Mishra Aashwin Ananda, Sanghi Sanjeev
Department of Mechanical Engineering, Indian Institute of Technology Delhi, Hauz-Khas, New Delhi-110016, India.
Chaos. 2006 Mar;16(1):013114. doi: 10.1063/1.2154792.
In this work, the asymmetric case of the Malkus waterwheel is studied, where the water inflow to the system is biasing the system toward stable motion in one direction, like a Pelton wheel. The governing equations of this system, when expressed in Fourier space and decoupled to form a closed set, can be mapped into a four-dimensional space where they form a quasi-Lorenz system. This set of equations is analyzed in light of analogues of the Rayleigh Bernard convection and conclusions are drawn. The properties and behavior of the equations are studied and correlated to the physical model. Phase space behavior and linear stability analysis are used for this. Spectral analysis is used as a qualitative measure of chaos. Chaotic behavior is quantified through the calculation of the Lyapunov exponents and these are further correlated to the bifurcation diagrams for a conclusive analysis of the dynamical behavior of the system.
在这项工作中，研究了马尔库斯水轮的非对称情况，即系统的水流偏向于使系统朝着一个方向做稳定运动，类似于冲击式水轮机。当该系统的控制方程在傅里叶空间中表示并解耦以形成一个封闭集时，可以映射到一个四维空间，在这个空间中它们形成一个准洛伦兹系统。根据瑞利 - 伯纳德对流的类似情况对这组方程进行分析并得出结论。研究了这些方程的性质和行为，并将其与物理模型相关联。为此使用了相空间行为和线性稳定性分析。光谱分析用作混沌的定性度量。通过计算李雅普诺夫指数对混沌行为进行量化，并且进一步将这些指数与分岔图相关联，以便对系统的动力学行为进行确定性分析。
