Chaotic Phenomena and Their Engineering Relevance

用数字模拟和基于动力系统几何理论的算法研究了表示两个耦合摆动动力学的常微分方程。这些方程是从一个简单的发电系统动力学稳定性的简单模型中推导出来的，该系统包括两个由传输线连接的发电机。该系统虽然不存在混沌吸引子，但存在分形盆地边界，这对确定工程稳定性具有重要意义。应用跨轨方法对分形盆地边界下伏构造的认识取得了实质性进展。然而，关于这种分形结构，仍有一些重要的未解问题，我们希望在来年对其进行研究。最近，研究人员发现了非线性动力学中的一个新的基本现象：灾难性分岔导致稳定性丧失，随后出现了更多的瞬态，导致几种可能的新吸引子结果之一。在这个项目中，根据初始分岔的形式，进一步探索具有不确定结果的其他类型的分岔。除了已知的不确定鞍节点和混沌蓝天突变外，还发现了不确定跨临界和亚临界分叉等新类型。这些出现在模拟势阱中强迫振荡的简单非线性振子原型中。工程应用包括在波涛汹涌的大海中倾覆船只。这些现象在工程问题中是普遍存在的。采用数字仿真和动力系统方法研究了另一种非线性振荡器，该振荡器描述了所提出的船舶稳定机制。这个振荡器包括一个延时项；研究发现，随着反馈增益的逐渐增大，系统的失稳将导致系统发展为近似周期振荡，进而产生混沌行为，这是倾覆的直接前兆。少

The study of ordinary differential equations representing the dynamics of two coupled swings has been pursued using digital simulations and algorithms based on the geometrical theory of dynamical systems. These equations were derived from a simple model of dynamical stability of a simple electric power generation system involving two generators connected by a transmission line. Although this system has no chaotic attractors, there is a fractal basin boundary which is of fundamental relevance in determining the engineering stability properties. Substantial progress has been made in understanding the underlying structure of this fractal basin boundary by applying the straddle orbit method. However, there remain some important unanswered questions about this fractal structure which we hope to investigate the coming year.Recently a new fundamental phenomenon in nonlinear dynamics has been discovered by the investigators : catastrophic bifurcations causing loss of stability followed by chao … More tic transients leading to one of several possible new attractor outcomes. In this project the ideas were pursued further to discover additional types of bifurcations with indeterminate outcome, according to the form of the initiating bifurcation. In addition to the already-known indeterminate saddle-node and chaotic blue sky catastrophe, new types include indeterminate transcritical and subcritical bifurcations have been discovered. These appear in simple prototypical nonlinear oscillators modelling forced oscillations in a potential well. Engineering applications include the capsize of ships in heavy seas. It appears that these phenomena are common and widespread in engineering problems.Another nonlinear oscillator describing a proposed ship stabilization mechanism was studied, again using digital simulation and dynamical systems approaches. This oscillator includes a time delay term ; it was found that the destabilization of the system by gradually increasing the feedback gain results in the development of almost periodic oscillations and then chaotic behavior which is an immediate precursor of capsize. Less