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Varin, Victor Petrovich
Statistics Math-Net.Ru
Total publications: 67
Scientific articles: 67

Number of views:
This page:782
Abstract pages:8490
Full texts:3172
References:1065
Senior Researcher
Candidate of physico-mathematical sciences (1992)
Speciality: 01.02.05 (Mechanics of fluids, gases and plasmas)
Birth date: 24.03.1959
E-mail:
Keywords: ordinary differential equations; computational mathematics; differential geometry; computer algebra.

Subject:

A new method of analysis of degeneracies of arbitrary co-dimension in nonlinear systems of ODEs depending on parameters was suggested. The method is based on the use of variational equations of higher order, which are obtained recursively from the original equations and each other. These equations may be computed with the help of computer algebra systems. Degeneracies are understood in a broad sense. Bifurcations, the loss of stability, geometric peculiarities in the phase space (folds and pleats of the families of solutions) etc.\ may be effectively studied by this method. As an application, the equation of oscillations of the satellite in the plane of its elliptic orbit was considered, for which some new degenerate periodic solutions were found. Besides, the classical center-focus problem was considered. It was shown that all cases ever studied for this problem (i.e. systems with one edge of the Newton polygon) may be investigated by the method. Within this problem, perturbed Hamiltonian system with two edges of the Newton polygon were studied for the first time. Explicit formulas for the asymptotics of the Poincare mapping were obtained.

Biography

Graduated from Faculty of Mathematics and Mechanics of Lomonosov Moscow State University (MSU) in 1981 (Department of Theory of Functions and Functional Analysis). Graduated from postgraduate studies in MSU in 1986 at Department of General Control Problems. Advisor Prof. K. I. Babenko. Ph.D. thesis was defended in 1992. At this time, I am a senior researcher at Keldysh Institute of Applied Mathematics. A list of my works contains 25 titles.

In 1999, I translated from Russian the book "Power Geometry" by Prof. A. D. Bruno. It appeared at "Elsevier", 2000, as "Power Geometry in Algebraic and Differential Equations".
   
Main publications:
  • Varin V. P. An analysis of periodic flows in the vicinity of the plane Poiseille flow (with A. L.Afendikov) // Eur. J. Mech., B/Fluids, 10:6, 1991, 577–603.
  • Varin V. P. Bifurcations of some viscous fluid flow and transition to turbulence (with A. L. Afendikov) // Eur. J. Mech., B/Fluids, 10:2, Suppl., 1991, 13–18.
  • Varin V. P. The limit problems for the equation of oscillations of a satellite (with A. D. Bruno) // Celestial Mechanics and Dynamical Astronomy, 1997, v. 67, no. 1, pp. 1–40.
  • Varin V.P. Degeneracies of periodic solutions to the Beletskii equation // Regular & Chaotic Dynamics, 2000, v. 5, no. 3, pp. 38–54.

https://www.mathnet.ru/eng/person9352
List of publications on Google Scholar
List of publications on ZentralBlatt
https://mathscinet.ams.org/mathscinet/MRAuthorID/244430
https://orcid.org/0000-0001-7324-1823

