2003/05/14 by Michael Schanz, Axel Pelster · 62 citations
Computer Science · Mathematics · Physics and Astronomy · #Bifurcation #Chaos control and synchronization #Computer science #Computer simulation #Constant (computer programming) #Control (management) #Control theory (sociology) #Geometry #Hopf bifurcation #Loop (graph theory) #Mathematical analysis #Mathematics #Mechanics #Nonlinear Dynamics and Pattern Formation #Nonlinear system #Numerical analysis #Orbit (dynamics) #Period-doubling bifurcation #Phase (matter) #Phase portrait #Physics #Quantum mechanics #Scaling #nlin.CD #stochastic dynamics and bifurcation
paper · pdf · doi:10.1103/physreve.67.056205
published in Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 67(5), 056205 (American Physical Society) · 12 pages
openalex publication_date 2003/05/14 · arxiv created 2005/01/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We derive the normal form for the delay-induced Hopf bifurcation in the first-order phase-locked loop with time delay by the multiple scaling method. The resulting periodic orbit is confirmed by numerical simulations. Further detailed numerical investigations demonstrate exemplarily that this system reveals a rich dynamical behavior. With phase portraits, Fourier analysis, and Lyapunov spectra it is possible to analyze the scaling properties of the control parameter in the period-doubling scenario, both qualitatively and quantitatively. Within the numerical accuracy there is evidence that the scaling constant of the time-delayed phase-locked loop coincides with the Feigenbaum constant delta approximately 4.669 in one-dimensional discrete systems.