Dynamics and Stability of Stochastic Nonlinear Auto-parametric Systems

自参数系统由至少一个振荡器组成，该振荡器被直接激励并以这样的方式非线性地耦合到子系统，使得振荡器可以在子系统保持静止的同时振动。单个振荡器或一个子系统的小幅振动可能使整个系统不稳定。这种不稳定是自参数共振。本文研究了含噪声自参数系统的平稳运动和平稳运动的稳定性。这种噪声系统的非线性动力学是非常丰富的，在很大程度上未被探索。自参数共振具有重要的工程应用。例如，在飞机中，连接发动机和机翼的“吊杆”的振动是由于湍流空气对机翼的作用力而发生的-这是随机的自参数“共振”。另一个例子是由搅动的沃茨产生的海船的垂荡-横摇运动;在极端情况下，倾覆的危险会出现。减震器也容易受到自参数共振的影响。虽然本研究的重点是上述的应用，本研究的结果将适用于生物学问题，如蓝斑，参与调制认知性能的脑核的尖峰神经元池的数学建模，该方法将是跨学科的，跨越频谱从分析技术和建模的数值和实验验证。第一个挑战是发展一个有效的，系统的方法来确定几乎肯定的稳定性的单模非线性解决方案。在存在分离的尺度，其中的噪声是渐近小，一个利用对称性使用最近的数学结果随机平均找到一个低维的系统描述。随机数值模拟将被用来验证通过分析得到的分叉行为，并提供线索，新的现象是全球性的，而不是通过理论分析检测。PI将从代表性不足的少数民族和妇女群体中招收学生。该项目将直接培训研究生和本科生。该提案的更广泛影响包括与意大利拉奎拉大学结构系建立伙伴关系，在那里进行实验验证。学生将与来自拉奎拉大学非线性动力学实验室的教授密切合作。拟议研究的理论部分将与计算和实验工作紧密结合。

Auto-parametric systems consist of at least one oscillator that is directly excited and coupled nonlinearly to a subsystem in such a way that the oscillator can vibrate while the subsystem remains stationary. Small amplitude vibration of the single oscillator or of one subsystem may destabilize the entire system. This destabilization is auto-parametric resonance. The purpose of this work is to examine the stationary motion and stability properties of stationary motion of noisy auto-parametric systems. The nonlinear dynamics of such noisy systems are extremely rich and largely unexplored. Auto-parametric resonance has many important engineering applications. For example, in aircraft, vibration of the "suspenders" connecting the engines and the wings occurs due to turbulent air forcing on the wings -- this is stochastic auto-parametric "resonance". Another example is the heave-roll motion of sea vessels produced by agitated waters; in extreme cases the danger of capsizing arises. Shock absorbers are also susceptible to auto-parametric resonance. While the focus of the present study is on the applications above the results of this research will be applicable even to biological problems such as the mathematical modeling of a pool of spiking neurons in the locus coeruleus, a brain nucleus involved in modulating cognitive performance.The approach will be cross-disciplinary, spanning the spectrum from analytical techniques and modeling to numerical and experimental verification. The first challenge is to develop an effective, systematic approach to determine the almost-sure stability of single mode nonlinear solutions. In the presence of a separation of scales, where the noise is asymptotically small, one exploits symmetries to use recent mathematical results concerning stochastic averaging to find a lower-dimensional description of the system. Stochastic numerical simulations will be used to verify the bifurcation behavior obtained through analysis and to provide clues to new phenomena which are global in nature and not detected through the theoretical analysis. The PI will recruit students from underrepresented minority and women groups. The project will directly train both graduate and undergraduate students. The broader impacts of the proposal include a partnership with the Structural Department of the University of L'Aquila, Italy, where experimental validations will be made. Students will work closely with professors from the University of L'Aquila's Nonlinear Dynamics Laboratory. The theoretical component of the proposed research will be tightly coupled to the computational and experimental efforts.