Eur. Phys. J. Special Topics 226, 1751-1764 (2017)
https://doi.org/10.1140/epjst/e2017-70055-y

Regular Article

Generalized Lorenz equations on a three-sphere

¹ Graduate School of Commerce and Management, Hitotsubashi University, Saiki, Japan
² JST, PRESTO, Sander and Yorke, USA
³ Department of Mathematical Sciences, George Mason University, 22030 Fairfax, USA
⁴ University of Maryland, College Park, Maryland 20742, USA

^a e-mail: esander@gmu.edu

Received: 13 February 2017
Revised: 27 March 2017
Published online: 21 June 2017

Abstract

Edward Lorenz is best known for one specific three-dimensional differential equation, but he actually created a variety of related N-dimensional models. In this paper, we discuss a unifying principle for these models and put them into an overall mathematical framework. Because this family of models is so large, we are forced to choose. We sample the variety of dynamics seen in these models, by concentrating on a four-dimensional version of the Lorenz models for which there are three parameters and the norm of the solution vector is preserved. We can therefore restrict our focus to trajectories on the unit sphere S³ in ℝ⁴. Furthermore, we create a type of Poincaré return map. We choose the Poincaré surface to be the set where one of the variables is 0, i.e., the Poincaré surface is a two-sphere S² in ℝ³. Examining different choices of our three parameters, we illustrate the wide variety of dynamical behaviors, including chaotic attractors, period doubling cascades, Standard-Map-like structures, and quasiperiodic trajectories. Note that neither Standard-Map-like structure nor quasiperiodicity has previously been reported for Lorenz models.