Constructing Vector Fields Research Articles

Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling

In this paper, we introduce a nonlinear time-varying coupling law, which can be designed in a fully decentralized manner and achieves approximate synchronization with arbitrary precision, under only mild assumptions on the individual vector fields and the underlying (undirected) graph structure. The proposed coupling law is motivated by the so-called funnel control method studied in adaptive control under the observation that arbitrary precision synchronization can be achieved for heterogeneous multi-agent systems by a high-gain coupling; consequently we call our novel synchronization method ‘(node-wise) funnel coupling.’ By adjusting the conventional proof technique in the funnel control study, we are even able to obtain asymptotic synchronization with the same funnel coupling law. Moreover, the emergent collective behavior that arises for a heterogeneous multi-agent system when enforcing arbitrary precision synchronization by the proposed funnel coupling law, is analyzed in this paper. In particular, we introduce a single scalar dynamics called ‘emergent dynamics’ which describes the emergent synchronized behavior of the multi-agent system under funnel coupling. Characterization of the emergent dynamics is important because, for instance, one can design the emergent dynamics first such that the solution trajectory behaves as desired, and then, provide a design guideline to each agent so that the constructed vector fields yield the desired emergent dynamics. We illustrate this idea via the example of a distributed median solver based on funnel coupling.

Constraints on primordial magnetic fields from inflation

We present generic bounds on magnetic fields produced from cosmic inflation. By investigating field bounds on the vector potential, we constrain both the quantum mechanical production of magnetic fields and their classical growth in a model independent way. For classical growth, we show that only if the reheating temperature is as low as Treh ≲ 102 MeV can magnetic fields of 10−15 G be produced on Mpc scales in the present universe. For purely quantum mechanical scenarios, even stronger constraints are derived. Our bounds on classical and quantum mechanical scenarios apply to generic theories of inflationary magnetogenesis with a two-derivative time kinetic term for the vector potential. In both cases, the magnetic field strength is limited by the gravitational back-reaction of the electric fields that are produced simultaneously. As an example of quantum mechanical scenarios, we construct vector field theories whose time diffeomorphisms are spontaneously broken, and explore magnetic field generation in theories with a variable speed of light. Transitions of quantum vector field fluctuations into classical fluctuations are also analyzed in the examples.

Realization problems for limit cycles of planar polynomial vector fields

We show that for any finite configuration of closed curves Γ⊂R2, one can construct an explicit planar polynomial vector field that realizes Γ, up to homeomorphism, as the set of its limit cycles with prescribed periods, multiplicities and stabilities. The only obstruction given on this data is the obvious compatibility relation between the stabilities and the parity of the multiplicities. The constructed vector fields are Darboux integrable and admit a polynomial inverse integrating factor.

Construction of codimension one homoclinic cycles

We give an explicit construction of families of Dm-equivariant polynomial vector fields in possessing a codimension one homoclinic cycle. The homoclinic cycle consists of m homoclinic trajectories all connected to the equilibrium at the origin. The constructed vector fields can provide a setting for a (numerical) bifurcation study of these homoclinic cycles, in particular for m equal to a multiple of 4, where the bifurcations form an open problem.

Monitoring of clouds on the basis of data provided by geostationary earth satellites

The process of constructing vector fields of displacements of clouds in the atmosphere from a series of consecutive images obtained by geostationary satellites is considered. The process is based on searching for the maximum of the coefficient of mutual correlation between reference objects (targets) found in the current image of the series and their positions in the next image of the series. In addition to the transfer transformation, the target can also be subjected to transformations of rotation and scaling. Interpolation of pixels is used to increase the accuracy of computations.

Transformation that changes the geometric structure of a vector field

We propose a method of constructing vector fields with certain vortex properties by means of transformations that change the value of the field vector at every point, the form of the field lines, and their mutual position. We discuss and give concrete examples of the prospects of using the method in applications involving solution of partial differential equations, including nonlinear ones.

