Analytical and numerical investigations of the phase-locked loop with time delay

Abstract

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 δ ≈ 4.669 in one-dimensional discrete systems.

Synergetic System Analysis for the Delay-Induced Hopf Bifurcation in the Wright Equation

“…In contrast to the corresponding analysis of the electronic system of a first-order PLL with time delay [11], this paper not only confirms the order parameter concept for delay systems but also represents a successful test for the slaving principle of synergetics, i.e., for the influence of the center manifold on the order parameter equations. Thus the validity of the circular causality chain of synergetics (see Figure 6.1) has been demonstrated for the Hopf bifurcation of a delay differential equation.…”

“…The predictions of the synergetic system analysis have been quantitatively tested by investigating the delay-induced Hopf bifurcation of the electronic system of a first-order phase-locked loop (PLL) [7]. The periodic orbit which results from the corresponding order parameter equation near the bifurcation point has been confirmed by both a multiple scale procedure and numerical simulations [10,11]. Although this application exemplarily proves the order parameter concept for delay systems, it does not allow us to draw conclusions about the slaving principle.…”

“…Furthermore, the Floquet theory for delay differential equations, and thus the linear stability analysis for a periodic reference state, was already established in [29,30,31]. However, a corresponding synergetic system analysis which derives the order parameter equations and the normal forms for bifurcations of oscillatory solutions is still missing [10,11,32].…”

Dynamics of an inertial two-neuron system with time delay

“…It is well known that time delay can cause a stable system to oscillate [16][17][18], and may lead to bifurcation scenarios resulting in chaotic dynamics [19,20]. Extending Hopfield's model, Marcus and Westervelt [23] considered the effect of including a temporal delay in the model to account for finite propagation and signal processing times, and they observed sustained oscillations resulting from the time delay.…”

An iteration method for calculating the periodic solution of time-delay systems after a Hopf bifurcation

“…The bifurcated periodic solution is refined from the iteration method as following The multiple scaling method is given in [20] Obviously, the high frequency terms contribute little to the bifurcated periodic solution if ε is small. Thus, it is sufficient to have an approximation x(t) ≈ x 0 (t) = r cos( π 2 t) in the vicinity of the Hopf bifurcation, with r determined from (11).…”