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}{% \vskip8pt \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}} \vskip16pt } \usepackage[absolute,overlay]{textpos} \makeatother \usepackage{lineno} \linenumbers \begin{document} \affil[1]{ } \renewcommand\Authands{ and } \date{\small \em Received: 1 January 1970 Accepted: 1 January 1970 Published: 1 January 1970} \maketitle \begin{abstract} \end{abstract} \keywords{} \begin{textblock*}{18cm}(1cm,1cm) % {block width} (coords) \textcolor{GJBlue}{\LARGE Global Journals \LaTeX\ JournalKaleidoscope\texttrademark} \end{textblock*} \begin{textblock*}{18cm}(1.4cm,1.5cm) % {block width} (coords) \textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}} \end{textblock*} \begin{textblock*}{10cm}(1.05cm,3cm) {{\textit{CrossRef DOI of original article:}} \underline{}} \end{textblock*}\let\tabcellsep& \par There are two universal methods for local study of nonlinear equations and systems of different kinds (algebraic, ordinary and partial differential): (a) normal form and (b) truncated equations.\par (a) Equations with linear parts can be reduced to their normal forms by local changes of coordinates. For algebraic equation, it is Implicit Function Theorem. For systems of ordinary differential equations (ODE), I completed the theory of normal forms, began by {\ref Poincaré (1879)} \hyperref[b55]{[Poincaré, 1928]} and {\ref Dulac (1912)} {\ref [Dulac, 1912]} for general systems {\ref [Bruno, 1964;} {\ref 1971]} and began by {\ref Birkhoff (1929)} \hyperref[b0]{[Birkhoff, 1966]} for Hamiltonian systems {\ref [Bruno, 1972;}\hyperref[b7]{1994]}.\par (b) Equations without linear part: I proposed to study properties of solutions to equations (algebraic, ordinary differential and partial differential) by studying sets of vector power exponents of terms of these equations. Namely, to select more simple ("truncated") equations {\ref [Bruno, 1962;}\hyperref[b6]{1989;}\hyperref[b8]{2000]} by means of generalization to polyhedrons the {\ref Newton (1678)} \hyperref[b53]{[Newton, 1964]} and the {\ref Hadamard (1893)} {\ref [Hadamard, 1893]} polygons.\par By means of power transformations {\ref [Bruno, 1962;}\hyperref[b6]{1989;} {\ref 2022b]} the normal forms and the truncated equations can be strongly simplified and often solved. Solutions to the truncated equations are asymptotically the first approximations of the solutions to the full equations. Continuing that process, we can obtain then {\ref (2.1)} \section[{II. SINGLE ALGEBRAIC EQUATION}]{II. SINGLE ALGEBRAIC EQUATION} \section[{The implicit function theorem:}]{The implicit function theorem:}\par London Journal of Research in Science: Natural and Formal Theorem 2.1. Let f (X, ?, T ) = ?a Q,r (T )X Q ? r , where 0 ? Q ? Z n , 0 ? r ? Z, the sum is finite and a Q,r (T ) are some functions of T = (t 1 , . . . , t m ), besides a 00 (T ) ? 0, a 01 (T ) ? ? 0. Then the solution to the equation f (X, ?, T ) = 0 has the form? = ?b R (T )X R def = b(T, X),\par where 0 ? R ? Z n , 0 < ?R?, the coefficients b R (T ) are functions on T that are polynomials from a Q,r (T ) with ?Q? + r ? ?R? divided by a\par 2?R?-1 01\par . The expansion b(T, X) is unique. Let g(X, ?, T ) = f (X, ? + b(T, X), T ), {\ref (2.2)} then g(X, 0, T ) ? 0.\par This is a generalization of Theorem 1.1 of {\ref [Bruno, 2000, Ch}. II] on the implicit function and simultaneously a theorem on reducing the algebraic equation {\ref (2.1)} to its normal form {\ref (2.2)} when the linear part a 01 (T ) ? ? 0 is nondegenerate. In it, we must exclude the values of T near the zeros of the function a 01 (T ).\par Let the point X 0 = 0 be singular. Write the polynomial in the formf (X) = ?a Q X Q , where a Q = const ? R, or C. Let S(f ) = \{Q : a Q ? = 0\}.\par The set S is called the support of the polynomial f (X). Let it consist of points Q 1 , . . . , Q k . The convex hull of the support S(f ) is the set {\ref (2.3)} which is called Newton's polyhedron.\par Its boundary ?Î?"(f ) consists of generalized faces Î?" (d) j , where d is its dimension of 0 ? d ? n -1 and j is the number. Each (generalized) face Î?" ( d ) j corresponds to its:\par ? boundary subset S (d)j = S ? Î?" (d) j , ? truncated polynomial f (d) j (X) = ?a Q X Q over Q ? S (d) j ,\par ? and normal coneU (d) j = P : ?P, Q ? ? = ?P, Q ?? ? > ?P, Q ??? ?, Q ? , Q ?? ? S (d) j , Q ??? ? S\textbackslash S (d) j\par , {\ref (2.4)} where P = (p 1 , . . . , p n ) ? R n Let X = (x 1 , . . . , x n ) ? R n or C n , and f (X) be a polynomial. A point X = X 0 , f (X 0 ) = 0 is called simple if the vector (?f /?x 1 , . . . , ?f /?x n ) in it is non-zero. Otherwise, the point X = X 0 is called singular or critical. By shifting X = X 0 + Y we move the point X 0 to the origin Y = 0. If at this point the derivative ?f /?x n ? = 0, then near X 0 all solutions to the equation f (X) = 0 have the form y n = ?b q 1 ,...q n-1 y q1 1 ? ? ? y q n-1 n-1 , that is, lie in (n -1)-dimensional space. Î?"(f ) = Q = k j=1 µ j Q j , µ j ? 0, k j=1 µ j = 1 ,\par Nonlinear Analysis as a Calculus\par At X ? 0 solutions to the full equation f (X) = 0 tend to non-trivial solutions of those truncated equations f (d) j (X) = 0 whose normal cone U (d) j intersects with the negative orthant P ? 0 in R n * .\par Remark 1. If in the sum {\ref (2.1)} all Q belong to a forward cone C:?Q, K i ? > c i , i = 1, . . . , m,\par then in the solution (2.2) of Theorem 2.1 all R belong to the same cone C: {\ref [Bruno, 1989, Part I, Chapter 1, § 3]}.?Q, K i ? > c i , i = 1, . . . , m,\par Let ln X def =(ln x 1 , . . . , ln x n ). The linear transformation of the logarithms of the coordinates(ln y 1 , . . . , ln y n ) def = ln Y = (ln X)?,\par (2.5) {\ref [Bruno, 1962;}\hyperref[b8]{2000:} where ? is a nondegenerate square n-matrix, is called power transformation. \section[{Power transformations}]{Power transformations}\par By the power transformation (2.5), the monomial X Q tranforms into the monomial Y R , where R = Q (? * ) -1 and the asterisk indicates a transposition.\par A matrix ? is called unimodular if all its elements are integers and det ? = ±1. For an unimodular matrix ?, its inverse ? -1 and transpose ? * are also unimodular.