American Institute of Physics, CHAOS 4 (1), pp. 75-83, 1994
Bifurcations in a system described by a nonlinear differential equation with delay

Yoshisuke Ueda and Hirofumi Ohta
Department of Electrical Engineering, Kyoto University, Kyoto 606, Japan

H. Bruce Stewart
Division of Applied Science, Brookhaven National Laboratory, New York 11973, USA

Computer simulations of a nonlinear differential equation with time delay have been carried out to determine the possible steady states over a wide range of parameter values. A variety of nonlinear phenomena, including chaotic attractors and multiple coexisting attractors, are observed. Precision of the solutions is verified by means of evaluating the computational error at each time step. A number of bifurcations are observed, and the involvement of unstable periodic orbits is confirmed. The phase space of the system is infinite-dimensional, but nonetheless all the bifurcation phenomena observed, including the blue sky disappearance (boundary crisis) of a chaotic attractor, show geometric structures which are consistent with familiar low-dimensional center-manifold descriptions.
I. INTRODUCTION (digest)
II. COMPUTER SIMULATION METHODS (digest)
  A. Taylor series method
  B. Validated numerical integration method
  C. Harmonic balance method of finding a periodic solution
III. SIMULATION RESULTS (digest)
  A. Overview of bifurcations
  B. Steady states and bifurcations for δ = 0.5
  C. Unstable limit cycles
  D. Further evidence concerning bifurcations
IV. CONCLUSIONS
Errata for the publication
Related papers