The Lorenz System has a Global Repeller at Infinity

DOI

10.1142/S1402925111001489How to use a DOI?

Keywords

Lorenz system; autonomous differential equations; ordinary differential equations; polynomial systems; asymptotic behavior; compactification; invariant set; strange attractor; vector fields; flows

Abstract

It is well known that the celebrated Lorenz system has an attractor such that every orbit ends inside a certain ellipsoid in forward time. We complement this result by a new phenomenon and by a new interpretation. We show that “infinity” is a global repeller for a set of parameters wider than that usually treated. We construct in a compacted space, a unit sphere that serves as the image of an ideal set at infinity. This sphere is shown to be the union of a family of periodic solutions. Each periodic solution is viewed as a limit cycle, or an isolated periodic orbit when restricted to a certain plane. The unconventional compactification y=x1−x†x is used.

Copyright

© 2011 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Harry Gingold
AU  - Daniel Solomon
PY  - 2021
DA  - 2021/01/07
TI  - The Lorenz System has a Global Repeller at Infinity
JO  - Journal of Nonlinear Mathematical Physics
SP  - 183
EP  - 189
VL  - 18
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925111001489
DO  - 10.1142/S1402925111001489
ID  - Gingold2021
ER  -