\par Theorem 2.2. For the face Î?" (d) j there exists a power transformation (2.5) with the unimodular matrix ? which reduces the truncated sumf (d) j (X) to the sum from d coordinates, that is, f (d) j (X) = Y S ?(d) j (Y ), where ?(d) j (Y ) = ?(d) j (y 1 , . . . , y d\par ) is a polynomial. Here S ? Z n . The additional coordinates y d+1 , . . . , y n are local (small).\par The article {\ref [Bruno, Azimov, 2023]} specifies an algorithm for computing the unimodular matrix ? of Theorem 2.2. \section[{Let Î?" (d) j}]{Let Î?" (d) j}\par be a face of the Newton polyhedron Î?"(f ). Let the full equation f (X) = 0 is changed into the equation g(Y ) = 0 after the power transformation of Theorem 2.2. Thus ?(d) j (y 1 , . . . , y d ) = g(y 1 , . . . , y d , 0, . . . , 0). \section[{Parametric expansion of solutions:}]{Parametric expansion of solutions:}\par London Journal of Research in Science: Natural and Formal\par Let the polynomial ?j be the product of several irreducible polynomials?(d) j = m k=1 h l k k (y 1 , . . . , y d ),\textbf{(2.6)}\par where 0 < l k ? Z. Let the polynomial h k be one of them. Three cases are possible:\par Case 1. The equation h k = 0 has a polynomial solution y d = ?(y 1 , . . . , y d-1 ). Then in the full polynomial g(Y ) let us substitute the coordinatesy d = ? + z d ,\par for the resulting polynomial h(y 1 , . . . , y d-1 , z d , y d+1 . . . , y n ) again construct the Newton polyhedron, separate the truncated polynomials, etc. Such calculations were made in \hyperref[b31]{[Bruno, Batkhin, 2012]} and were shown in {\ref [Bruno, 2000, Introduction]}.\par Case 2. The equation h k = 0 has no polynomial solution, but has a parametrization of solutionsy j = ? j (T ), j = 1, . . . , d, T = (t 1 , . . . , t d-1 ).\par Then in the full polynomial g ( Y ) we substitute the coordinatesy j = ? i (T ) + ? j ?, j = 1, . . . , d,\textbf{(2.7)}\par where ? j = const, ? |? j | ? = 0, and from the full polynomial g(Y ) we get the polynomialh = ?a Q ?? ,r (T )Y ?? Q ?? ? r ,\textbf{(2.8)}\par whereY ?? = (y d+1 , . . . , y n ), 0 ? Q ?? = (q d+1 , . . . , q n ) ? Z n-d , 0 ? r ? Z. Thus a 00 (T ) ? 0, a 01 (T ) = d j=1 ? j ?? (d) j /?y j (T ).\par If in the expansion (5.7) l k = 1, then a 01 ? ? 0. By Theorem 2.1, all solutions to the equation h = 0 have the form i.e., according to (2.7) the solutions to the equation g = 0 have the form? = ?b Q ?? (T )Y ?? Q ?? , London Journal ofy j = ? j (T ) + ? j ?b Q ?? (T )Y ?? Q ?? , j = 1, . . . , d.\par Such calculations were proposed in \hyperref[b20]{[Bruno, 2018a]}.\par If in (5.7) l k > 1, then in (2.8) a 01 (T ) ? 0 and for the polynomial (2.8) from Y ?? , ? we construct a Newton polyhedron by support S(h) = \{Q ?? , r : a Q ?? ,r (T ) ? ? 0\}, separate the truncations and so on.\par Case 3. The equation h k = 0 has neither a polynomial solution nor a parametric one. Then, using Hadamard's polyhedron \hyperref[b20]{[Bruno, 2018a;} {\ref 2019a]}, one can compute a piece-wise approximate parametric solution to the equation h k = 0 and look for an approximate parametric expansion.\par Similarly, one can study the position of an algebraic manifold in infinity.\par Here we consider an ordinary differential equation of the formf x, y, y ? , . . . , y (n) = 0,\textbf{(3.1)}\par where x is independent variable, y is the dependent variable, y ? = dy/dx and f is a polynomial of its arguments. Near x 0 = 0 or ? we look for solutions of equation {\ref (3.1)} in the form of asymptotic seriesy = ? k=1 b k x s k ,\textbf{(3.2)}\par III. SINGLE ODE \hyperref[b43]{[BRUNO, 2004]} 3.1. Setting of the problem:\par where b k are functions of log x and ?s k > ?s k+1 with? = -1, if x 0 = 0, 1, if x 0 = ?. (3.3)\par We set X = (x, y). By a differential monomial a(x, y) we mean the product of an ordinary monomial To every differential monomial a (X) one assigns its (vector) exponent Q(a) = (q 1 , q 2 ) ? R 2 by the following rules. For a monomial of the form {\ref (3.4)} let {\ref r 1 ,} {\ref r 2 )}; for a derivative of the form (3.5) let Q d l y/dx l = (-l, 1).cx r 1 y r 2 def = cX R ,\textbf{(3.4}Q cX R = R, that is, Q (cx r 1 y r 2 ) = (\par When differential monomials are multiplied, their exponents are summed as vectors:Q (a 1 a 2 ) = Q (a 1 ) + Q (a 2 ) .\par The set S(f ) of exponents Q (a i ) of all the differential monomials a 2 (X) in a differential sum of the form {\ref (3.6)} is called the support of the sum f (X). Obviously, S(f ) ? R 2 . The closure Î?"(f ) of the convex hull of the support S(f ) is referred to as the polygon of the sum f (X). The boundary ?Î?"(f ) of the polygon Î?"(f ) consists of vertices Î?" f (d) j (X) = a i (X) over Q (a i ) ? S (d) j . (3.7)\par Let R 2 * be the plane conjugate to the plane R 2 so that the inner (scalar) product?P, Q? def = p 1 q 1 + p 2 q 2\par is defined for anyP = (p 1 , p 2 ) ? R 2 * and Q = (q 1 , q 2 ) ? R 2 . Corresponding to any face Î?" (d) j are its normal cone, U (d) j = P : ?P, Q? = ?P, Q ? ? , Q, Q ? ? S (d) j ?P, Q? > ?P, Q ?? ? , Q ?? ? S(f )\textbackslash S (d) j\par and the truncated sum (3.7). All these constructions are applicable to equation {\ref (3.1)}, where f is a differential sum.\par Let x ? 0 or x ? ? and suppose that a solution of the equation (3.1) has the formy = c r x r + o |x| r+? ,\textbf{(3.8)}\par where c r is a coefficient, c r = const ? C, c r ? = 0, the exponents r and ? are in R, and ?? < 0. Then we say that the expressiony = c r x r , c r ? = 0 (3.9)\par gives the power-law asymptotic form of the solution (3.8).\par Thus, corresponding to any faceÎ?" ( d ) j are the normal cone U ( d ) j in R 2\par * and the truncated equation f (d) j (X) = 0. (\textbf{3}= 0. We set g(X) def = X -Q f (0) j (X).\par Then the solution (3.7), (3.10) satisfies the equation \section[{Solution of the truncated equation:}]{Solution of the truncated equation:}\par London Journal of Research in Science: Natural and Formal g(X) = 0\par Substituting y = cx r into g(X), we see that g (x, cx r ) does not depend on x, c and is a polynomial in r, that is,g (x, cx r ) ? ?(r),\par where ?