Grant: Discovery Grants Program - Individual

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.