Strange Attractors in a System Described by Nonlinear Differential-Difference Equation

  • Y. Ueda
  • H. Ohta
Part of the Springer Series in Synergetics book series (SSSYN, volume 24)


This report deals with strange attractors which occur in a system described by the following differential-difference equation:
$$\frac{{d\theta (t)}}{{dt}} + \sin \theta (t - L) = \delta$$
. This equation is a mathematical model of phase-locked loops (PLL) with time delay. Synchronized states of the PLL are represented by the equilibrium points of the equation. The pull-in region, i.e., the parameter region in which all initial conditions lead to quiescent steady states, was already reported with some regions correlated with asynchronized steady states [1]. This report surveys various types of steady states, especially chaotic steady states, in computer-simulated systems of Eq. (1).


Librium Point Strange Attractor Ordinary Differential Equation Stable Limit Cycle Stable Equilibrium Point 
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  1. Y. Ueda: IEEE 24th Midwest Symposium on CAS Proc., pp. 549–553 (1981)Google Scholar
  2. Y. Ueda: J. Statistical Physics, 20, 2, pp. 181–196 (1979)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Y. Ueda
    • 1
  • H. Ohta
    • 1
  1. 1.Department of Electrical EngineeringKyoto UniversityKyotoJapan

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