(r) is the characteristic polynomial of the differential sum f (0) j (X). Hence, in a solution (3.9) of the equation (3.10) the exponent r is a root of the characteristic equation {\ref (3.11)} and the coefficient c r is arbitrary. Among the roots r i of the equation {\ref (3.11)}, one must single out only those for which one of the vectors ? {\ref (1, r)}, where ? = ±1, belongs to the normal cone U (0)?(r) def = g (x, x r ) = 0,j of the vertex Î?" (0) j .\par In this case the value of ? uniquely determined. The corresponding expressions of the sum with an arbitrary constant c r are candidates for the role of truncated solutions of the equation {\ref (3.1)}. Moreover, by {\ref (3.3)}, if ? = -1, then x ? 0, and if ? = 1, then x ? ?.\par Complex roots r to characteristic equation {\ref (3.11}) may bring to exotic expansions of solutions {\ref (3.2)}, where coefficients b k are power series in x ?i with real ? ? R and i 2 = -1. \section[{Corresponding to an edge Î?"}]{Corresponding to an edge Î?"}\par (1)j is a truncated equation (3.10) with d = 1 whose normal cone U (1) j is a ray \{?N j , ? > 0\}. If ?(1, r) ? U (1)\par j , this condition uniquely determines the exponent r of the truncated solution (3.9) and the value ? = ±1 in {\ref (3.3)}. To find the coefficient c r , one must substitute the expression (3.9) into the truncated equation (3.10). After cancelling a factor which is a power of x, we obtain an algebraic defining equation for the coefficient c r ,f (c r ) def = x -s f (1) j (x, c r x r ) = 0 Corresponding to every root c r = c (i)\par r ? = 0 of this equation is an expression of the form (3.9) which is a candidate for the role of a truncated solution of the equation {\ref (3.1)}. Moreover, by {\ref (3.3)}, if in the normal cone U\par (1) j one has p 1 < 0, then x ? 0, and if p 1 > 0, then x ? ?. From the polygon Î?" of the initial equation (3.1) we take a vertex or an edge Î?" (d) j . Then we found a power solution y = b 1 x P 1 of the truncated equation f (d) j (X) = 0, as it was described above, puty = b 1 x P 1 + z and obtain new equation g(x, z) = 0.\par We construct the polygon 1 Î?" for the new equation, take a vertex or an edge 1 Î?" (e) k , solve the truncated equation?(e) k (x, z) = 0,\par and obtain the second term b 2 x P 2 of expansion (3.2) and so on. \section[{Computation of solution to equation (3.1) as expansion (3.2)}]{Computation of solution to equation (3.1) as expansion (3.2)}\par London Journal of Research in Science: Natural and Formal\par We construct the polygon 1 Î?" for the new equation, take a vertex or an edge 1 Î?" (e)\par k , solve the truncated equation?(e) k (x, z) = 0,\par and obtain the second term b 2 x P 2 of expansion (3.2) and so on.\par In \hyperref[b43]{[Bruno, 2004]} there are some properties, that simplify computation. Thus, we can obtain the 4 types of expansions (3. {\ref [Bruno, 2006;} {\ref 2018b]}; 4. Exotic, when all b k are power series in x i? \hyperref[b45]{[Bruno, 2007]}.\par Except expansions (3.2) of solutions y(x) of equation {\ref (3.1)}, there are exponential expansionsy = ? k=1 b k (x) exp [k?(x)] ,\par where b k (x) and ?(x) are power series in x {\ref [Bruno, 2012a,b]}.\par Also there are solutions in the form of transseries \hyperref[b26]{[Bruno, 2019b]}. These results were applied to 6 Painlevè equations \hyperref[b19]{[Bruno, 2015;} {\ref 2018b,c;} {\ref Bruno, Goruchkina, 2010]}. Written as differential sums they are: Equation P 5 :Equation P 1 : f (x, y) def = -y ?? + 3y 2 + x = 0. Equation P 2 : f (x,f (z, w) def = -z 2 w(w -1)w ?? + z 2 3 2 w - 1 2 (w ? ) 2 -zw(w -1)w ? + + (w -1) 3 (?w 2 + ?) + ?zw 2 (w -1) + ?z 2 w 2 (w + 1) = 0. Equation P 6 : f (x, y) def = 2y ?? x 2 (x -1) 2 y(y -1)(y -x) -(y ? ) 2 [ x 2 (x -1) 2 (y -1)(y -x)+ + x 2 (x -1) 2 y(y -x) + x 2 (x -1) 2 y(y -1) ]+ + 2y ? [ x(x -1) 2 y(y -1)(y -x) + x 2 (x -1)y(y -1)(y -x)+ + x 2 (x -1) 2 y(y -1) ] -[ 2?y 2 (y -1) 2 (y -x) 2 + 2?x(y -1) 2 (y -x) 2 + + 2?(x -1)y 2 (y -x) 2 + 2?x(x -1)y 2 (y -1) 2 ] = 0.\par Here a, b, c, d and ?, ?, ?, ? are complex parameters. If all they are nonzero, then polygons for these equations are shown in Figures \hyperref[fig_20]{1,}\hyperref[fig_8]{2,}\hyperref[fig_10]{3}. \section[{Supports and polygons for equations}]{Supports and polygons for equations}q 2 q 1 0 1 1 P 1 q 2 q 1 0 1 1 P 2\par Nonlinear Analysis as a Calculus\par Then there exists such power series ?(x) with integral increasing exponents, that after substitutiony = z + ?(x) (3.13) the transformed differential sum g(x, z) = f (x, z + ?(x)) (3.14) for z = z ? = ? ? ? = z (n) = 0 (3.15)\par has only resonant terms b m x m , wherem = v + ? k ? Z (3.16)\par and m ? ?.\par So here the eigenvalue? k is resonant if ? -v ? ? k ? Z.\par 3) truncated differential sum f (0) 1 (X) have eigenvalues ? 1 , . . . , ? l , 0 ? l ? n; 4) the most left point of the support S(f ) in the axis q 2 = 0 be (?, 0). Evidently ? ? Z.\par Supports and polygons for equations P 3 (left), P 4 (right). -1 0 1 q 1 q 2 P 3 q 2 q 1 0 1 1 P 4\textbf{2}\par Nonlinear Analysis as a Calculus\par Theorem 3.3. Let 1) f x, y, y ? , . . . , y (n) be a polynomial in x, y, y ? , . . . , y (n) ; 2) its Newton polygon Î?"(f ) have a vertex Î?" (0) 1 = (v, 1) at the right side of its boundary ?Î?";\par 3) truncated differential sum f (0) j (X) have eigenvalues ? 1 , . . . , ? l , 0 ? l ? n; 4) the most right point of the support S(f ) in the axis q 2 = 0 be (?, 0). Evidently ? ? Z.\par Then there exists such power series ?(x) with integral decreasing exponents, that after substitution (3.13), the differential sum (3.14) for identities {\ref (3.15)} has only resonant terms b m x m , where equality {\ref (3.16}) is true, and m ? ?. f (0) j (X) has no integral eigenvalue ? k ? ?-v (for Theorem 3.2) or ? k ? ? -v (for Theorem 3.3), then the initial equation f (X) = 0 has formal solution y = ?(x). If the truncated sum f (0) j (X) contains the derivation y (n) , then the series ?(x) converges according to Theorem 3.4 in \hyperref[b43]{[Bruno, 2004]}. \section[{So here the eigenvalue ?}]{So here the eigenvalue ?}k is resonant if ? -v ? ? k ? Z. Equations g(x, z) = 0 for (3.\par Remark 2. If the truncated sum f (0) j (X) has integral eigenvalue ? k ? ?v (for Theorem 3.2) or ? k ? ? -v (for Theorem 3.3), then the initial equation f (X) = 0 Supports and polygons for equations P 5 (left), P 6 (right). q 2 q 1 0 1 1 P 5 q 2 q 1 0 1 1 P 6\par Nonlinear Analysis as a Calculus\par We will consider such a generalization of the power function cx r which preserves their main properties. The real numberp ? (?