73 citations · 945 across the 70 of their papers we have counts for
8 papers · 1 filter
The Integrability of a knife-edge Billiard in a Disk
Alejandro Bravo-Doddoli, Huaidian Hou, William Clark +1
This paper proves the integrability of a nonholonomic billiard defined by a knife-edge in a disk. The paper begins by parametrizing the impact space using coordinates that reveal t…
Invariant Forms in Hybrid and Impact Systems and a Taming of Zeno
William Clark, Anthony Bloch
Hybrid (and impact) systems are dynamical systems experiencing both continuous and discrete transitions. In this work, we derive necessary and sufficient conditions for when a give…
Existence of invariant volumes in nonholonomic systems subject to nonlinear constraints
William Clark, Anthony Bloch
We derive conditions for a nonholonomic system subject to nonlinear constraints (obeying Chetaev's rule) to preserve a smooth volume form. When applied to affine constraints, these…
The Bouncing Penny and Nonholonomic Impacts
William Clark, Anthony Bloch
The evolution of a Lagrangian mechanical system is variational. Likewise, when dealing with a hybrid Lagrangian system (a system with discontinuous impacts), the impacts can also b…
Families of periodic orbits: closed 1-forms and global continuability
Matthew D. Kvalheim, Anthony M. Bloch
We investigate global continuation of periodic orbits of a differential equation depending on a parameter, assuming that a closed 1-form satisfying certain properties exists. We be…
Poincaré-Bendixson Theorem for Hybrid Systems
William Clark, Anthony Bloch, Leonardo Colombo
The Poincaré-Bendixson theorem plays an important role in the study of the qualitative behavior of dynamical systems on the plane; it describes the structure of limit sets in such…