 | volume  = 9
 | pages   = 189–208
 | doi     = 10.1016/0167-2789(83)90298-1
| bibcode=1983PhyD....9..189G
}}

{{cite journal
 | first1  = Mikhail I.
 | last1   = Rabinovich
 | first2  = A. L.
 | last2   = Fabrikant
 | title   = Stochastic Self-Modulation of Waves in Nonequilibrium Media
 | journal = Sov. Phys. JETP
 | year    = 1979
 | volume  = 50
 | pages   = 311
|bibcode = 1979JETP...50..311R }}

{{cite book
 | last      = Sprott
 | first     = Julien C.
 | title     = Chaos and Time-series Analysis
 | publisher = [[Oxford University Press]]
 | year      = 2003
 | pages     = 433
 | isbn      = 0-19-850840-9
}}

{{cite journal
 | first1    = Marius-F.
 | last1     = Danca
 | first2    = Miguel
 | last2     = Romera
 | title     = Algorithm for Control and Anticontrol of Chaos in Continuous-Time Dynamical Systems
 | journal   = Dynamics of Continuous, Discrete and Impulsive Systems
 | series    = Series B: Applications & Algorithms
 | volume    = 15
 | year      = 2008
 | pages     = 155–164
 | publisher = Watam Press
 | id       = {{hdl|10261/8868}}
 | issn      = 1492-8760
}}

{{cite journal
 | first1    = Marius-F.
 | last1     = Danca
 | first2    = Guanrong
 | last2     = Chen
 | title     = Birfurcation and Chaos in a Complex Model of Dissipative Medium
 | journal   = International Journal of Bifurcation and Chaos
 | volume    = 14
 | issue     = 10
 | year      = 2004
 | pages     = 3409–3447
 | publisher = World Scientiﬁc Publishing Company
 | doi       = 10.1142/S0218127404011430
|bibcode = 2004IJBC...14.3409D }}

==外部链接==
* Weisstein, Eric W. [http://mathworld.wolfram.com/Rabinovich-FabrikantEquation.html "Rabinovich–Fabrikant Equation."] {{Wayback|url=http://mathworld.wolfram.com/Rabinovich-FabrikantEquation.html |date=20131226105252 }} From MathWorld—A Wolfram Web Resource.
*Chaotics Models a more appropriate approach to the chaotic graph of the system [http://www.robert-doerner.de/Glossar/glossar.html#R "Rabinovich–Fabrikant Equation"] {{Wayback|url=http://www.robert-doerner.de/Glossar/glossar.html#R |date=20131228040004 }}

{{吸引子}}
{{混沌理论}}

[[Category:非线性常微分方程]]
[[Category:混沌理论]]