(x)) = ? lim x ? ?? log |?(x)| ? log |x| ,\par where arg x = const ? [0, 2?), is called the order of the function ?(x) on the ray when x ? 0 or x ? ?. The order p ? (?) is not defined on the ray arg x = const, where the limit point x = 0 or x = ? is a point of accumulation of poles of the function ?(x).\par In Subsections 3.2-3.4 it was shown that as x ? 0 (? = -1) or as x ? ? (? = 1) solutions y = ?(x) to the ODE f (x, y) = 0, where f (x, y) is a differential sum, can be found by means of algorithms of Plane PG, ifp ? (?(x)) -l = p ? d l ?/dx l , l = 1, . . . , n,\par where n is the maximal order of derivatives in f (x, y). Here we introduce algorithms, which allow calculate solutions y = ?(x) with the propertyp ? (?(x)) + l? ? = p ? d l ?/dx l , l = 1, . . . , n, where ? ? ? R, ? = ±1. Lemma 3.3.1. If p ? (?(x)) = -? ? + p ? (? ? (x)) = -2? ? + p ? (? ?? (x)), then ? + ?? ? ? 0.\par Note, that in Plane PG we had ? ? = -1, i. e. ? +?? ? = 0. So, new interesting possibilities correspond to ? + ?? ? > 0.\par We consider the ODEf (x, y) = i a i (x, y) = 0,\par where f (x, y) is a differential sum. To each differential monomial a i (x, y), we assign its (vector) power exponent Q(a i ) = (q 1 , q 2 , q 3 ) ? R 3 by the following rules: power exponent of the product of differential monomials is the sum of power exponents of factors:Q(a 1 a 2 ) = Q(a 1 ) + Q(a 2 ).\par The set S(f ) of power exponents Q(a i ) of all differential monomials a i (x, y) presented in the differential sum f (x, y) is called the space support of the sum f (x, y). Obviously, S(f ) ? R 3 . The convex hull Î?"(f ) of the support S(f ) is called the polyhedron of the sum f (x, y). The boundary ?Î?"(f ) of the polyhedron Î?"(f ) consists of the vertices Î?" (0) j , the edges Î?" f (d) j (x, y) = a i (x, y) over Q(a i ) ? Î?" (d) j ? S(f ).\par Support and polyhedron for equation P 1 . The approach allows to obtain solutions with expansions {\ref (3.2)}, where coefficients b k (x) are all periodic or all elliptic functions {\ref [Bruno, 2012c,d;} {\ref Bruno, Parusnikova, 2012]}.\par Expansions of solutions to more complicated equations such as hierarchies Painlevé see in {\ref [Anoshin, Beketova, (et al.)}, 2023; Bruno, For P 1 -P 5 with all parameters nonzero, their polyhedrons are shown in {\ref Figures 4,} {\ref 5,} {\ref 6,} {\ref 7,} {\ref 8} correspondingly.\par Here we consider the system ?i = f i (X), i = 1, . . . , n, {\ref (4.1)} where?= d/d t, X = (x 1 , . . . , x n ) ? C n or R n , all f i (X) are polynomials from X. A point X = X 0 = const is called singular if all f i (X 0 ) = 0, i = 1, . . . , n.\par Let the point X 0 = 0 be a singular point. Then the system (4.1) has the linear part? = XA,\par where A is a square n-matrix. Let ? = (? 1 , . . . , ? n ) be a vector of its eigenvalues.\par Theorem {\ref 4.1 ([Bruno, 1964;} {\ref 1971} {\ref , 1972]}). There exists an invertible formal change of coordinatesx i = ? i (Y ), i = 1, . . . , n,\par where ? i (Y ) are power series from Y = (y 1 , . . . , y n ) without free terms, which reduces the system (4.1) to normal form?i = y i g i (Y ) = y i Y Q , i = 1, . . . , n,\textbf{(4.2)}\par IV. AUTONOMOUS ODE SYSTEM \section[{Normal form:}]{Normal form:}\par Support and polyhedron for equation P 2 . Here y i g i (Y ) are power series on Y without free terms.Let N i = \{Q ? Z n : q j ? 0, j ? = i, q i ? -1\} , i = 1, . . . , n,\par andN = N 1 ? N 2 ? ? ? ? ? N n .\par Then the number k of linearly independent Q ? N satisfying the equation ( \hyperref[formula_59]{4}.3) is called multiplicity of resonance.\par Theorem 4.2. Let k be the multiplicity of resonance of the system (4.1). Then there exists a power transformationln Z = (ln Y ) ?\par with unimodular matrix ? which reduces the normal form (4.2), ( \hyperref[formula_59]{4}.3) to the system(ln z i ) = h i (y 1 , . . . , y k ), i = 1, . . . , n,\par in which the first k coordinates form a closed subsystem without a linear part, and the remaining nk coordinates are expressed via them by means of integrals.\par Thus, if ? ? = 0, then the original system (4.1) of order n can be reduced to a system of order k, but without the linear part. Support and polyhedron for equation P 3 . \section[{Figure 6:}]{Figure 6:}\par Let's write the system (4.1) as {\ref (4.4)} and put A Q = (a 1Q , . . . , a nQ ).(ln x i ) = a iQ X Q , i = 1, . . . , n,\par The setS = \{Q : A Q ? = 0\}\par is called the support of the system {\ref (4.4)}. Its convex hull Î?" (2.3) is its Newton's polyhedron. Its boundary ?Î?" consists of generalized faces Î?" ? boundary subset S (d) j = Î?" (d) j ? S, ? truncated system (ln X) = Â(d) j (X) = A Q X Q over Q ? S (d) j , (4.5) ? normal cone U ( d ) j ? R n * (2.4) and ? tangent cone T (d) j .\par According to {\ref [Bruno, 2000, Chapt. 1, §2]} let d > 0 and Q be the interior point of a face Î?" (d) j , that is, Q does not lie in a face of smaller dimension. If d = 0, then 4.2: Newton's polyhedron {\ref [Bruno, 1962;}\hyperref[b8]{2000]}. Support and polyhedron for equation P 4 . Q = Î?" (0) j . The conic hull of the set S -Q T (d) j = Q = µ 1 Q 1 -Q + ? ? ? + µ k Q k -Q , µ 1 , . . . , µ k ? 0, Q 1 , . . . , Q k ? S is called the tangent cone of the face Î?" (d) j , 0 ? d ? n -1, T (d) j ? R n .d ? = X R d t,\par R ? Z n , which reduce the system (4.4) to the formd (ln Y ) /d ? = B(Y ),\textbf{(4.6)}\par where the systemd (ln Y ) /d ? = B(d) j (Y ) ? B(d) j (y 1 , . . . , y d ) = B(y 1 , . . . , y d , 0, . . . , 0),\textbf{(4.7)}\par corresponds to the truncated system (4.5). be singular for the truncated system (4.7). Near the point (4.8), the local coordinates arez i =y i -y 0 i , i = 1, . . . , d, z j =y j , j = d + 1, . . . , n.\par Let at the point Z = (z 1 , . . . , z n ) = 0 the eigenvalues of the matrix of the linear part of the system (4.7) are ? = ?1 , . . . , ?n , where ?1 , . . . , ?d are the eigenvalues of the subsystem of the first d equations.