Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

On a Reversible Three-dimensional System Containing Attractor and Lorenz Repeller

Author(s):

Alexander Sergeevich Gonchenko

Researcher
Nizhny Novgorod National Research University named after N.I. Lobachevsky,
23 Gagarina Ave., Nizhny Novgorod, 603022, Russia

agonchenko@mail.ru

Alexander Gennadievich Korotkov

Lead Engineer
Nizhny Novgorod National Research University named after N.I. Lobachevsky,
23 Gagarina Ave., Nizhny Novgorod, 603022, Russia

koral81@bk.ru

Evgenia Alexandrovna Samylina

graduate student
National Research University "Higher School Economics",
st. Bolshaya Pecherskaya, 25/12, Nizhny Novgorod, 603155, Russia

samylina_evgeniya@mail.ru

Abstract:

We study the problem on the existence of Lorenz-like attractors and repellers in three-dimensional time-reversible systems, as well as the structure of bifurcation scenarios for their emergence. In this connection, we consider a system that is the flow normal form for reversible bifurcations of a fixed point with the triplet (-1,-1,+1) of multipliers. The bifurcation set itself of the indicated reversible bifurcation should be extremely complex (the normal form contains 7 independent parameters). However, we are mainly interested here in bifurcations leading to the appearance of a symmetric pair "Lorenz attractor -- Lorenz repeller", which, as we show, can be studied within the framework of two-parameter subfamilies. In the paper, two main bifurcation scenarios for the emergence of such a pair are described, and also a quite unusual scenario is outlined for the appearance of Lorenz-like attractor and repeller in the case when the system has only two equilibria. The corresponding phenomenon seems new -- for comparison, we note that even the Lorenz system has three equilibria: one of them belongs to the attractor, and the other two reside in its "holes".

Keywords

• Lorenz attractor
• reversible system
• strange attractor

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