The global structural stability and application of the toroidal differential equations \(\dot x=a_ 1[\sin(x-y)-\sin x]-\Delta\omega\), \(\dot y=a_ 2[\sin(y-x)-\sin y]-\Delta\omega\)
type: Item
instance of: scholarly article
title: The global structural stability and application of the toroidal differential equations \(\dot x=a_ 1[\sin(x-y)-\sin x]-\Delta\omega\), \(\dot y=a_ 2[\sin(y-x)-\sin y]-\Delta\omega\)
zbMATH Open document ID: 0724.58038
DOI: 10.1007/BF02454510
author: Rongxing Zhou
published in: Applied Mathematics and Mechanics. (English Edition)
publication date: 1988-01-01T00:00:00Z
review text: \textit{F. A. Cassara, T. S. Sundresh} and \textit{H. Schachter} [IEEE Trans. Commun., COM-25, 1480-1485 (1977)] obtained a model of CCPLL describing the problem how the capacity of information can be increased and the problem of the common frequence channel resisted. The model is of the form \[ {\dot \theta}_ 1=a_ 1\{\sin (\Delta \omega t-\theta_ 1)- \sin (\theta_ 2-\theta_ 1)\},\quad {\dot \theta}_ 2=a_ 2\{\sin (\Delta \omega t-\theta_ 2)-\sin (\theta_ 1-\theta_ 2)\}. \] Let \(x=\theta_ 1-\Delta \omega t\), \(y=\theta_ 2-\Delta \omega t\). Then the model becomes \[ (1)\quad \dot x=a_ 1[\sin (x-y)-\sin x-\beta_ 1],\quad \dot y=a_ 2[\sin (y-x)-\sin y-\beta_ 2]. \] The author analyzes the number and types of the singular points of the system (1) on the torus [-\(\pi\),\(\pi\) ]\(\times [-\pi,\pi]\); (1) has only saddle, node and saddle-node points and at most six singular points. He gives the bifurcation curve on the parameter plane \((\beta_ 1,\beta_ 2)\) which is just the set of all the points \((\beta_ 1,\beta_ 2)\) such that there are saddle nodes. He proves the structural stability of the system (1) on the torus and the existence and uniqueness of the second-kind limit cycle of (1) in some cases.
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 94C05
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34D30
zbMATH DE Number: 4194771
zbMATH Keywords: toroidal differential equation
zbMATH Keywords: global structural stability
MaRDI profile type: Publication
cites work: Q5582924
cites work: Structural stability on two-dimensional manifolds
description: scientific article; zbMATH DE number 4194771