Dynamics of mechanical systems

对未来的预测和控制是科学技术的中心主题。我们都想知道未来会带来什么。我们还希望控制未来，以便要么消除不想要的状态(例如，防止疾病传播)，要么达到想要的状态(例如，将卫星的相机重定向到目标)。这一提议涉及机械系统的动力学，即它们如何随着时间的推移而移动或演变。虽然计算机算法可以在一定程度上预测系统的未来演变，但从理论上理解可能的动力学特征可以提高预测，并为控制它的策略奠定基础。如果许多系统显示出相似的特征，则可以一起进行分析。对于机械系统，这些特征可能指的是环境的几何形状(例如，在平面上的运动，而不是在空间中的运动)、物理定律(例如，能量守恒的运动，而不是有摩擦的运动)或对称性(例如，一个没有标记的完美圆球，从各个方向看都是相同的)。然后，系统被组织成特定的类别，在每个类别中，我们试图描述可能的动态行为。特别令人感兴趣的是稳态(其中系统保持“冻结”)和其他特殊的“期货”，它们的稳定性，以及当某些物理特征或参数变化时可能发生的变化(“分叉”)。我的研究增加了动力系统的基础知识，并在机械工程(机械设备的控制和稳定)、空间科学(例如小行星系统的稳定性、月球-行星系统的自旋轨道耦合)和物理化学(例如多原子分子的旋转振动态)中有应用。我的重点是理解当参数变化时机械系统的演化。我使用数学形式主义，特别是微分方程组(描述系统的演化)、微分几何和拓扑学(描述空间)和分叉理论(描述动力学的可能分支)的方法。我还在使用数据驱动的方法、机器学习算法以及强大的计算和模拟。

Prediction and control of the future are central themes of science and technology. We all want to know what the future will bring. We also want to control the future so that either undesired states are eliminated (for example, preventing the spreading of a disease), or desired states are attained (for example, redirecting the camera of a satellite towards a target).  This proposal concerns the dynamics of mechanical systems, that is, how they move or evolve over time. While the future evolution of a system can to some degree be predicted using computer algorithms, a theoretical understanding of possible dynamical features can improve prediction and also lay the groundwork for strategies to control it.  Many systems may be analyzed together if they display similar characteristics. For mechanical systems, these characteristics may refer to the geometry of the environment (e.g., motion on a flat surface as opposed to motions in space), the physical laws (for example, motions where the energy is conserved, as opposed to motions with friction), or symmetries (for example, a perfectly round ball, without markings on it, looks the same from all directions). Systems are then organized in specific categories, and within each one, we attempt to describe the possible dynamical behaviours. Of particular interest are the steady states (in which the system remains "frozen") and other special "futures", their stability, and the changes ("bifurcations") that may occur as certain physical features or parameters vary.  My research adds to the fundamental knowledge of dynamical systems and has applications in mechanical engineering (control and stabilization of mechanical devices), space science (for example, stability of asteroidal systems, spin-orbit coupling of moon-planet systems) and physical chemistry (for example, roto-vibrational states of polyatomic molecules). My focus is on the understanding of mechanical systems evolution when parameters change. I use the mathematical formalism, and in particular, methods of differential equations (to describe the evolution of the system), differential geometry and topology (to describe the space) and bifurcation theory (to describe the possible branches of the dynamics). I am also using data-driven methods, machine learning algorithms and powerful computations and simulations.