Propagation of Geometric Structures in Smooth Hybrid Systems

Geometric Hybrid Dynamical Systems: Part I - Modeling and Stability

We present a framework for the modeling and analysis of geometric hybrid dynamical systems as hybrid inclusions on $C^1$-manifolds. Using tools from nonsmooth and set-valued analysis on manifolds, we derive coordinate-independent sufficient conditions for the existence of nontrivial solutions to this type of systems. W...

Variational Structure at the Edge of Stability

When discrete-time optimizers operate at the edge of stability, they exhibit near-two-periodic behavior. These oscillatory dynamics are reminiscent of conservative systems, such as the dynamics generated by symplectic integrators. However, a precise formulation of the connection between discrete-time optimizers at the...

Geometry, Topology, and Symmetry of Open Smooth and Hybrid Systems

A Dynamical Systems view of Feedback Synthesis

Control theory and dynamical systems are closely intertwined fields. Pontryagin's maximum principal offers a strong connection by providing a constructive way to synthesize feedback control laws via lifting the control system to a (Hamiltonian) dynamical system. Feedback laws can then be encoded as invariant manifolds...

On the Structural Properties of Discrete-Time and Sampled-Data Hamiltonian Dynamics

While continuous-time Hamiltonian dynamics are naturally energy preserving with a symplectic flow, their discrete-time counterparts enhance either geometric or energy preservation properties, but rarely both within a unified framework. It is the object of this paper to more deeply investigate this question. In both lin...

Asymptotically Autonomous Maps: Heteroclinic connections, Splitting and Phase-induced tipping

We consider discrete-time dynamical systems generated by the iteration of asymptotically autonomous maps, whose asymptotic behavior may be either conservative or dissipative. These systems are of particular interest in the modeling of complex phenomena with time-dependent parameter variation. With an appropriate time c...