Pyragas time-delayed feedback is a control scheme designed to stabilize unstable periodic orbits, which occur naturally in many nonlinear dynamical systems. It has been successfully implemented in a ...number of applications, including lasers and chemical systems. The control scheme targets a specific unstable periodic orbit by adding a feedback term with a delay chosen as the period of the unstable periodic orbit. However, in an experimental or industrial environment, obtaining the exact period or setting the delay equal to the exact period of the target periodic orbit may be difficult. This could be due to a number of factors, such as incomplete information on the system or the delay being set by inaccurate equipment. In this paper, we evaluate the effect of Pyragas control on the prototypical generic subcritical Hopf normal form when the delay is close to but not equal to the period of the target periodic orbit. Specifically, we consider two cases: first, a constant, and second, a linear approximation of the period. We compare these two cases to the case where the delay is set exactly to the target period, which serves as the benchmark case. For this comparison, we construct bifurcation diagrams and determine any regions where a stable periodic orbit close to the target is stabilized by the control scheme. In this way, we find that at least a linear approximation of the period is required for successful stabilization by Pyragas control.
We investigate an optimal velocity car-following model for
n cars on a circular single-lane road, where reaction-time delay of drivers is taken into account. The stability of the uniform flow ...equilibrium is studied analytically, while bifurcating periodic solutions for different wave numbers are investigated with numerical continuation techniques. This reveals that the periodic solution with the smallest wave number may be stable, and all other periodic solutions are unstable.
As
n is increased, periodic solutions develop stop- and go-fronts that correspond to rapid deceleration and acceleration between regions of uniformly flowing and stagnant traffic. In terms of the positions of all cars on the ring these fronts are associated with traffic jams. All traffic jams form a traffic pattern that evolves under time, due to slow motion of the fronts. The traffic pattern corresponding to the stable periodic motion of cars is the only stable one. However, we find that other periodic orbits may be unstable only so weakly that they give rise to transient traffic jams that may persist for long times. Eventually, such traffic jams either merge with one another or disperse, until the stable traffic pattern is reached.
The stability and nonlinear dynamics of two semiconductor lasers coupled side to side via evanescent waves are investigated by using three different models. In the composite-cavity model, the ...coupling between the lasers is accurately taken into account by calculating electric field profiles (composite-cavity modes) of the whole coupled-laser system. A bifurcation analysis of the composite-cavity model uncovers how different types of dynamics, including stationary phase-locking, periodic, quasiperiodic, and chaotic intensity oscillations, are organized. In the individual-laser model, the coupling between individual lasers is introduced phenomenologically with ad hoc coupling terms. Comparison with the composite-cavity model reveals drastic differences in the dynamics. To identify the causes of these differences, we derive a coupled-laser model with coupling terms which are consistent with the solution of the wave equation and the relevant boundary conditions. This coupled-laser model reproduces the dynamics of the composite-cavity model under weak-coupling conditions.
We demonstrate a method for tracking the onset of oscillations (Hopf bifurcation) in nonlinear
dynamical systems. Our method does not require a mathematical model of the dynamical system
but instead ...relies on feedback controllability. This makes the approach potentially applicable in an
experiment. The main advantage of our method is that it allows one to vary parameters directly along
the stability boundary. In other words, there is no need to observe the transient oscillations of the
dynamical system for a long time to determine their decay or growth. Moreover, the procedure automatically
tracks the change of the critical frequency along the boundary and is able to continue the
Hopf bifurcation curve into parameter regions where other modes are unstable.We illustrate the basic
ideas with a numerical realization of the classical autonomous dry friction oscillator.
A modern real time flood forecasting system requires itsmathematical model(s) to handle highly complex rainfall runoffprocesses. Uncertainty in real time flood forecasting willinvolve a variety of ...components such as measurement noise fromtelemetry systems, inadequacy of the models, insufficiency ofcatchment conditions, etc. Probabilistic forecasting is becomingmore and more important in this field. This article describes a novel attempt to use a Fuzzy Logic approach for river flow modelling based on fuzzy decision trees. These trees are learntfrom data using the MA-ID3 algorithm. This is an extension of Quinlan's ID3 and is based on mass assignments. MA-ID3 allows for the incorporation of fuzzy attribute and class values intodecision trees aiding generalisation and providing a framework for representing linguistic rules. The article showed that with only five fuzzy labels, the FDT model performed reasonably welland a comparison with a Neural Network model (Back Propagation)was carried out. Furthermore, the FDT model indicated that therainfall values of four or five days before the prediction time are regarded as more informative to the prediction than the morerecent ones. Although its performance is not as good as the neural network model in the test case, its glass box nature couldprovide some useful insight about the hydrological processes.PUBLICATION ABSTRACT
An optically injected semiconductor laser can produce excitable multipulses. Homoclinic bifurcation curves confine experimentally accessible regions in parameter space where the laser emits a certain ...number of pulses after being triggered from its steady state by a single perturbation. This phenomenon is organized by a generic codimension-two homoclinic bifurcation and should also be observable in other systems.
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