Ein gesuchter, dennoch bislang unbekannter elementarer Satz

TL;DR

This paper characterizes quasiergodic sets in continuous dynamical systems with piecewise differentiable trajectories, showing they form a partition of the phase-space and can be constructed via a specific PDE, with implications for understanding attractors.

Contribution

It provides an elementary theorem describing quasiergodic sets as a phase-space partition and introduces a PDE for constructing these sets in finite-dimensional systems.

Findings

Quasiergodic sets form a partition of the phase-space.

A first-order PDE describes invariants for constructing quasiergodic sets.

Quasiergodic sets are sensitive attractors unless they are closed trajectories or fixed points.

Abstract

If and only if each point of a set of the phase-space is in the topological hull of a trajectory running through any other point of this set, we call this set a quasiergodic set. But which are these so defined quasiergodic sets in the case of a given continuous dynamical system, which has piecewise differentiable trajectories in a finit-dimensional real phase-space, which is compact? Let its trajectories define a field of normed tangents, which is continuous in almost each point of the phase-space: Then the topological hulls of all trajectories of the phase-space form a partition of it. Thus the elements of this partition are the quasiergodic sets of the given continuous dynamical system. This is the important but rather trivial statement of the elementary theorem 1.1, which we present in this tractatus. We call this theorem elementary, because it is limited to finit-dimensional real phase-spaces. We also find, that there is a linear homogeneous partial differential equation of first order, describing the invariants of the system, which allow us the construction of the quasiergodic sets. Furthermore we show, that any quasiergodic set is a sensitive attractor, if it is neither a closed trajectory or a fixed point.