Alexander D. Bruno, Professor, Keldysh Institute of Applied Mathe-
matics (Moscow),
email: abruno@keldysh.ru
Abstract:
1. Calculus. There are two universal methods for the local study of nonlinear equations and systems of different kinds (algebraic, ordinary, and partial differential): (a) normal form and (b) truncated equations.
(a) Equations with a linear part can be reduced to their normal form by a local change of coordinates. For algebraic equations, it is Implicit Function Theorem. For systems of ordinary differential equations (ODE), I completed the theory of normal forms, which began by Poincaré (1879) and Dulac (1912) for general systems [1] and began by Birkhoff (1929) for Hamiltonian systems [3].
(b) Equations without linear part: I proposed to study properties of solutions to equations (algebraic, ordinary differential, and partial differential) by studying sets of vectorial power exponents in terms of these equations.
Namely to select more simple (“truncated”) equations [0, 2, 4] by means of generalization to polyhedrons, the Newton (1678) and the Hadamard (1893) polygons. By means of power transformations [0, 2, 13] the truncated equations can be strongly simplified and often solved. Solutions of the truncated equations are asymptotically the first approximations of the solutions to the full equations. Continuing that process, we can obtain approximations of any precision to solutions of initial equations. Based on the developed Asymptotic Nonlinear Analysis, I proposed algorithms for solutions to a wide set of singular problems. In particular, for the computation of six different types of asymptotic expansions of solutions to ODE [7, 10, 11], including expansions into trans-series [12].
2. Applications in complicated problems of (a) Mathematics, (b)Mechanics, (c) Celestial Mechanics, and (d) Hydromechanics.
(a) In Mathematics: together with my students, I found all asymptotic expansions of five types of solutions to the Painlevé equations (1906) [10,16] and also gave a very effective method of determining of integrability of the ODE system [17, 18].
(b) In Mechanics: I computed with the high precision influence of small mutation oscillations on the velocity of the precession of a gyroscope [2] and also studied values of parameters of a centrifuge, ensuring the stability of its rotation [19].
(c) In Celestial Mechanics: together with my students, I studied periodic solutions of the Beletsky equation (1956) [5,6], describing the motion of a satellite around its mass center, moving along an elliptic orbit. I found new families of periodic solutions, which are important for passive orientation of the 1satellite [2], including cases with big values of the eccentricity of the orbit, inducing a singularity. Besides, simultaneously with Hénon (1997), I found all regular and singular generating families of periodic solutions of the restricted three-body problem and studied bifurcations of generated families. It allowed explaining some singularities of motions of small bodies of the Solar System [9]. In particular, I found orbits of periodic flies around planets with a close approach to Earth [20].
(d) In Hydromechanics: I studied small surface waves on water [4] and a boundary layer on a needle [8], where equations of a flow have a singularity.
All that is subject to five following one-year courses: 1. Algebraic equations; 2. One ODE of finite order; 3. System of ODEs; 4. Hamiltonian systems; 5. Partial differential equations.
Each course is more complicated than the previous one. For example, power transformations are enough for algebraic equations, but differential equations also demand logarithmic transformations. The main ideas will be explained in the simplest case: algebraic equations [14,15].
Main References (all by A.D.Bruno)
[0] The asymptotic behavior of solutions of nonlinear systems of differential equations. // Soviet Math. Dokl. 3 (1962) 464–467.
[1] Analytical form of differential equations (I). // Trans. Moscow Math. Soc. 25 (1971) 131–288. (II) // Ibid. 26 (1972) 199–239.
[2] Local Methods in Nonlinear Differential Equations. Springer-Verlag: Berlin, 1989. 350 p.
[3] The Restricted 3-Body Problem: Plane Periodic Orbits. Walter de Gruyter, Berlin, 1994. 362 p.
[4] Power Geometry in Algebraic and Differential Equations. Elsevier, Amsterdam, 2000. 385 p.
[5] Families of periodic solutions to the Beletsky equation // Cosmic Research 40:3 (2002) 274–295.
[6] Classes of families of generalized periodic solutions to the Beletsky equation (with V.P. Varin) // Celestial Mechanics and Dynamical Astronomy, 88:4 (2004), 325–341.
[7] Asymptotics and expansions of solutions to an ordinary differential equation
// Russian Mathem. Surveys 59:3 (2004) 429–480.
[8] Axisymmetric boundary layer on a needle (with T.V. Shadrina) // Transactions of Moscow Math. Soc. 68 (2007) 201–259.
[9] Periodic solutions of the restricted three-body problem for small mass ratio (with V.P. Varin) // J. Appl. Math. Mech. 71:6 (2007) 933–960.
[10] Asymptotic expansions of solutions of the sixth Painleve equation (with I.V.Goruchkina)// Transactions of Moscow Math. Soc. 71 (2010) 1–104.
[11] Complicated and exotic expansions of solutions to the Painleve equations // Formal and Analytic Solutions of Diff. Equations, G. Filipuk etal. (eds.), Springer Proceedings in Mathematics & Statistics 256, 2018, pp. 103-145 https://doi.org/10.100/978-3-319-99148-1_7
[12] Power-exponential transseries as solutions to ODE // Journal of Mathematical Sciences: Advances and Applications 2019, Vol. 59, pp. 33-60 DOI:
http://dx.doi.org/10.18642/jmsaa_7100122093
[13] On the generalized normal form of ODE systems // Qual. Theory Dyn.Syst. 21, 1 (2022). https://doi.org/10.1007/s12346-021-00531-4
[14] Algorithms and software for solving a polynomial equation in one or two variables (with A.B.Batkhin) // Programming and Computer Software, 2021, Vol. 47:5, 353–373. DOI: 10.1134/S0361768821050042
[15] Resolution of algebraic singularity by algorithms of Power Geometry (with A.B. Batkhin)// Programming and Computer Software 38:2 (2012) 57–72.
[16] Power geometry and expansions of solutions to the Painleve equations // Transnational Journal of Pure and Applied Mathematics, 1:1, (2018), 43–61.
[17] Algorithmic analysis of local integrability (with V.F. Edneral) // Doklady Mathematics 79:1 (2009) 48–52.
[18] On new integrals of the Algaba-Gamero-Garcia system (with V.F. Edneral, and V.G. Romanovski) // Proceedings CASC 201, LNCS 10490, V.P. Gerdt etal (Eds.), Springer, 2017, pp.40–50. DOI: 10.1007/978-3-319-66320-3_4
[19] Sets of stability of multiparameter Hamiltonian systems (with A.B. Batkhin and V.P. Varin) // J. Appl. Math. Mech. 76:1 (2012) 56–92.
[20] On periodic flybys of the moon // Celestial Mechanics, 24:3 (1981), 255–268.
The full list of my publications is at my site http://brunoa.name.