Nonlinear Analysis as a Calculus
Abstract
In the last 60 years, there was formed a universal nonlinear analysis, whose unified algorithms allow to find asymptotic forms and asymptotic expansions of solutions to nonlinear equations and systems of different types: algebraic, ordinary differential (ODE), partial differential (PDE) and systems of mixed-type equations. This calculus contains two main algorithms: (a) Reducing equations to the normal form and (b) Separating truncated equations, and two kinds of transformations of coordinate can be used to simplify the obtained equations: (A) Power and (B) Logarithmic. Here we show that for algebraic equation, single ODE, autonomous system of ODE’s, Hamiltonian system, single PDE. Some applications are mentioned as well.
References
V. I. Anoshin, A. D. Beketova, A. Parusnikova, E. D. Prokopenko (2023) Convergence of formal solutions to the second term of the fourth Painlevé hierarchy in a neighborhood of origin. 63(1), 86-95. https://doi.org/10.1134/S0965542523010049
A. B. Batkhin, A. D. Bruno, V. P. Varin (2012) Stability sets of multiparameter Hamiltonian systems. 76(1), 56-92. https://doi.org/10.1016/j.jappmathmech.2012.03.006
G. D. Birkhoff (1966) Dynamical Systems. 9, 305.
A. D. Bruno (1962) The asymptotic behavior of solutions of nonlinear systems of differential equations. 3, 464-467.
A. D. Bruno (1964) Normal form of differential equations. 5, 1105-1108.
A. D. Bruno (1971) Analytical form of differential equations (I). 25, 131-288.
A. D. Bruno (1972) Analytical form of differential equations (II). 26, 199-239.
A. D. Bruno (1981) On periodic flybys of the Moon. 24(3), 255-268. https://doi.org/10.1007/BF01229557
A. D. Bruno (1989) Local Methods in Nonlinear Differential Equations.
A. D. Bruno (1994) The Restricted 3-body Problem: Plane Periodic Orbits.
A. D. Bruno (2000) Power Geometry in Algebraic and Differential Equations.
A. D. Bruno (2002) Families of periodic solutions to the Beletsky equation. 40(3), 274-295. https://doi.org/10.1023/A:1015981105366
A. D. Bruno (2004) Asymptotics and expansions of solutions to an ordinary differential equation. 59(3), 429-480. https://doi.org/10.1070/RM2004v059n03ABEH000736
A. D. Bruno (2006) Complicated expansions of solutions to an ordinary differential equation. 73(1), 117-120. https://doi.org/10.1134/S1064562406010327
A. D. Bruno (2007) Exotic expansions of solutions to an ordinary differential equation. 76(2), 729-733. https://doi.org/10.1134/S1064562407050237
A. D. Bruno (2012) Exponential expansions of solutions to an ordinary differential equation. 85(2), 259-264. https://doi.org/10.1134/S1064562412020287
A. D. Bruno (2012) Power-exponential expansions of solutions to an ordinary differential equation. 85(3), 336-340. https://doi.org/10.1134/S106456241203009X
A. D. Bruno (2015) Power geometry and elliptic expansions of solutions to the Painlevé equations. 2015, 340715. https://doi.org/10.1155/2015/340715
A. D. Bruno (2018) Algorithms for solving an algebraic equation. 44(6), 533-545. https://doi.org/10.1134/S0361768819100013
A. D. Bruno (2018) Complicated and exotic expansions of solutions to the Painlevé equations. 103-145. https://doi.org/10.1007/978-3-319-99148-1_7