Lyapunov Vector Fields for Autonomous Unmanned Aircraft Flight Control

General techniques for constructing vector fields for unmanned aircraft guidance are provided that incorporate Lyapunov stability properties to produce simple, globally stable vector fields in three dimensions. Use of these fields to produce circular loiter pattern attractors is illustrated, along with a simple switching algorithm to enable following of arbitrary way point sequences. Alternatively, attractor shape variations are developed by warping the circular loiter, preserving global stability guarantees, and accurate path tracking. An example of this technique is provided that produces a racetrack loiter pattern, and three different variations in the warping technique are compared. Finally, tracking of the vector field is considered, using Lyapunov techniques to show global stability of heading and path position for several types of tracking control laws that are compatible with low cost unmanned aircraft avionics.

On the astrometric behaviour of binary microlensing events

Despite the suspected binarity for a significant fraction of Galactic lenses, the current photometric surveys detected binary microlensing events only for a small fraction of the total events. The detection efficiency is especially low for non-caustic crossing events, which comprise the majority of the binary lensing events, as a result of the absence of distinctive features in their light curves combined with small deviations from the standard light curve of a single point-mass event. In addition, even if they are detected, it will be difficult to determine the solution of the binary lens parameters owing to the severe degeneracy problem. In this paper, we investigate the properties of binary lensing event expected when they are astrometrically observed by using high-precision interferometers. For this, we construct vector field maps of excess centroid shifts, which represent the deviations of the binary lensing centroid shifts from those of a single lensing event as a function of source position. From the analysis of the maps, we find that the excess centroid shifts are substantial in a considerably large area around caustics. In addition, they have characteristic sizes and directions depending strongly on the source positions with respect to the caustics and the resulting trajectories of the light centroid (astrometric trajectories) have distinctive features, which can be distinguished from the deviations caused by other reasons. We classify the types of the deviations and investigate where they occur. Because of the strong dependence of the centroid shifts on the lens system geometry combined with the distinctive features in the observed astrometric trajectories, astrometric binary lensing observations will provide an important tool that can probe the properties of the Galactic binary lens population.

Properties of planet-induced deviations in the astrometric microlensing centroid shift trajectory

An extrasolar planet can be detected by microlensing because it can distort the smooth lensing light curve created by the primary lens. As a new method to search for and characterize extrasolar planets, Safizadeh, Dalal & Griest proposed to detect the planet-induced distortions in the trajectory of the microlensed source star's centroid motion (astrometric curve), which is observable by using the next generation of high-precision interferometers. In this paper we investigate the properties of the planet-induced deviations in the astrometric curves (excess centroid shifts Δδ) and the correlations of Δδ with the photometric deviations. For this, we construct vector field maps of Δδ, which represent the difference of the centroid shift from that expected in the absence of the planet as a function of source positions. From this investigation, we find that significant astrometric deviations occur not only in the region near the caustics but also in the region close to the planet–primary axis between the caustics. However, owing to the difference in the locations of the caustics between the two types of systems with the planet–primary separations (normalized by the angular Einstein ring radius) and , the locations of the major deviation regions of the two systems are different from each other. For systems with , the major deviation vectors have orientations, in most cases, pointing towards the planet, while they point away from the planet for systems with . The major deviation region is surrounded by the region of moderate deviations, within which the orientation of Δδ is reversed compared with the orientation in the major deviation region. We also find that the astrometric deviation is closely correlated with the photometric one, as discussed by Safizadeh et al. The astrometric deviation increases as the photometric deviation increases, and Δδ is directed towards the planet when the light curve has positive deviation, and vice versa. We also present excess centroid shift maps for lens systems with various values of the planetary separation, planet/primary mass ratio, and source size to show the changes in the pattern of Δδ with these parameters.

Vector fields generating more than one flow

Vector fields generating more than one flow on a manifold are constructed. For one‐dimensional case a complete description of the set of flows is given. For dimensions larger than one a method of constructing vector fields with dense or open branched sets sets is given. Density of vector fields with nonempty sets of branched points is studied.