Publications in Math-Net.Ru Citations
2024
1. V. P. Varin, “Rational coefficients of orthogonal expansions of some functions”, Keldysh Institute preprints, 2024, 059, 23 pp.  mathnet
2. V. P. Varin, “Spectral methods for polynomial interpolation and approximation”, Keldysh Institute preprints, 2024, 027, 20 pp.  mathnet
2023
3. V. P. Varin, “Rational arithmetic with a round-off”, Keldysh Institute preprints, 2023, 071, 25 pp.  mathnet
4. V. P. Varin, “Approximation of differential operators with boundary conditions”, Zh. Vychisl. Mat. Mat. Fiz., 63:8 (2023),  1251–1271  mathnet  elib; Comput. Math. Math. Phys., 63:8 (2023), 1381–1400 5
5. V. P. Varin, “Functional summation of series”, Zh. Vychisl. Mat. Mat. Fiz., 63:1 (2023),  16–30  mathnet  elib; Comput. Math. Math. Phys., 63:1 (2023), 14–28 2
2022
6. V. P. Varin, “Approximation of differential operators with boundary conditions”, Keldysh Institute preprints, 2022, 077, 31 pp.  mathnet 1
7. V. P. Varin, “Functional summation of series”, Keldysh Institute preprints, 2022, 028, 23 pp.  mathnet 3
8. V. P. Varin, “Some applications of Brun's criterion”, Keldysh Institute preprints, 2022, 014, 22 pp.  mathnet
9. V. P. Varin, “Sequence transformations in proofs of irrationality of some fundamental constants”, Zh. Vychisl. Mat. Mat. Fiz., 62:10 (2022),  1587–1614  mathnet  elib; Comput. Math. Math. Phys., 62:10 (2022), 1559–1585 7
10. V. P. Varin, “Invariant curves of some discrete dynamical systems”, Zh. Vychisl. Mat. Mat. Fiz., 62:2 (2022),  199–216  mathnet  elib; Comput. Math. Math. Phys., 62:2 (2022), 201–217  isi  scopus 4
2021
11. V. P. Varin, “Sequence transformations in proofs of irrationality of some fundamental constants”, Keldysh Institute preprints, 2021, 084, 32 pp.  mathnet 1
12. V. P. Varin, “On interpolation of some recurrent sequences”, Zh. Vychisl. Mat. Mat. Fiz., 61:6 (2021),  913–925  mathnet  elib; Comput. Math. Math. Phys., 61:6 (2021), 901–913  isi  scopus 4
2020
13. V. P. Varin, “Invariant curves of some discrete dynamical systems”, Keldysh Institute preprints, 2020, 078, 27 pp.  mathnet
14. V. P. Varin, “On intrpolation of some recurrent sequences”, Keldysh Institute preprints, 2020, 024, 19 pp.  mathnet 1
15. V. P. Varin, “Computation of periodic solutions to pendulum type systems with a small parameter”, Zh. Vychisl. Mat. Mat. Fiz., 60:12 (2020),  2055–2072  mathnet  elib; Comput. Math. Math. Phys., 60:12 (2020), 1990–2006  isi  scopus
16. V. P. Varin, “Nonlocal singularities on families of periodic solutions to ordinary differential equations”, Zh. Vychisl. Mat. Mat. Fiz., 60:1 (2020),  57–69  mathnet  elib; Comput. Math. Math. Phys., 60:1 (2020), 53–64  isi  scopus
2019
17. V. P. Varin, “Pendulum type systems from computer algebra viewpoint”, Keldysh Institute preprints, 2019, 103, 28 pp.  mathnet
18. V. P. Varin, “Integration of ODEs on Riemann surfaces with an arbitrary precision”, Keldysh Institute preprints, 2019, 001, 24 pp.  mathnet  elib
19. V. P. Varin, “Integration of ordinary differential equations on Riemann surfaces with unbounded precision”, Zh. Vychisl. Mat. Mat. Fiz., 59:7 (2019),  1158–1173  mathnet  elib; Comput. Math. Math. Phys., 59:7 (2019), 1105–1120  isi  scopus 3
2018
20. V. P. Varin, “Factorial transformation for some classical combinatorial sequences”, Zh. Vychisl. Mat. Mat. Fiz., 58:11 (2018),  1747–1770  mathnet  elib; Comput. Math. Math. Phys., 58:11 (2018), 1687–1707  isi  scopus 12
21. V. P. Varin, “Asymptotic expansion of Crocco solution and the Blasius constant”, Zh. Vychisl. Mat. Mat. Fiz., 58:4 (2018),  530–540  mathnet  elib; Comput. Math. Math. Phys., 58:4 (2018), 517–528  isi  scopus 6
