2023/09/22 by Maria Oprea, Aden Shaw, Oprea, Maria +9
Engineering · Mathematics · Physics and Astronomy · #Algebra over a field #Astro and Planetary Science #Computer science #Coset #Discrete mathematics #Dynamical systems theory #Dynamics and Control of Mechanical Systems #Hamiltonian (control theory) #Hamiltonian system #Homogeneous space #Hybrid system #Invariant (physics) #Lie group #Mathematical analysis #Mathematical optimization #Mathematical physics #Mathematics #Operator (biology) #Physics #Poisson distribution #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #Reduction (mathematics) #Term (time)
paper · pdf · doi:10.48550/arxiv.2309.12569
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2023/09/22 · openalex created_date 2023/09/26 · openalex updated_date 2026/08/06
Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. As typical in dynamical analysis, an essential goal is to study the long-term behavior of these systems. In this work, we present two different novel approaches for studying these systems. The first approach is based on constructing an analog of the Frobenius-Perron (transport) operator for hybrid systems. Rather than tracking the evolution of a single trajectory, this operator encodes the asymptotic nature of an ensemble of trajectories. The second approach presented applies to an important subclass of hybrid systems, mechanical impact systems. We develop an analog of Lie-Poisson(-Suslov) reduction for left-invariant impact systems on Lie groups. In addition to the Hamiltonian (and constraints) being left-invariant, the impact surface must also be a right coset of a normal subgroup. This procedure allows a reduction from a 2n-dimensional system to an (n+1)-dimensional one. We conclude the paper by presenting numerical results on a diverse array of applications.