Mechanical Mechanism and Energy Evolution of the Waterwheel Chaotic Rotation
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摘要: 为了揭示水轮混沌旋转的生成机制，采用力矩分析方法研究了水轮混沌旋转的力学机理与能量转换问题。把Malkus水轮的数学模型转换为Kolmogorov系统，基于惯性力矩、内力矩、耗散力矩和外力矩的不同耦合模式，利用理论分析和数值仿真相结合的方法，分析探讨了Malkus水轮混沌旋转的主要影响因素和内在的力学机理。研究了水轮系统Hamilton能量。动能和势能之间的相互转换，讨论了能量与Rayleigh数之间的关系。影响水轮系统混沌生成的主要因素是外力矩和耗散力矩。通过分析和仿真得知：力矩缺失模式并不能使系统生成混沌，全力矩模式才能使系统产生混沌，即混沌发生时四种力矩缺一不可，与此同时只有耗散和外力相匹配时系统才能产生混沌，此时水轮发生混沌旋转。引进Casimir函数分析水轮系统的动力学行为和能量转换，并估计混沌吸引子的界。Casimir函数反映了能量转换和轨道与平衡点间的距离，数值结果仿真刻画了它们之间的关系。
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关键词:
- Kolmogorov系统 /
- 混沌 /
- 混沌水轮 /
- 力学机理
Abstract: In order to reveal the mechanism of water wheel chaotic rotation, the Mechanical Mechanism and energy conversion of water wheel chaotic rotation are studied by the method of moment analysis. The mathematical model of the Malkus water wheel rotation is transformed into Kolmogorov system. Based on the different coupling modes of inertia moment, internal moment, dissipation moment and external moment, the main influencing factors and internal mechanical mechanism of the Malkus water wheel chaotic rotation are analyzed and discussed by using the method of theoretical analysis and numerical simulation. The conversion among Hamiltonian energy, kinetic energy and potential energy is investigated. The relationship between the energies and the Rayleigh number is discussed. The main factors affecting chaotic rotation are internal energy, kinetic energy and Hamiltonian energy. Through analysis and simulation, it is found that the lack of torque mode can not make the system generate chaos, but the full torque mode can make the system produce chaos, at the same time, the system can produce chaos only when the dissipation and external force match, at this time the water wheel is in a chaotic rotating state. The Casimir function is introduced to analyze the system dynamics. The bound of chaotic attractor is obtained by the Casimir function. The Casimir function reflects the energy conversion and the distance between the orbit and the equilibria. These relationships are illustrated by numerical simulations.-
Key words:
- Kolmogorov system /
- Chaos /
- The waterwheel system /
- Dynamical Mechanism
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图 2 分岔和最大Lyapunov指数图：(a) 分岔图；(b) 最大Lyapunov指数
Figure 2. bifurcation diagram and Lyapunov exponent spectrum: (a) bifurcation diagram; (b) Lyapunov exponent spectrum
图 3 分岔图：(a)
$r=r_1 $ 处的音叉分岔；(b)$P^{\pm} $ 点$r=r_{\rm{h}} $ 处的亚临界Hopf分岔Figure 3. bifurcation diagram: (a) Pitchfork bifurcation at
$r=r_1 $ ; (b) Subcritical bifurcation when$r=r_{\rm{h}} $ at$P^{\pm} $ 
图 4 系统在惯性力矩下：(a) 周期轨道的3维视图；(b) 状态变量z的轨迹
Figure 4. System is under inertial torques: (a) 3-D view of periodic orbit; (b) The trajectory of state z
图 5 系统在惯性力矩和内力矩下：(a) 周期轨道的3维视图；(b) 状态变量z的轨迹
Figure 5. System is under inertial and internal torques: (a) 3-D view of periodic orbit; (b) The trajectory of state z
图 7 系统在惯性力矩，内力矩和外力矩下：(a) 螺旋状轨道的3维视图；(b) 状态变量z的轨迹
Figure 7. The system is under inertial, internal and external torques: (a) 3-D view of spiral-like orbit; (b) The trajectory of state z
图 10 系统(2.5)的动力学：(a) 拟周期轨道的3维视图；b) 动能和势能
Figure 10. The dynamics of system (17): (a) 3-D view of quasi periodic attractor; (b) Kinetic Energy and Potential Energy
图 11 Casimir函数及混沌吸引子的边界：(a) Casimir能量时间演化；(b) 混沌吸引子的边界
Figure 11. Time evolution of Casimir energy and the boundary of chaotic attractor: (a) Time evolution of Casimir energy; (b) The boundary of chaotic attractor
图 12 函数与距离 D1，D2的关系。
Figure 12. The relationship between Casimir function and the D1 and D2.
图 13 能量和Casimir函数相对于
$ r$ 的演化：(a) 能量；b) Casimir函数Figure 13. Evolution of energy and Casimir function with
$ r$ increase: (a) energy; (b) Casimir function
图 14
$D_1$ 与$D_2$ 的距离和与 Casimir函数关于$r$ 的演化：(a)$D_1 $ 与$D_2 $ 的距离和；(b) Casimir函数Figure 14. The sum of distances
$D_1$ and$D_2$ and Casimir function with$r$ increase: (a) The sum of distances$D_1 $ and$D_2 $ ; (b) Casimir function表 1
$ \sigma =5,r$ 取不同值时水轮系统(1)的动力学行为与能量演化及其相应的旋转状态Table 1. Dynamics behavior and energy evolution of system (1) and corresponding actual rotation of the waterwheel for
$\sigma=5$ and different$r$ values$r$-value $0<r<1$
r = 0, $r_1=1$$r>1$
$r_{\rm{e}}=1.058\;45,\quad r_{\rm{g}}=13.965\;6,\quad r_{\rm{h}}=$15.041 2equilibria $O$ stable node Saddle node
(one direction is unstable,
the other two directions are stability)equilibria $P^{\pm}$ inexistence stable node stable focus stable focus saddle point trajectory of system (4) tends to stable equilibria $O$ tends to stable equilibria $P^{+}$ or $P^{-}$ spiral line tends to $P^{+}$ or $P^{-}$ same as left, but closer to $r_{\rm{h}}$, jumping back and forth between $P^{+}$ and $P^{-}$, a transient chaos, ultimately tends to $P^{+}$ or $P^{-}$ unstable limit cycles (subcritical Hopf bifurcation) lead to chaos the kinetic energy minim grow bigger increases maintain increaseing the Casimir function minimum gradual increases increases keep increasing the trend the sum of $D_1$ and $D_2$ inexistence monotone increases increases grow bigger the state of the water wheel motionless irregular rotation fig. 1(a)、(b) unstable rotation fig. 1(b)、(c) chaotic rotation fig. 1(c) 
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