On Forced Oscillations in a Relay System with Hysteresis

This paper discusses the phenomenon associated with the forced synchronization (“entrainment of a self-sustained oscillator by an external force”) in a relay system with hysteresis, which manifests itself in the occurrence of periodic motions close to the rhythmic activity of neurons, when packets o...

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Bibliographic Details
Published inAutomation and remote control Vol. 85; no. 4; pp. 377 - 386
Main Authors Zhusubaliyev, Zh. T., Sopuev, U. A., Bushuev, D. A.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 2024
Springer Nature B.V
Subjects
Bifurcations
CAE) and Design
Calculus of Variations and Optimal Control; Optimization
Computer-Aided Engineering (CAD
Control
Dynamic structural analysis
Entrainment
Forced vibration
Hysteresis
Intervals
Isomorphism
Mapping
Mathematics
Mathematics and Statistics
Mechanical Engineering
Mechatronics
Nonlinear Systems
Oscillations
Relay systems
Robotics
Synchronism
Systems Theory
discontinuous circle mapping
forced relay system with hysteresis
border-collision and period adding bifurcations
system with two time scales
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Summary:This paper discusses the phenomenon associated with the forced synchronization (“entrainment of a self-sustained oscillator by an external force”) in a relay system with hysteresis, which manifests itself in the occurrence of periodic motions close to the rhythmic activity of neurons, when packets of fast oscillations are interspersed with intervals of the slow dynamics. To study this phenomenon, we introduce a circle mapping, which, depending on the parameters, can be a circle diffeomorphism or discontinuous map (“gap map”). In both cases, this mapping demonstrates the so-called period-adding bifurcation structure. It is demonstrated that packets number of fast oscillations in the period of periodic motion is determined by the rotation number, and the length of the intervals between the packets may be found of the boundaries of the absorbing interval. The change in the number of pulses in the packet occurs through the border-collision bifurcation.
ISSN:0005-1179
1608-3032
DOI:10.1134/S0005117924040088