A. D. Bruno (2018) Power geometry and expansions of solutions to the Painlevé equations. 1(1), 43-61.
A. D. Bruno (2019) On the parametrization of an algebraic curve. 106(6), 885-893. https://doi.org/10.1134/S0001434619110233
A. D. Bruno (2019) Power-exponential transseries as solutions to ODE. 59, 33-60. https://doi.org/10.18642/jmsaa_7100122093
A. D. Bruno (2022) Families of periodic solutions and invariant tori of Hamiltonian systems. https://doi.org/10.1142/q0335
A. D. Bruno (2022) On the generalized normal form of ODE systems. 21(1). https://doi.org/10.1007/s12346-021-00531-4
A. D. Bruno, A. A. Azimov (2023) Computation of unimodular matrices of power transformations. 49(1), 32-41. https://doi.org/10.1134/S0361768823010036
A. D. Bruno, A. B. Batkhin (2012) Resolution of an algebraic singularity by power geometry algorithms. 38(2), 57-72. https://doi.org/10.1134/S036176881202003X
Bruno A. D., Aleksey B. Batkhin (2021) Survey of Eight Modern Methods of Hamiltonian Mechanics. 10(4). https://doi.org/10.3390/axioms10040293
Bruno A. D., Aleksey B. Batkhin (2023) Asymptotic forms of solutions to system of nonlinear partial differential equations. 9(1), 35. https://doi.org/10.3390/universe9010035
Bruno A. D., Vladimir F. Enderal (2009) Algorithmic analysis of local integrability. 79(1), 48-52. https://doi.org/10.1134/S1064562409010141
Bruno A. D., Vladimir F. Enderal, Valery G. Romanovski (2017) On new integrals of the Algaba-Gamero-Garcia system. 10490. https://doi.org/10.1007/978-3-642-32973-9
Bruno A. D., Irina V. Goruchkina (2010) Asymptotic expansions of solutions of the sixth Painlevé equation. 71, 1-104. https://doi.org/10.1090/S0077-1554-2010-00186-0
Bruno D., Nikolai Kudryashov (2009) Expansions of solutions to the equation P12 by algorithms of power geometry. 6(3), 311-337.
Bruno D., Tatiana Shadrina (2007) Axisymmetric boundary layer on a needle. 68, 201-259. https://doi.org/10.1090/S0077-1554-07-00165-3
Bruno D., Vladimir Varin (2004) Classes of families of generalized periodic solutions to the Beletsky equation. 88(4), 325-341. https://doi.org/10.1023/B:CELE.0000023390.25801.f9
Bruno D., Vladimir Varin (2007) Periodic solutions of the restricted three-body problem for small mass ratio. 71(6), 933-960. https://doi.org/10.1016/j.jappmathmech.2007.12.012
Bruno D. (2012) Space Power Geometry for one ODE and P1%E2%80%93P4, P6. 41-52. https://doi.org/10.1515/9783110275667.41
Bruno D. A. (2012) Regular Asymptotic Expansions of Solutions to One ODE and P1%E2%80%93P5. 67%E2%80%9382. https://doi.org/10.1515/9783110275667.67
Bruno D. A., Anna V. Parusnikova (2012) Elliptic and Periodic Asymptotic Forms of Solutions to P5. 53%E2%80%9366. https://doi.org/10.1515/9783110275667.53
H. Dulac (1912) Solutions d'un syst%C3%A8me d'%C3%A9quations diff%C3%A9rentielles dans le voisinage des valeurs singuli%C3%A8res. 40, 324-383.
R. R. Gontsov, I. V. Goryuchkina (2019) Convergence of formal Dulac series satisfying an algebraic ordinary differential equation. 210(9), 1207%E2%80%931221. https://doi.org/10.1070/SM9064
J. Hadamard (1893) Etude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann. 9, 171-216.
M. Hénon (1997) Generating Families in the Restricted Three-Body Problem. 278.
I. Newton (1964) A treatise of the method of fluxions and infinite series, with its application to the geometry of curve lines. 1, 27-137.
O. A. Oleinik, V. N. Samokhin (1999) Mathematical Models in Boundary Layer Theory. 528. https://doi.org/10.1201/9780203749364
H. Poincaré (1928) Sur les propriétés des fonctions définies par les équations aux différance partielles. I, XLIX-CXXIX.
A. D. Polyanin, A. I. Zhurov (2021) Separation of Variables and Exact Solutions to Nonlinear PDEs. 401.
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