\par Theorem 4.4. There exists an invertible formal change of coordinates {\ref [Bruno, 2022b]}: where W = (w 1 , . . . , w n ) which reduces the system (4.6) to the generalized normal formz i = ? i (W ), i = 1, . . . , n, \section[{Generalized normal form}]{Generalized normal form}?i = w i c i (W ) = w i c iQ W Q , i = 1, . . . , n,\textbf{(4.9)}\par whereQ, ? = 0 and Q ? T (d) j ? Z n .\par (4.10)Here ? i = w i ? iQ W Q , i = 1, . . . , n, where Q ? T (d) j ? Z n .\par The system (4.9), (4.10) is reduced to a system of lower order by the power transformation of Theorem 4.2 .\par Let X = X 0 be a singular point of the sys tem {\ref (4.1)}. Two cases are possible: Case 1. ? ? = 0, then by Theorem 4.1 we reduce the system to a normal form, then by Theorem 4.2 we reduce the normal form to a subsystem of order k < n without linear part and obtain the problem of studying its singular points.\par Case 2. ? = 0, then we compute the Newton polyhedron and separate truncated systems in which the normal cone U (d) j intersects the negative orthant of P ? 0. Each of them is reduced to the form (4.6), (4.7) by the transformation of Theorem 4.3. For each singular point (4.8), we apply Theorem 4.4 and obtain a subsystem of smaller order.\par Continuing this branching process, after a finite number of resolution of singularities we come to an explicitly solvable system from which we can understand the nature of solutions of the original system. But Theorem 4.3 can be applied to the original system (4.1), i.e. to each of the generalized faces Î?" (d) j of its Newton polyhedron Î?". Then to each singular point (4.8) we apply Theorems 4.4, 4.2 and reduce the order of the system. Here also through a finite number of steps of the singularity resolution we come to an explicitly solvable system. This allows us to study the singularities of the original system in infinity. This is the basis of the integrability criterion in \hyperref[b37]{[Bruno, Enderal, 2009;}\hyperref[b39]{Bruno, Enderal, Romanovski, 2017]}.\par The normal form can be computed in the neighborhood of a periodic solution or invariant torus {\ref [Bruno, 1972, II, §11]}, {\ref [Bruno, 2022a]}.\par See {\ref [Bruno, Batkhin, 2023]} for similar computations for a system of partial differential equations. \section[{Analysis of singularities: tem}]{Analysis of singularities: tem}\par and is defined by one Hamiltonian function H(x, y), where x = (x 1 , . . . , x n ), y = (y 1 , . . . , y m ). Here the normal form of the system (4.11) corresponds to the normal form of one Hamiltonian function. See details in \hyperref[b33]{[Bruno, Batkhin, 2021]}.\par Let X = ( x 1 , . . . , x n ) ? C n or R n independent variables and y ? C or R be a dependent one. Consider Z = (z 1 , . . . , z n , z n+1 ) = (x 1 , . . . , x n , y).\par Differential monomial a(Z) is called a product of an ordinary monomialcZ R = cz r 1 1 ? ? ? z r n+1 n+1 ,\par where c = const, and a finite number of derivatives of the following form? l y ?x l 1 1 ? ? ? ? l n x n def = ? l y ?X L , 0 ? l j ? Z, n j=1 l j = l, L = (l 1 , . . . , l n ) .\par Vector power exponent Q(a) ? R n+1 corresponds to the differential monomial a(Z), it is constructed according to the following rules:Q(c) = 0, if c ? = 0, Q Z R = R, Q ? l y j /?X L = (-L, 1).\par The product of monomials corresponds to the sum of their vector power exponents:Q(ab) = Q(a) + Q(b).\par Differential sum is the sum of differential monomials V. ONE PARTIAL DIFFERENTIAL EQUATION 5.1. Support \hyperref[b8]{[Bruno, 2000} Ch. be 6-8]:f (Z) = a k (Z). (\textbf{5}\par Let the support S(f ) of the differential sum (5.1) consists of one point E n+1 = (0, . . . , 0, 1). Then the substitution y = cX P , P = (p 1 , . . . , p n ) ? R n\par (5.2) in the differential sum f (Z) gives the monomial c? P (P )X P\par where ? P (P ) is a polynomial of P which coefficients depend on P .\par Monomial (5.2) will be called resonant for f (Z) if for it ? P (P ) = 0.\par Let µ k be the maximal order of the derivative over x k in f (Z), k = 1, . . . , n.If in P = (p 1 , . . . , p n ) p k ? µ k , k = 1, . . . , n, (5.3) then f (Z) = c?(P )X P ,\par where ?(P ) is the characteristic polynomial of the sum of f (Z) and its coefficients do not depend on P . But if the inequalities (5.3) are not satisfied, then ? P (P ) ? = ?(P ).Example. Let n = 2, f (Z) = x 1 ?y ?x 1 + x 2 2 ? 2 y ?x 2 2 . If P = (1, 1), then f (x 1 , x 2 , cx 1 x 2 ) = cx 1 x 2 . If P = (1, 2), then f (x 1 , x 2 , cx 1 x 2 2 ) = c x 1 x 2 2 + x 1 ? x 2 2 ? 2 = c ? 3x 1 x 2 2 . Generally here for p 1 ? 1, p 2 ? 2 we have f (x 1 , x 2 , cx 1 x 2 ) = c[p 1 + p 2 (p 2 - 1)]x p 1 1 x p 2 2 and ?(P ) = p 1 + p 2 (p 2 -1).\par For a differential sum f (Z) we denote by f k (Z) the sum of all differential monomials of f (Z) which have n + 1 coordinate q n+1 of vector power exponents Q = (q 1 , . . . , q n , q n+1 ) equal to k:q n+1 = k. Denote Z n + = \{P : 0 ? P ? Z n \}.\par Consider the PDE f (Z) = 0.\par (5.4) \section[{Normal form:}]{Normal form:}\par 5.2. Resonant monomials:\par 1. f 0 (Z) = ?(X) is a power series from X without a free term, 2. f 1 (Z) = a(Z)+b(Z), where S(a) = E n+1 = (0, . . . , 0, 1), S(b) ? Z n+1 + \textbackslash 0 × \{q n+1 = 1\}.\par Then there exists a substitution y = ? + (X), where (X) is a power series from X without a free term, which transforms the equation (5.4) to the normal form g(X, ?) = 0,\par (5.5)\par where g 0 (X) = c P X P is a power series without a free term, P ? Z n + containing only resonant monomials c P X P for sum a(Z). is the formal solution to the equation (5.4).\par If in equation (5.4) differential sum does not contain derivatives, thena(Z) = const ? z n+1 = const ? y.\par Closure of a convex hull where the space R n+1 * is conjugate to the space R n+1 , ??, ?? is the scalar product, and truncated sumÎ?"(f ) = Q = ? j Q j , Q j ? S, ? j ? 0, ? j = 1 of the support S(f ) is called the polyhedron of sum f (Z). The boundary ?Î?" of the polyhedron Î?"(f ) consists of generalized faces Î?" (d) j , where d = dim Î?" (d) j . Each face Î?" (d) j corresponds to normal cone U (d) j = P ? R n+1 * : ?P, Q ? ? = ?P, Q ? ? > ?P, Q ? ?, where Q, Q ? ? Î?" (d) j , Q ? ? Î?"\textbackslash Î?"f (d) j (Z) = a k (Z)by Q(a k ) ? Î?" (d) j S.\par Consider the equationf (Z) = 0, (5.6)\par where f is the differential sum. In the solution of equation (5.6)y = ?