A direct method for computation of simple bifurcations
A direct method for computation of simple bifurcations
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DOI：
10.1016/s0021-9991(95)90068-3
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发表时间：
1995-10
影响因子：
4.1
通讯作者：
M. Poliashenko;C. Aidun
中科院分区：
文献类型：
--
作者：
M. Poliashenko;C. Aidun
在本研究中，我们专注于偏微分方程的计算分析，重点是平衡态的稳定性及其分支。在实际应用中，仅在参数空间中的一点上获得平衡解是不够的。平衡解分支、它们的稳定性特征，特别是从一个状态到另一个状态的过渡的临界点（例如，分叉点）是理解问题的物理学所必需的。原则上，平衡状态的线性稳定性可以通过求解特征值问题来研究，从而可以检测出分叉点。我们审查替代技术检测分岔点是直接和数值效率，因此，更实用。从一个大尺寸的动力系统，它代表了一组耦合偏微分方程的投影到一个基函数，我们讨论的时间演化方法，测试函数的方法，和直接方法的相对有效性。然后，我们将扩展直接方法，以实现更实用和更有效的实现。用这种技术，我们计算的序列从稳态到混沌流在一个二维盖驱动的腔的纵横比0.8，1.0，和1.5的过渡。我们证明了这种技术的有效性，通过计算有趣的新的动力学在这个相对简单的流体动力学系统。特别是，我们表明，取决于纵横比，从稳态的第一个过渡可能是通过超临界或亚临界的Hopf分岔导致系统的时间周期性状态。我们构造了失稳扰动结构，并得出了主定态的第一次分叉是由于主涡的离心不稳定性。向混沌过渡的机制是低维的。在二次Hopf分岔后，系统发生向混沌的过渡.
In the present study, we focus on the computational analysis of partial differential equations with emphasis on the stability of the equilibrium states and on their bifurcations. In practical applications, it is not sufficient to obtain an equilibrium solution at a point in the parameter space. The equilibrium solution branches, their stability characteristics, and particularly the critical points of transition from one state to another (e.g., bifurcation points), are required for understanding the physics of the problem. In principle, the linear stability of an equilibrium state can be investigated by solving an eigenvalue problem, and consequently, the points of bifurcations can be detected. We review alternative techniques for detecting bifurcation points which are direct and numerically efficient and, therefore, more practical. Starting with a large dimension dynamical system, which represents a projection of a set of coupled partial differential equations onto a basis function, we discuss the relative effectiveness of the time evolution approach, the test function approach, and the direct method. We will then extend the direct method for a more practical and efficient implementation. With this technique, we compute the sequence of transitions from steady state to chaotic flow in a two-dimensional lid-driven cavity of aspect ratio 0.8, 1.0, and 1.5. We demonstrate the effectiveness of this technique by computing interesting new dynamics in this relatively simple hydrodynamic system. In particular, we show that depending on the aspect ratio, the first transition from steady state could be through a supercritical or a subcritical Hopf bifurcation leading the system to a time periodic state. We construct the destabilizing disturbance structure and conclude that the first bifurcation of the primary steady state is due to the centrifugal instability of the primary eddy. The mechanism of transition to chaos is low-dimensional. The transition to chaos occurs after a secondary Hopf bifurcation.