# 拉比诺维奇-法布里康特方程

${\displaystyle {\dot {x}}=y(z-1+x^{2})+\gamma x\,}$
${\displaystyle {\dot {y}}=x(3z+1-x^{2})+\gamma y\,}$
${\displaystyle {\dot {z}}=-2z(\alpha +xy),\,}$

Danca and Chen[2]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

### 平衡点

${\displaystyle {\tilde {\mathbf {x} }}_{0}=(0,0,0)}$
${\displaystyle {\tilde {\mathbf {x} }}_{1,2}=\left(\pm q_{-},-{\frac {\alpha }{q_{-}}},1-\left(1-{\frac {\gamma }{\alpha }}\right)q_{-}^{2}\right)}$
${\displaystyle {\tilde {\mathbf {x} }}_{3,4}=\left(\pm q_{+},-{\frac {\alpha }{q_{+}}},1-\left(1-{\frac {\gamma }{\alpha }}\right)q_{+}^{2}\right)}$

${\displaystyle q_{\pm }={\sqrt {\frac {1\pm {\sqrt {1-\gamma \alpha \left(1-{\frac {3\gamma }{4\alpha }}\right)}}}{2\left(1-{\frac {3\gamma }{4\alpha }}\right)}}}}$

1. [0,0,0]
2. [.46748585798513339859, -2.3530123557983251267, .95430463972208895291]
3. [-.46748585798513339859, -2.3530123557983251267, .95430463972208895291]
4. [1.3347123182858183570, -.82414763460993508052, .62751354209609286530]
5. [-1.3347123182858183570, -.82414763460993508052, .62751354209609286530]

### γ = 0.87, α = 1.1

γ = 0.87 and α = 1.1，初始条件为(−1, 0, 0.5).[4] The 关联维数 为 2.19 ± 0.01.[5] 李雅普诺夫指数, λ 约为 0.1981, 0, −0.6581 卡普兰 - 约克量纲, DKY ≈ 2.3010[4]

### γ = 0.1

Rabinovich Fabricant xy plot alpha=-0.05

Rabinovich Fabricant xy plot alpha=0.05

Rabinovich Fabricant xy plot alpha=0

Rabinovich Fabricant xy plot alpha=0.25

Danca and Romera[6]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

## 参考文献

1. ^ Rabinovich, Mikhail I.; Fabrikant, A. L. (1979). "Stochastic Self-Modulation of Waves in Nonequilibrium Media". Sov. Phys. JETP 50: 311. Bibcode:1979JETP...50..311R.
2. ^ Danca
3. ^ name="DancaChen"
4. Sprott
5. ^ Grassberger
6. ^ Danca

Grassberger, P.; Procaccia, I. Measuring the strangeness of strange attractors. Physica D. 1983, 9: 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1.

Rabinovich, Mikhail I.; Fabrikant, A. L. Stochastic Self-Modulation of Waves in Nonequilibrium Media. Sov. Phys. JETP. 1979, 50: 311. Bibcode:1979JETP...50..311R.

Sprott, Julien C. Chaos and Time-series Analysis. Oxford University Press. 2003: 433. ISBN 0-19-850840-9.

Danca, Marius-F.; Romera, Miguel. Algorithm for Control and Anticontrol of Chaos in Continuous-Time Dynamical Systems. Dynamics of Continuous, Discrete and Impulsive Systems. Series B: Applications & Algorithms (Watam Press). 2008, 15: 155–164. ISSN 1492-8760. hdl:10261/8868.

Danca, Marius-F.; Chen, Guanrong. Birfurcation and Chaos in a Complex Model of Dissipative Medium. International Journal of Bifurcation and Chaos (World Scientiﬁc Publishing Company). 2004, 14 (10): 3409–3447. Bibcode:2004IJBC...14.3409D. doi:10.1142/S0218127404011430.