(X), (5.7)\par where ? is a series on the powers of x k and their logarithms, the series ? corresponds to its support, polyhedron, normal cones u i and truncations. The logarithm ln x i has a zero power exponent on x i . The truncated solution y = ? corresponds to the normal coneu ? R n+1 * .\par Theorem 5.2. If the normal cone u intersects with the normal cone (5.2), then the truncation y = ?(X) of the solution (5.3) satisfies the truncated equationf (d) j (Z) = 0.\par (5.8)\par To simplify the truncated equation (5.8), it is convenient to use a power transformation. Let ? be a square real nondegenerate block matrix of dimension n + 1 of the form? = ? 11 ? 12 0 ? 22 ,\par where ? 11 and ? 22 are square matrices of dimensions n and 1, respectively. We denote ln Z = (ln z 1 , . . . , ln z n+1 ), and by the asterisk * we denote transposition.\par Variable change. ln W = (ln Z) ? (5.9) is called the power transformation.\par Theorem 5. {\ref 3 ([Bruno, 2000]}). The power transformation (5.5) reduces a differential monomial a(Z) with a power exponent Q(a) into a differential sum b(W ) with a power exponent Q(b):R = Q(b) = Q(a)? -1 * . \section[{Power transformations:}]{Power transformations:}\par London Journal of Research in Science: Natural and Formal\par Corollary 5.3.1. The power transformation (5.9) reduces the differential sum (2.1) with support S(f ) to the differential sum g(W ) with support S(g) = S(f )? -1 * , i.e. \section[{S(f ) = S(g)? *}]{S(f ) = S(g)? *}\par Theorem 5.4. For the truncated equationf (d) j (Z) = 0\par there is a power transformation (5.9) and monomial Z T that translates the equation above into the equationg(W ) = Z T fj (Z) = 0,\par where g(W ) is a differential sum whose support has n + 1d zero coordinates.\par Let z j be one of the coordinates x k or y. Transformation ? j = ln z j is called logarithmic.\par Theorem 5.5. Let f (Z) be a differential sum such that all its monomials have a jth component q j of the vector exponent of degree Q = (q 1 , . . . , q m+n ) equal to zero, then the logarithmic transformation (5.1) reduces the differential sum f (Z) into a differential sum from z 1 , . . . , ? j , . . . , z n . \section[{Logarithmic transformation:}]{Logarithmic transformation:}\par A truncated equation 5.7. Calculating asymptotic forms of solutions: f (n) j (Z) = 0 is taken. If it cannot be solved, then a power transformation of the Theorem 5.4 and then a logarithmic transformation of the Theorem 5.5 should be performed. Then a simpler equation is obtained. In case it is not solvable again, the above procedure is repeated until we get a solvable equation. Having its solutions, we can return to the original coordinates by doing inverse coordinate transformations. So the solutions written in original coordinates are the asymptotic forms of solutions to the original equation (5.2).\par In {\ref [Bruno, Batkhin, 2023}] method of selecting truncated equations was applied to systems of PDE.\par Traditional approach to PDE see in \hyperref[b54]{[Oleinik, Samokhin, 1999;}\hyperref[b56]{Polyanin, Zhurov, 2021]}.\par Here we provide a list of some applications in complicated problems of (c) Mathematics, (d) Mechanics, (e) Celestial Mechanics and (f) Hydromechanics.\par (c) In Mathematics: together with my students I found all asymptotic expansions of five types of solutions to the six Painlevé equations {\ref (1906)} {\ref [Bruno, 2018c;} {\ref Bruno, Goruchkina, 2010]} and also gave very effective method of determination of integrability of ODE system \hyperref[b37]{[Bruno, Enderal, 2009;}\hyperref[b39]{Bruno, Enderal, Romanovski, 2017]}.\par (d) In Mechanics: I computed with high precision influence of small mutation oscillations on velocity of precession of a gyroscope \hyperref[b6]{[Bruno, 1989]} and also studied values of parameters of a centrifuge, ensuring stability of its rotation \hyperref[b31]{[Batkhin, Bruno, (et al.), 2012]}.\par (e) In Celestial Mechanics: together with my students I studied periodic solutions of the Beletsky equation ( {\ref 1956}) \hyperref[b9]{[Bruno, 2002;}\hyperref[b43]{Bruno, Varin, 2004]}, describing motion of 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 satellite \hyperref[b6]{[Bruno, 1989]}, including cases with big values of the eccentricity of the orbit, inducing a singularity. Besides, simultaneously with \hyperref[b52]{[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 to explain some singularities of motions of small bodies of the Solar System \hyperref[b45]{[Bruno, Varin, 2007]}. In particular, I found orbits of periodic flies round planets with close approach to the Earth {\ref [Bruno, 1981]}.\par (f) In Hydromechanics: I studied small surface waves on a water {\ref [Bruno, 2000, Chapter 5}], a boundary layer on a needle {\ref [Bruno, Shadrina, 2007]}, where equations of a flow have a singularity, and an one-dimensional model of turbulence bursts {\ref [Bruno, Batkhin, 2023]}. \begin{figure}[htbp] \noindent\textbf{6}\includegraphics[]{image-2.png} \caption{\label{fig_1}) 6 ©}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-3.png} \caption{\label{fig_2}j}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-4.png} \caption{\label{fig_3}}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-5.png} \caption{\label{fig_4}}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-6.png} \caption{\label{fig_5}}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-7.png} \caption{\label{fig_6}}\end{figure} \begin{figure}[htbp] \noindent\textbf{1}\includegraphics[]{image-8.png} \caption{\label{fig_7}Figure 1 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{2}\includegraphics[]{image-9.png} \caption{\label{fig_8}Figure 2 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-10.png} \caption{\label{fig_9}}\end{figure} \begin{figure}[htbp] \noindent\textbf{3}\includegraphics[]{image-11.png} \caption{\label{fig_10}Figure 3 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-12.png} \caption{\label{fig_11}}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-13.png} \caption{\label{fig_12}}\end{figure} \begin{figure}[htbp] \noindent\textbf{4}\includegraphics[]{image-14.png} \caption{\label{fig_13}Figure 4 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{5}\includegraphics[]{image-15.png} \caption{\label{fig_14}Figure 5 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-16.png} \caption{\label{fig_15}j}\end{figure} \begin{figure}[htbp] \noindent\textbf{43}\includegraphics[]{image-17.png} \caption{\label{fig_16}Theorem 4 . 