Phase Control for the Dynamics of Connected Rotators
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PICAL ISSUE
Phase Control for the Dynamics of Connected Rotators D. S. Khorkin∗,a , M. I. Bolotov∗,b, L. A. Smirnov∗,∗∗,c , and G. V. Osipov∗,d ∗
∗∗
Lobachevsky Nizhny Novgorod State University, Nizhny Novgorod, Russia Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia e-mail: a [email protected], b [email protected], c smirnov [email protected], d [email protected] Received July 23, 2019 Revised October 18, 2019 Accepted January 30, 2020
Abstract—We study the dynamics of rotational motions in a system of two asymmetrically coupled pendulum-type systems, investigating the mechanisms of losing stability for in-phase rotational motion. We analyze the scenario where chaotic dynamics appears depending on the values of the control parameters. Keywords: rotator, phase control, synchronization, rotational mode, chaos DOI: 10.1134/S0005117920080111
1. INTRODUCTION The study of collective behavior in networks of connected elements is an attractive and important area of nonlinear dynamics, relevant from the point of view of both theory and applications [1–4]. It is known that even with a weak coupling, elements of ensembles can tend to achieve a common rhythm of operation, i.e., to synchronize [1]. A rather wide class of objects considered in physics, radio engineering, electronics, and other fields of natural science can be described using models of coupled pendulum systems [5]. Despite the simplicity of these models, they are used not only to describe mechanical objects [6] but also for various processes in molecular biology [7–9], semiconductor structures [10], and more. This model can also be considered as a basic model in theoretical studies of coupled Josephson contacts [11–13], as well as phase synchronization systems [3, 4, 14, 15]. 2. MODEL DESCRIPTION In this work, we consider the behavior of an ensemble of two partial phase synchronization systems connected in parallel through phase mismatch signals [3, 4]. A block diagram of this ensemble is shown in Fig. 1. The mathematical model of a system of two such objects can be represented as a system of pendulum-type equations: ϕ¨1 + λϕ˙1 + sin ϕ1 = γ + κ1 sin (ϕ2 − ϕ1 ), ϕ¨2 + λϕ˙2 + sin ϕ2 = γ + κ2 sin (ϕ1 − ϕ2 ).
(1)
Here λ is the signal attenuation coefficient, γ is the ratio of the initial to maximum frequency mismatch, κ1 , κ2 are the signal gain parameters characterizing the strength of coupling between systems. An ensemble of two symmetrically connected identical pendulums was considered in [16]. The dynamics of non-identical pendulums (with different values of γ) were studied in [17]. Note that system (1) can be used to describe the behavior of an ensemble of globally coupled rotators where 1499
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Fig. 1. Block diagram of a pair of phase synchronization systems (PSS1 and PSS2) connected in parallel via phase mismatch signals (PD—phase detector).
two clusters are formed with different numbers of mutually synchronous elements (N1 and N2 ) [18]. Since N1 and N2 are different, the coupling
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