2017
22. V. P. Varin, “Factorial transformation for some classical combinatorial sequences”, Keldysh Institute preprints, 2017, 114, 31 pp.  mathnet
23. V. P. Varin, “Special solutions to Chazy equation”, Zh. Vychisl. Mat. Mat. Fiz., 57:2 (2017),  210–236  mathnet  elib; Comput. Math. Math. Phys., 57:2 (2017), 211–235  isi  scopus 2
2016
24. V. P. Varin, “Asymptotic expansion of Crocco solution and Blasius constant”, Keldysh Institute preprints, 2016, 106, 20 pp.  mathnet 1
2015
25. V. P. Varin, “Special solutions to Chazy equation”, Keldysh Institute preprints, 2015, 043, 39 pp.  mathnet
26. V. P. Varin, “Flat expansions and their applications”, Zh. Vychisl. Mat. Mat. Fiz., 55:5 (2015),  807–821  mathnet  mathscinet  zmath  elib; Comput. Math. Math. Phys., 55:5 (2015), 797–810  isi  elib  scopus 4
2014
27. V. P. Varin, “Flat expansions and their applications”, Keldysh Institute preprints, 2014, 023, 25 pp.  mathnet
28. V. P. Varin, “A solution of the Blasius problem”, Zh. Vychisl. Mat. Mat. Fiz., 54:6 (2014),  1022  mathnet  mathscinet  elib; Comput. Math. Math. Phys., 54:6 (2014), 1025–1036  isi  elib  scopus 9
2013
29. V. P. Varin, “A solution of the Blasius problem”, Keldysh Institute preprints, 2013, 040, 21 pp.  mathnet 2
30. V. P. Varin, “An analysis of a vacuum diode model”, Zh. Vychisl. Mat. Mat. Fiz., 53:2 (2013),  263  mathnet  zmath  elib; Comput. Math. Math. Phys., 53:2 (2013), 194–204  isi  elib  scopus 1
2012
31. V. P. Varin, “An analysis of a vacuum diode model”, Keldysh Institute preprints, 2012, 008, 19 pp.  mathnet
2011
32. A. B. Batkhin, A. D. Bruno, V. P. Varin, “Sets of stability of multiparameter Hamiltonian systems”, Keldysh Institute preprints, 2011, 042, 32 pp.  mathnet 3
33. V. P. Varin, “Spectral properties of the period-doubling operator”, Keldysh Institute preprints, 2011, 009, 20 pp.  mathnet
2010
34. V. P. Varin, “Flat Expansions of Solutions to ODEs at Singularities”, Keldysh Institute preprints, 2010, 064, 12 pp.  mathnet 4
35. A. D. Bruno, A. B. Batkhin, V. P. Varin, “Computation of the Sets of Stability in Multiparameter Problems”, Keldysh Institute preprints, 2010, 023, 22 pp.  mathnet 4
36. A. D. Bruno, A. B. Batkhin, V. P. Varin, “The stability set of a gyroscopic problems”, Keldysh Institute preprints, 2010, 004, 30 pp.  mathnet 7
2009
37. V. P. Varin, A. G. Petrov, “A hydrodynamic model of human cochlea”, Zh. Vychisl. Mat. Mat. Fiz., 49:9 (2009),  1708–1723  mathnet  zmath; Comput. Math. Math. Phys., 49:9 (2009), 1632–1647  isi  scopus 5
2008
38. V. P. Varin, A. G. Petrov, “A mathematical model of human cochlea”, Keldysh Institute preprints, 2008, 096, 26 pp.  mathnet
39. A. D. Bruno, V. P. Varin, “The families c and i of periodic solutions of the restricted problem for $\mu=5\cdot10^{-5}$”, Keldysh Institute preprints, 2008, 022, 26 pp.  mathnet
40. V. P. Varin, “Closed families of periodic solutions of the restricted three-body problem”, Keldysh Institute preprints, 2008, 016, 27 pp.  mathnet 2
41. V. P. Varin, “Methods without saturation for boundary value problems”, Keldysh Institute preprints, 2008, 001, 13 pp.  mathnet 1
2007
42. A. D. Bruno, V. P. Varin, “Family $c$ of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2007, 051, 14 pp.  mathnet 1
43. A. D. Bruno, V. P. Varin, “Complicated families of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2007, 035, 18 pp.  mathnet
44. A. D. Bruno, V. P. Varin, “Periodic solutions of the restricted three-body problem for small $\mu$”, Keldysh Institute preprints, 2007, 034, 22 pp.  mathnet