3 .}\end{figure} \begin{figure}[htbp] \noindent\textbf{7}\includegraphics[]{image-18.png} \caption{\label{fig_17}Figure 7 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{8}\includegraphics[]{image-19.png} \caption{\label{fig_19}Figure 8 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{1}\includegraphics[]{image-20.png} \caption{\label{fig_20}. 1 )}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-21.png} \caption{\label{fig_21}}\end{figure} \begin{figure}[htbp] \noindent\textbf{4}\includegraphics[]{image-22.png} \caption{\label{fig_22}4 .}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-23.png} \caption{\label{fig_23}London}\end{figure} \label{foot_0}\footnote{\label{foot_0} © 2023 London Journals Press} \label{foot_1}\footnote{\label{foot_1} © 2023 London Journals Press Volume 23 | Issue 5 | Compilation 1.0} \backmatter \begin{bibitemlist}{1} \bibitem[]{b10}\label{b10} \textit{}, \xref{http://dx.doi.org/10.1023/A:1015981105366}{10.1023/A:1015981105366}. p. . \bibitem[]{b13}\label{b13} \textit{}, \xref{http://dx.doi.org/10.1134/S1064562406010327}{10.1134/S1064562406010327}. 117. \bibitem[]{b15}\label{b15} \textit{}, \xref{http://dx.doi.org/10.1134/S1064562407050237}{10.1134/S1064562407050237}. p. . \bibitem[]{b17}\label{b17} \textit{}, \xref{http://dx.doi.org/10.1134/S1064562412020287}{10.1134/S1064562412020287}. p. . \bibitem[]{b21}\label{b21} \textit{}, \xref{http://dx.doi.org/10.1134/S0361768819100013}{10.1134/S0361768819100013}. p. . \bibitem[]{b30}\label{b30} \textit{}, \xref{http://dx.doi.org/10.1134/S0361768823010036}{10.1134/S0361768823010036}. p. . \bibitem[]{b32}\label{b32} \textit{}, \xref{http://dx.doi.org/10.1134/S036176881202003X}{10.1134/S036176881202003X}. p. . \bibitem[]{b34}\label{b34} \textit{}, \xref{http://dx.doi.org/10.3390/axioms10040293}{10.3390/axioms10040293}. 10. \bibitem[]{b36}\label{b36} \textit{}, \xref{http://dx.doi.org/10.3390/universe9010035}{10.3390/universe9010035}. 35. \bibitem[]{b38}\label{b38} \textit{}, \xref{http://dx.doi.org/10.1134/S1064562409010141}{10.1134/S1064562409010141}. 79 p. . \bibitem[Bruno and Shadrina]{b42}\label{b42} \textit{}, A D Bruno , T V Shadrina . \xref{http://dx.doi.org/10.1090/S0077-1554-07-00165-3}{10.1090/S0077-1554-07-00165-3}. Moscow Math. Soc. -2007. 68 p. . \bibitem[]{b44}\label{b44} \textit{}, \xref{http://dx.doi.org/10.1023/B:CELE.0000023390.25801.f9}{10.1023/B:CELE.0000023390.25801.f9}. p. . \bibitem[Birkhoff ()]{b0}\label{b0} \textit{}, G D Birkhoff . \textit{Dynamical Systems} 1966. Providence, Rhode Island: Colloquim Publications. 9. \bibitem[Newton ()]{b53}\label{b53} ‘A treatise of the method of fluxions and infinite series, with its ap-plication to the geometry of curve lines’. I Newton . The Mathematical Works of Isaac Newton H Woolf (ed.) 1964. New York -London: Johnson Reprint Corp. 1 p. . \bibitem[Bruno and Enderal ()]{b37}\label{b37} \textit{Algorithmic analysis of local integrability}, A D Bruno , V F Enderal . 2009. \bibitem[Bruno ()]{b20}\label{b20} \textit{Algorithms for solving an algebraic equation // Programming and Computer Software}, A D Bruno . 2018a. \bibitem[Bruno]{b3}\label{b3} ‘Analytical form of differential equations (I)’. A D Bruno . Trans. Moscow Math. Soc. -1971. \bibitem[Bruno]{b4}\label{b4} ‘Analytical form of differential equations (II)’. A D Bruno . Trans. Moscow Math. Soc. -1972. \bibitem[Bruno and Goruchkina]{b40}\label{b40} ‘Asymptotic expansions of solutions of the sixth Painlevé equation // Transactions of’. A D Bruno , I V Goruchkina . \xref{http://dx.doi.org/10.1090/S0077-1554-2010-00186-0}{10.1090/S0077-1554-2010-00186-0}. Moscow Math. Soc.-2010. 71 p. . \bibitem[Bruno and Batkhin]{b35}\label{b35} \textit{Asymptotic forms of solutions to system of nonlinear partial differential equations // Universe. -2023}, A D Bruno , A B Batkhin . \bibitem[Bruno]{b11}\label{b11} ‘Asymptotics and expansions of solutions to an ordinary differential equation’. A D Bruno . \xref{http://dx.doi.org/10.1070/RM2004v059n03ABEH}{10.1070/RM2004v059n03ABEH}. Russian Mathem. Surveys. -2004. 59 (3) p. . \bibitem[Bruno et al. (eds.) ()]{b46}\label{b46} A D Bruno , Chapter . \xref{http://dx.doi.org/10.1515/9783110275667.41}{10.1515/9783110275667.41}. \textit{Proceedings of the International Conference}, A D Bruno , A B Batkhin , Berlin (eds.) (the International ConferenceSaint Petersburg, Russia; Boston) June 17-23, 2011. 2012. De Gruyter. (Space Power Geometry for one ODE and P1-P4) \bibitem[Bruno and Varin ()]{b43}\label{b43} \textit{Classes of families of generalized periodic solutions to the Beletsky equation // Celestial Mechanics and Dynamical Astronomy}, A D Bruno , V P Varin . 2004. 88. \bibitem[Bruno et al. ()]{b22}\label{b22} \textit{Complicated and exotic expansions of solutions to the Painlevé equations // Formal and Analytic Solutions of Diff}, A D Bruno , G Filipuk , A Lastra , S Michalik , Cham . \xref{http://dx.doi.org/10.1007/978-3-319-99148-1_7}{10.1007/978-3-319-99148-1\textunderscore 7}. 2017. 2018. Springer. 256 p. . \bibitem[Bruno]{b12}\label{b12} ‘Complicated expansions of solutions to an ordinary differential equa-tion’. A D Bruno . Doklady Mathematics. -2006. 73. \bibitem[Bruno and Azimov]{b29}\label{b29} \textit{Computation of unimodular matrices of power trans-formations // Programming and Computer Software.-2023}, A D Bruno , A A Azimov . 49. \bibitem[Bruno et al. ()]{b39}\label{b39} ‘Computer Algebra in Scientific Computing’. A D Bruno , V F Enderal , V G Romanovski . \xref{http://dx.doi.org/10.1007/978-3-642-32973-9}{10.1007/978-3-642-32973-9}. \textit{On new integrals of the Algaba-Gamero-Garcia system}, Lecture Notes in Computer Science V P Gerdt (ed.) (Berlin Heidelberg) 2017. 2017. Springer. 10490. (Proceedings CASC) \bibitem[Gontsov and Goryuchkina]{b50}\label{b50} ‘Convergence of formal Dulac series satisfying an algebraic ordinary differential equation’. R R Gontsov , I V Goryuchkina . \xref{http://dx.doi.org/10.1070/SM9064}{10.1070/SM9064}. Sb. Math.