Parabolic Torus Research Articles

Quasi-periodic Hamiltonian pitchfork bifurcation in a phenomenological model with 3 degrees of freedom

Litvak-Hinenzon et al. developed the phenomenological model to simulate the horizontal motion of particles in the atmosphere, and a series of their papers (see e.g., Litvak-Hinenzon et al. (Phys. D 164:213-250, 2002; Nonlinearity 15:1149-1177, 2002; SIAM J. Appl. Dyn. Syst. 3:525-573, 2004)) focused on studying the fate of parabolic resonance lower dimensional tori (as part of a quasi-periodic Hamiltonian pitchfork bifurcation (HPB) scenario in the unperturbed phenomenological model) under perturbations. However, the structural stability of quasi-periodic HPB involved therein has not been fully exposed theoretically (Litvak-Hinenzon et al. have only given some numerical explanations). Based on BCKV singularity theory established by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993), we consider a more general quasi-periodic HPB triggered by the Z2-invariant universal unfolding Nbckv=ay22−(λ+bI1)x22+cx44with respect to Z2-equivariant BCKV-restricted morphisms of the planar singularity a2y2+c4x4 (the coefficients a,b,c≠0, the I1 is regarded as distinguished parameter with respect to the external parameter λ). We prove a KAM (Kolmogorov–Arnold–Moser) theorem concerning parabolic tori in such quasi-periodic HPB, by which Diophantine parabolic tori (and the whole corresponding Diophantine HPB scenario) survive non-integrable and Z2-invariant Hamiltonian perturbations, parametrized by pertinent large Cantor sets. In the context of Z2-symmetry, our results can be seen as rigorous proof of the structural stability problem of bifurcations of Floquet-tori triggered by the universal unfolding Nbckv with distinguished parameters which is proposed by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993). Ultimately, we similarly obtain the structural stability result of quasi-periodic HPB in the phenomenological model mentioned above, which can be utilized as a starting point for a deeper understanding of the various resonances and chaotic dynamics (in the gaps of the Cantor sets), just as Litvak-Hinenzon et al. did for normally parabolic tori undergoing a HPB.

On symplectic fillings of virtually overtwisted torus bundles

We use Menke's JSJ-type decomposition theorem for symplectic fillings to reduce the classification of strong and exact symplectic fillings of virtually overtwisted torus bundles to the same problem for tight lens spaces. For virtually overtwisted structures on elliptic or parabolic torus bundles, this gives a complete classification. For virtually overtwisted structures on hyperbolic torus bundles, we show that every strong or exact filling arises from a filling of a tight lens space via round symplectic 1-handle attachment, and we give a condition under which distinct tight lens space fillings yield the same torus bundle filling.

Whiskered Parabolic Tori in the Planar $$(n+1)$$-Body Problem

The planar \((n+1)\)-body problem models the motion of \(n+1\) bodies in the plane under their mutual Newtonian gravitational attraction forces. When \(n\ge 3\), the question about final motions, that is, what are the possible limit motions in the planar \((n+1)\)-body problem when \(t\rightarrow \infty \), ceases to be completely meaningful due to the existence of non-collision singularities. In this paper we prove the existence of solutions of the planar \((n+1)\)-body problem which are defined for all forward time and tend to a parabolic motion, that is, that one of the bodies reaches infinity with zero velocity while the rest perform a bounded motion. These solutions are related to whiskered parabolic tori at infinity, that is, parabolic tori with stable and unstable invariant manifolds which lie at infinity. These parabolic tori appear in cylinders which can be considered “normally parabolic”. The existence of these whiskered parabolic tori is a consequence of a general theorem on parabolic tori developed in this paper. Another application of our theorem is a conjugation result for a class of skew product maps with a parabolic torus with its normal form generalizing results of Takens (Ann Inst Fourier 23(2):163–195, 1973), and Voronin (Funktsional Anal i Prilozhen 15(1):1–17, 96, 1981).

Strong symplectic fillability of contact torus bundles

In this paper, we study strong symplectic fillability and Stein fillability of some tight contact structures on negative parabolic and negative hyperbolic torus bundles over the circle. For the universally tight contact structure with twisting $\pi$ in $S^1$-direction on a negative parabolic torus bundle, we completely determine its strong symplectic fillability and Stein fillability. For the universally tight contact structure with twisting $\pi$ in $S^1$-direction on a negative hyperbolic torus bundle, we give a necessary condition for it being strongly symplectically fillable. For the virtually overtwisted tight contact structure on the negative parabolic torus bundle with monodromy $-T^n$ ($n<0$), we prove that it is Stein fillable. By the way, we give a partial answer to a conjecture of Golla and Lisca.

On the uniqueness of the contact structure approximating a foliation

According to a theorem of Eliashberg and Thurston a $C^2$-foliation on a closed 3-manifold can be $C^0$-approximated by contact structures unless all leaves of the foliation are spheres. Examples on the 3-torus show that every neighbourhood of a foliation can contain non-diffeomorphic contact structures. In this paper we show uniqueness up to isotopy of the contact structure in a small neighbourhood of the foliation when the foliation has no torus leaf and is not a foliation without holonomy on parabolic torus bundles over the circle. This allows us to associate invariants from contact topology to foliations. As an application we show that the space of taut foliations in a given homotopy class of plane fields is not connected in general.