2006
45. A. D. Bruno, V. P. Varin, “The generating family $i$ of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2006, 036  mathnet 1
2005
46. A. D. Bruno, V. P. Varin, “The family $h$ of periodic solutions of the restricted problem for small $\mu$”, Keldysh Institute preprints, 2005, 067  mathnet 1
47. A. D. Bruno, V. P. Varin, “The family $h$ of periodic solutions of the restricted problem for big $\mu$”, Keldysh Institute preprints, 2005, 064  mathnet 1
48. A. D. Bruno, V. P. Varin, “The family $h$ of periodic solutions of the restricted problem for small $\mu$”, Keldysh Institute preprints, 2005, 048  mathnet
49. A. D. Bruno, V. P. Varin, “On families of periodic solutions to the restricted three-body problem”, Keldysh Institute preprints, 2005, 010  mathnet
2004
50. V. P. Varin, “Poincaré map for some polynomial systems of differential equations”, Mat. Sb., 195:7 (2004),  3–20  mathnet  mathscinet  zmath; Sb. Math., 195:7 (2004), 917–934  isi  scopus 16
2003
51. V. P. Varin, “Asymptotics of the Poincaré mapping for some plane polynomial systems”, Keldysh Institute preprints, 2003, 034, 24 pp.  mathnet
2002
52. V. P. Varin, “Wave fronts and caustics of vector fields”, Keldysh Institute preprints, 2002, 066  mathnet
53. A. D. Bruno, V. P. Varin, “Classes of families of generalized periodic solutions to the Beletsky equation”, Keldysh Institute preprints, 2002, 064  mathnet
54. V. P. Varin, “Degenerate bifurcation of periodic solutions to an ODE caused by symmetry”, Keldysh Institute preprints, 2002, 051  mathnet
2001
55. V. P. Varin, “The isolated generating periodic solutions to the Beletsky equation”, Keldysh Institute preprints, 2001, 076  mathnet
56. V. P. Varin, “The center conditions for nearly hamiltonian systems”, Keldysh Institute preprints, 2001, 048  mathnet
57. V. P. Varin, “The center-focus problem and the variational equations”, Keldysh Institute preprints, 2001, 009  mathnet
2000
58. V. P. Varin, “The Symmetric Anomaly and its Computational Application”, Keldysh Institute preprints, 2000, 057  mathnet
59. V. P. Varin, “Degeneracies of Periodic Solutions to the Beletsky Equation”, Regul. Chaotic Dyn., 5:3 (2000),  313–328  mathnet  mathscinet  zmath 11
1999
60. V. P. Varin, “Localization of Degeneracies in Families of Periodic Solutions to an ODE and their Regularization”, Keldysh Institute preprints, 1999, 023  mathnet
61. V. P. Varin, “Localization of Degeneracies in Families of Periodic Solutions to an ODE and their Regularization”, Keldysh Institute preprints, 1999, 022  mathnet
1997
62. V. P. Varin, “Generalized Periodic Solutions to the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1997, 097  mathnet
63. V. P. Varin, “The critical subfamilies of the family $K_0$ of periodic solutions to the equation of oscillations of a satellite”, Keldysh Institute preprints, 1997, 020  mathnet 1
1996
64. V. P. Varin, “The Critical Families of Periodic Solutions to the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1996, 101  mathnet
1995
65. A. D. Bruno, V. P. Varin, “The Second Limit Problem for the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1995, 128  mathnet
66. A. D. Bruno, V. P. Varin, “The First Limit Problem for the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1995, 124  mathnet
1990
67. A. L. Afendikov, V. P. Varin, “A study of self-oscillating regimes that are close to the Poiseuille flow in a two-dimensional channel”, Dokl. Akad. Nauk SSSR, 313:6 (1990),  1407–1412  mathnet  mathscinet  zmath; Dokl. Math., 35:8 (1990), 720–723

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