-2019. 210 (9) p. . \bibitem[Bruno et al. ()]{b48}\label{b48} ‘Elliptic and Periodic Asymptotic Forms of Solutions to P5’. A D Bruno , A V Parusnikova , Chapter . \xref{http://dx.doi.org/10.1515/9783110275667.53}{10.1515/9783110275667.53}. \textit{Proceedings of the International Conference}, A D Bruno , A B Batkhin , Berlin (eds.) (the International ConferenceSaint Petersburg, Russia; Boston) June 17-23, 2011. De Gruyter. \bibitem[Hadamard]{b51}\label{b51} ‘Etude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann’. J Hadamard . \textit{Journal de mathématiques pures et appliquées 4e série. -1893. -T. 9}, p. . \bibitem[Bruno]{b14}\label{b14} ‘Exotic expansions of solutions to an ordinary differential equation’. A D Bruno . Doklady Mathematics.-2007. 76. \bibitem[Bruno and Kudryashov]{b41}\label{b41} ‘Expansions of solutions to the equation P12 by algorithms of power geometry’. A D Bruno , N A Kudryashov . Ukrainian Mathematical Bulletin. -2009. 6 (3) p. . \bibitem[Bruno]{b16}\label{b16} \textit{Exponential expansions of solutions to an ordinary differential equa-tion // Doklady Mathematics}, A D Bruno . 85. \bibitem[Bruno (ed.)]{b27}\label{b27} \textit{Families of periodic solutions and invariant tori of Hamiltonian systems // Formal and Analytic Solutions of Differential Equations}, A D Bruno . \xref{http://dx.doi.org/10.1142/q0335}{10.1142/q0335}. G. Filipuk, A. Lastra, S. Michalik. -WORLD SCIENTIFIC (ed.) \bibitem[Bruno ()]{b9}\label{b9} \textit{Families of periodic solutions to the Beletsky equation}, A D Bruno . 2002. 40. \bibitem[Hénon ()]{b52}\label{b52} ‘Generating Families in the Restricted Three-Body Problem’. M Hénon . Lecture Note in Physics.Monographs 1997. Berlin, Heidelber, New York: Springer. 52. \bibitem[Bruno ()]{b6}\label{b6} \textit{Local Methods in Nonlinear Differential Equations. -Berlin, Hei-delberg}, A D Bruno . 1989. New York, London, Paris, Tokyo: Springer-Verlag. \bibitem[Oleinik and Samokhin ()]{b54}\label{b54} \textit{Mathematical Models in Boundary Layer The-ory}, O A Oleinik , V N Samokhin . \xref{http://dx.doi.org/10.1201/9780203749364}{10.1201/9780203749364}. 1999. New York: Chapman \& Hall/CRC. \bibitem[Nonlinear Analysis as a Calculus]{b25}\label{b25} \xref{http://dx.doi.org/10.1134/S0001434619110233}{10.1134/S0001434619110233}. \textit{Nonlinear Analysis as a Calculus}, p. . \bibitem[Bruno]{b2}\label{b2} ‘Normal form of differential equations’. A D Bruno . Soviet Math. Dokl. -1964. 5 p. . \bibitem[Bruno]{b5}\label{b5} \textit{On periodic flybys of the Moon // Celestial Mechanics. -1981}, A D Bruno . \xref{http://dx.doi.org/10.1007/BF01229557}{10.1007/BF01229557}. 24 p. . \bibitem[Bruno]{b28}\label{b28} ‘On the generalized normal form of ODE systems // Qual’. A D Bruno . \xref{http://dx.doi.org/10.1007/s12346-021-00531-4}{10.1007/s12346-021-00531-4}. \textit{Theory Dyn. Syst. -2022b} 21 (1) . \bibitem[Bruno]{b24}\label{b24} ‘On the parametrization of an algebraic curve’. A D Bruno . Mathematical Notes.-2019a. 106. \bibitem[Bruno and Varin ()]{b45}\label{b45} ‘Periodic solutions of the restricted three-body problem for small mass ratio’. A D Bruno , V P Varin . \xref{http://dx.doi.org/10.1016/j.jappmathmech.2007.12.012}{10.1016/j.jappmathmech.2007.12.012}. \textit{J. Appl. Math. Mech} 2007. 71 (6) p. . \bibitem[Bruno ()]{b19}\label{b19} ‘Power geometry and elliptic expansions of solutions to the Painlevé equations’. A D Bruno . \xref{http://dx.doi.org/10.1155/2015/340715}{10.1155/2015/340715}. \textit{International Journal of Differential Equations} 2015. 2015. 340715. \bibitem[Bruno]{b23}\label{b23} ‘Power geometry and expansions of solutions to the Painlevé equa-tions’. A D Bruno . Transnational Journal of Pure and Applied Mathematics. -2018c. 1 (1) p. . \bibitem[Bruno ()]{b8}\label{b8} \textit{Power Geometry in Algebraic and Differential Equations}, A D Bruno . 2000. Elsevier Science. (-Ams-terdam) \bibitem[Bruno]{b18}\label{b18} \textit{Power-exponential expansions of solutions to an ordinary differential equation // Doklady Mathematics.-2012b}, A D Bruno . 3. -P. 336-340. -DOI: 10.1134 S106456241. 85 p. 203009. \bibitem[Bruno ()]{b26}\label{b26} ‘Power-exponential transseries as solutions to ODE’. A D Bruno . \xref{http://dx.doi.org/10.18642/jmsaa_7100122093}{10.18642/jmsaa\textunderscore 7100122093}. \textit{Journal of Mathematical Sciences: Advances and Applications} 2019b.. 59 p. . \bibitem[Bruno and Chapter ()]{b47}\label{b47} ‘Regular Asymptotic Expansions of Solutions to One ODE’. A D Bruno , Chapter . \xref{http://dx.doi.org/10.1515/9783110275667.67}{10.1515/9783110275667.67}. Compilation 1.0. \textit{Proceedings of the International Conference}, A D Bruno , A B Batkhin , Berlin (eds.) (the International ConferenceSaint Petersburg, Russia; Boston) June 17-23, 2011. 2012. 2023. London Journals Press. 15 p. 31. \bibitem[Bruno and Batkhin ()]{b31}\label{b31} \textit{Resolution of an algebraic singularity by power geometry algorithms // Programming and computer software}, A D Bruno , A B Batkhin . 2012. 38. \bibitem[Polyanin and Zhurov ()]{b56}\label{b56} \textit{Separation of Variables and Exact Solutions to Nonlinear PDEs}, A D Polyanin , A I Zhurov . 2021. New York: Chapman \& Hall/CRC. (401 p. Nonlinear Analysis as a Calculus) \bibitem[Dulac]{b49}\label{b49} ‘Solutions d'un système d'équations différentialles dans le voisinage des valeures singulières’. H Dulac . 1912. -T. 40. -P. 324-383. \textit{Bull. Soc. Math. France} \bibitem[Poincaré ()]{b55}\label{b55} ‘Sur les propriétés des fonctions définities par les équations aux diffé-rence partielles // Oeuvres de Henri Poincaré’. H Poincaré . \textit{T. I} 1928. Paris: XLIX-CXXIX. \bibitem[Bruno and Batkhin ()]{b33}\label{b33} \textit{Survey of Eight Modern Methods of Hamiltonian Mechanics // Axioms}, A D Bruno , A B Batkhin . 2021. \bibitem[Bruno]{b1}\label{b1} ‘The asymptotic behavior of solutions of nonlinear systems of differ-ential equations’. A D Bruno . Soviet Math. Dokl. -1962. \bibitem[Bruno ()]{b7}\label{b7} \textit{The Restricted 3-body Problem: Plane Periodic Orbits}, A D Bruno . 1994. Berlin: Walter de Gruyter. \end{bibitemlist} \end{document}