Compensation for gravity deformation via subreflector motion of 65 m shaped Cassegrain antenna

Compensation for degrading electromagnetic performances is one of the most important concerns for engineers involved in the structural design of a large antenna. A method is proposed to compensate the degrading electromagnetic performances induced by the main reflector deformation of the large and shaped Cassegrain antenna by adjusting the subreflector position in real time. Segmented parabola fitting based on the least squares method of the theoretical discrete data of the main shaped reflector yields a standard group of parabolic torus that is later used to fit the deformed main reflector, with all the parabolic toruses confocal in the fitting. The optimised best-fit parameters and the matching between the main reflector and subreflector calculate the adjustment quantity of the subreflector. The adjustment quantities, stored in the database of the subreflector, working under various conditions can be called in real time to compensate the gravity deformation of the main reflector. The application of the method to a 65 m aperture-reflector antenna was successful, thus, laying a good foundation for its application in engineering practice.

Bifurcations of normally parabolic tori in Hamiltonian systems

We consider perturbations of integrable Hamiltonian systems in the neighbourhood of normally parabolic invariant tori. Under appropriate transversality conditions the tori in the unperturbed system bifurcate according to a (generalized) cuspoid catastrophe. Combining techniques of KAM theory and singularity theory, we show that such bifurcation scenarios survive the perturbation on large Cantor sets. Applications to rigid body dynamics and forced oscillators are pointed out.

Design of multibeam parabolic torus reflector antennas

Design formulas, principles, and procedures for a multibeam parabolic torus reflector antenna (PTRA) are presented. A six-beam PTRA with 15° geostationary arc coverage for satellite communication is described. Experimental and theoretical radiation patterns are given. Six identical beams are obtained with low sidelobe and low cross-polarization levels. A comparison between the calculated and measured patterns of the central beam shows good agreement. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 27: 343–347, 2000.

The Quasi-Periodic Centre-Saddle Bifurcation

Nearly integrable families of Hamiltonian systems are considered in the neighbourhood of normally parabolic invariant tori. In the integrable case such tori bifurcate into normally elliptic and normally hyperbolic invariant tori. With a KAM-theoretic approach it is shown that both the normally parabolic tori and the bifurcation scenario survive a non-integrable perturbation, parametrised by pertinent large Cantor sets. These results are applied to rigid body dynamics.

RF characterization of an inflatable parabolic torus reflector antenna for space-borne applications

Space-borne satellite applications provide a vast array of services extending from global connectivity to Earth observation systems. The soil moisture radiation mission is a proposed space-borne passive microwave system complementary to the existing Earth observing system operating at low microwave frequencies and requiring an antenna with multibeam, high-beam efficiency, and dual polarization capabilities. To achieve both the large reflector size and the multibeam pattern at the operational frequencies an innovative multibeam reflector antenna design was needed. The advances in inflatable antenna technology has been proposed to overcome the launch vehicle size and weight restrictions. This paper describes a novel offset parabolic torus reflector antenna design that produces the desired multibeam pattern and is compatible with the inflatable antenna technology. Using the system requirements of this mission as an example, the design process for an inflatable parabolic torus reflector antenna is outlined, the development of suitable distortion models is given, and representative RF characteristics are presented. These RF characteristics include far-field patterns, beam contour patterns, beam efficiency, and other key performance parameters. The development of an advanced analytical modeling/numerical tool in support of the design effort is also detailed.

Theoretical analysis of a parabolic torus reflector antenna with multibeam

The parametric equations and the formulas of unit normal vector and surface element for a parabolic torus reflector antenna are derived and the mechanism of producing multibeam is proposed, Based on physical optics, the radiation pattern formulas for the antenna are given, with which the effects of geometric parameters on the antenna are studied. The good agreement between the calculated patterns and the measured ones shows that the theory is helpful for designing parabolic torus antennas.

Parabolic torus transreflector antenna

The parabolic torus transreflector antenna is a circularly symmetric antenna which can be scanned through 360° by rotation of its feed alone. The reflector consists of a radome carrying 45° angled wires so that 45° polarised radiation is reflected on one side and transmitted by the other. A theoretical model has been developed to predict the radiation properties of this antenna for various feed configurations. An experimental torus has been constructed and tested, showing good agreement between theory and experiment. Good radiation patterns (sidelobes ≤ – 20 dB) are obtained with limited aperture efficiency (about 25% of the horizontal aperture). The theory shows how the phase aberrations and amplitude distortions in the radiating aperture, due to the antenna geometry, limit the sidelobe performance for wider aperture illuminations with a single horn feed. A linear array feed, designed to correct phase aberrations in one plane, is found to offer only a partial improvement if the full vertical aperture is required. It is concluded that greater aperture efficiency (≃45%), with sidelobes ≤ – 20 dB, requires either a torus with a restricted vertical aperture, (≤5 λ0), or a planar array feed.