Gaia Early Data Release 3: The celestial reference frame (Gaia-CRF3)

Context. Gaia-CRF3 is the celestial reference frame for positions and proper motions in the third release of data from the Gaia mission, Gaia DR3 (and for the early third release, Gaia EDR3, which contains identical astrometric results). The reference frame is defined by the positions and proper motions at epoch 2016.0 for a specific set of extragalactic sources in the (E)DR3 catalogue. Aims. We describe the construction of Gaia-CRF3 and its properties in terms of the distributions in magnitude, colour, and astrometric quality. Methods. Compact extragalactic sources in Gaia DR3 were identified by positional cross-matching with 17 external catalogues of quasi-stellar objects (QSO) and active galactic nuclei (AGN), followed by astrometric filtering designed to remove stellar contaminants. Selecting a clean sample was favoured over including a higher number of extragalactic sources. For the final sample, the random and systematic errors in the proper motions are analysed, as well as the radio-optical offsets in position for sources in the third realisation of the International Celestial Reference Frame (ICRF3). Results. Gaia-CRF3 comprises about 1.6 million QSO-like sources, of which 1.2 million have five-parameter astrometric solutions in Gaia DR3 and 0.4 million have six-parameter solutions. The sources span the magnitude range G = 13-21 with a peak density at 20.6 mag, at which the typical positional uncertainty is about 1 mas. The proper motions show systematic errors on the level of 12 μas yr-1 on angular scales greater than 15 deg. For the 3142 optical counterparts of ICRF3 sources in the S/X frequency bands, the median offset from the radio positions is about 0.5 mas, but it exceeds 4 mas in either coordinate for 127 sources. We outline the future of Gaia-CRF in the next Gaia data releases. Appendices give further details on the external catalogues used, how to extract information about the Gaia-CRF3 sources, potential (Galactic) confusion sources, and the estimation of the spin and orientation of an astrometric solution

Gaia Early Data Release 3: acceleration of the solar system from Gaia astrometry

Stars and planetary system

Quadratic Planar Systems With Two Parallel Invariant Straight Lines

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Slow-fast Systems On Algebraic Varieties Bordering Piecewise-smooth Dynamical Systems

This article extends results contained in Buzzi et al. (2006) [4], Llibre et al. (2007, 2008) [12,13] concerning the dynamics of non-smooth systems. In those papers a piecewise C k discontinuous vector field Z on Rn is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider F:U→R a polynomial function defined on the open subset U⊂Rn. The set F -1(0) divides U into subdomains U1,U2,. . .,Uk, with border F -1(0). These subdomains provide a Whitney stratification on U. We consider Zi:Ui→Rn smooth vector fields and we get Z=(Z 1, . . ., Z k) a discontinuous vector field with discontinuities in F -1(0). Our approach combines several techniques such as ε-regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an ε-regularization of Z (see Sotomayor and Teixeira, 1996 [18]; Llibre and Teixeira, 1997 [15]). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 [16]), in systems with hysteresis (Seidman, 2006 [17]) and in mechanical systems with impacts (di Bernardo et al., 2008 [5]). © 2011 Elsevier Masson SAS.1364444462Alexander, J.C., Seidman, T.I., Sliding modes in intersecting switching surfaces. I. Blending (1998) Houston J. Math., 24, pp. 545-569Alexander, J.C., Seidman, T.I., Sliding modes in intersecting switching surfaces. II. Hysteresis (1999) Houston J. Math., 25, pp. 185-211Andronov, A.A., Vitt, A.A., Khaikin, S.E., (1966) Theory of Oscillators, , Dover, New YorkBuzzi, C., Silva, P.R., Teixeira, M.A., A singular approach to discontinuous vector fields on the plane (2006) J. Differential Equations, 231, pp. 633-655di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P., Piecewise-Smooth Dynamical Systems. Theory and Applications (2008) Appl. Math. Sci., 163. , Springer-Verlag, LondonDumortier, F., Roussarie, R., Canard cycles and center manifolds (1996) Mem. Amer. Math. Soc., 121Fenichel, N., Geometric singular perturbation theory for ordinary differential equations (1979) J. Differential Equations, 31, pp. 53-98Filippov, A.F., Differential Equations with Discontinuous Right-Hand Sides (1988) Math. Appl. (Soviet Ser.), , Kluwer Academic Publishers, DordrechtJones, C., Geometric singular perturbation theory (1995) Lecture Notes in Math., 1609. , Springer-Verlag, Heidelberg, C.I.M.E. LecturesKozlova, V.S., Structural stability of a discontinuous system (1984) Vestnik Moskov. Univ. Ser. I Mat. Mekh., 5, pp. 16-20Kuznetsov, Y.A., Rinaldi, S., Gragnani, A., One-parameter bifurcations in planar Filippov systems (2003) Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (8), pp. 2157-2188Llibre, J., Silva, P.R., Teixeira, M.A., Regularization of discontinuous vector fields via singular perturbation (2007) J. Dynam. Differential Equations, 19 (2), pp. 309-331Llibre, J., Silva, P.R., Teixeira, M.A., Sliding vector fields via slow fast systems (2008) Bull. Belg. Math. Soc. Simon Stevin, 15, pp. 851-869Llibre, J., Silva, P.R., Teixeira, M.A., Study of singularities in non smooth dynamical systems via singular perturbation (2009) SIAM J. Appl. Dyn. Syst., 8, pp. 508-526Llibre, J., Teixeira, M.A., Regularization of discontinuous vector fields in dimension three (1997) Discrete Contin. Dyn. Syst., 3, pp. 235-241Minorsky, N., (1969) Theory of Nonlinear Control Systems, p. 331. , McGraw-Hill, New York, London, Sydney, xxSeidman, T., (2006) Proc. Dover Conf., , Aspects of modeling with discontinuities, in: G. N'Guerekata (Ed.), Advances in Applied and Computational MathematicsSotomayor, J., Teixeira, M.A., Regularization of discontinuous vector fields (1996), 95, pp. 207-223. , in: International Conference on Differential Equations, Lisboa, EquadiffTeixeira, M.A., Stability conditions for discontinuous vector fields (1990) J. Differential Equations, 88, pp. 15-24Teixeira, M.A., Generic bifurcation of sliding vector fields (1993) J. Math. Anal. Appl., 176, pp. 436-457Teixeira, M.A., Perturbation theory for non-smooth systems (2009) Encyclopedia of Complexity and Systems Science, vol. 22 (Perturbation Theory), pp. 6697-6719. , Springer-Verlag, New York, R. Meyers (Ed.)Whitney, H., Elementary structure of real algebraic varieties (1957) Ann. of Math., 66, pp. 545-55

