On d-Collision-Free Dynamical Systems

IFAC-PapersOnLine, vol. 55, pp. 25–30

Abstract

We present several recent results and generalizations that have been obtained in the theory of collision-freeness studied in Zerz and Herty (2019). A nonlinear ODE system ẋ(t) = f(x(t)) is called collision-free if the solution to the initial value problem with x(0) = x0 has distinct components for all times t whenever the initial state x0 has distinct components. This is an important structural property of particle systems. Here, we address the case where the state of the i-th particle has d components xi1, ..., xid, and a collision occurs if there exists some time t and i ≠ j such that xik(t) = xjk(t) for all 1 ≤ k ≤ d. Thus, the already known case corresponds to d = 1. In this paper, we characterize d-collision-freeness for linear, polynomially nonlinear, and general C1-systems. Moreover, we show that d-collision-freeness implies D-collision-freeness for all multiples D of d that divide the overall state space dimension. In particular, 1-collision-freeness implies d-collision-freeness. Finally, we address the connection between permutation symmetry and d-collision-freeness.

Authors 4

1. Melanie Harms corresponding Aachen Lehrstuhl für Algebra und Zahlentheorie

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen University, Aachen, Germany

2. Simone Bamberger corresponding Aachen Lehrstuhl für Algebra und Zahlentheorie

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen University, Aachen, Germany

3. Eva Zerz corresponding Aachen Lehrstuhl für Algebra und Zahlentheorie

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen University, Aachen, Germany

4. Michaël Herty corresponding Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Germany