Birth Of Limit Cycles Bifurcating From A Nonsmooth Center

This paper is concerned with a codimension analysis of a two-fold singularity of piecewise smooth planar vector fields, when it behaves itself like a center of smooth vector fields (also called nondegenerate σ-center). We prove that any nondegenerate σ-center is σ-equivalent to a particular normal form Z0. Given a positive integer number k we explicitly construct families of piecewise smooth vector fields emerging from Z0 that have k hyperbolic limit cycles bifurcating from the nondegenerate σ-center of Z0 (the same holds for k=∞). Moreover, we also exhibit families of piecewise smooth vector fields of codimension k emerging from Z0. As a consequence we prove that Z0 has infinite codimension. © 2013 Elsevier Masson SAS.10213647Arrowsmith, D.K., Place, C.M., (1990) An Introduction to Dynamical Systems, , Cambridge University PressBuzzi, C.A., de Carvalho, T., Teixeira, M.A., On three-parameter families of Filippov systems - the fold-saddle singularity (2012) Int. J. Bifurc. Chaos, 22 (12), p. 18. , 1250291Buzzi, C.A., de Carvalho, T., Teixeira, M.A., On 3-parameter families of piecewise smooth vector fields in the plane (2012) SIAM J. Appl. Dyn. Syst., 11 (4), pp. 1402-1424Caubergh, M., (2004), Limit cycles near centers, Thesis Limburgh University, DiepenbeckCaubergh, M., Dumortier, F., Hopf-Takens bifurcations and centers (2004) J. Differ. Equ., 202, pp. 1-31Chow, S.N., Hale, J.K., (1982) Methods of Bifurcation Theory, , Springer-VerlagCeragioli, F., (1999) Discontinuous ordinary differential equations and stabilization, , http://calvino.polito.it/~ceragioli, PhD thesis, University of Firenze, Italy, Electronically available atCortés, J., Discontinuous dynamical systems: A tutorial on solutions, nonsmooth analysis, and stability, , arxiv:0901.3583, posted inDumortier, F., Singularities of Vector Fields (1978) Monografias de Matematica, 32. , Instituto de Matematica Pura e Aplicada, Rio de JaneiroEkeland, I., Discontinuits de champs hamiltoniens et existence de solutions optimales en calcul des variations (1977) Publ. Math. Inst. Hautes Études Sci., 47, pp. 5-32. , (in French)Filippov, A.F., Differential Equations with Discontinuous Right-Hand Sides (1988) Math. Appl., Sov. Ser., , Kluwer Academic Publishers, DordrechtGavrilov, L., Horozov, E., Limit cycles of perturbations of quadratic Hamiltonian vector fields (1993) J. Math. Pures Appl., 72 (2), pp. 213-238Golubitski, M., Guillemin, V., (1973) Stable Mappings and Their Singularities, , Springer-VerlagGuardia, M., Seara, T.M., Teixeira, M.A., Generic bifurcations of low codimension of planar Filippov systems (2011) J. Differ. Equ., 250, pp. 1967-2023Kuznetsov, Y.A., Rinaldi, S., Gragnani, A., One-parameter bifurcations in planar Filippov systems (2003) Int. J. Bifurc. Chaos, 13, pp. 2157-2188Takens, F., Unfoldings of certain singularities of vector fields generalised Hopf bifurcations (1973) J. Differ. Equ., 14 (3), pp. 476-493Teixeira, M.A., Perturbation theory for non-smooth systems, Meyers: encyclopedia of complexity and systems (2008) Science, 15