An Approximation Algorithm for Quadratic Dynamic Systems Based on N. Chomsky s Grammar for Taylor s Formula

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1 ISSN , Proceedings of the Steklov Institute of Mathematics, 2016, Vol. 293, Suppl. 1, pp. S17 S21. c Pleiades Publishing, Ltd., Original Russian Text c A.A. Azamov, M.A. Bekimov, 2015, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2015, Vol. 21, No. 2. An Approximation Algorithm for Quadratic Dynamic Systems Based on N. Chomsky s Grammar for Taylor s Formula A. A. Azamov 1 and M. A. Bekimov Received February 16, 2015 Abstract Single-step methods for the approximate solution of the Cauchy problem for dynamic systems are discussed. It is shown that a numerical integration algorithm with a high degree of accuracy based on Taylor s formula can beproposedinthecaseofquadraticsystems. An explicit estimate is given for the remainder. The algorithm is based on N. Chomsky s generative grammar for the language of terms of Taylor s formula. Keywords: dynamic system, quadratic system of equations, Cauchy problem, numerical solution, Taylor s formula, remainder term, error estimate, algorithm, context-free grammar. DOI: /S In the modern theory of dynamical systems, many of the results are based on the numerical solution of the Cauchy problem dx/dt = f(x), x(0) = x 0, (0.1) where x R d (see [1,11 13]). Leaving aside the linear case, quadratic systems (see, e.g., [7] as well as reviews [10, 17] on planar systems), in which the components of f are defined as f i (x) = d j,k=1 a jk i x j x k + d b j i x j + c i, (0.2) where x =(x 1,x 2,...,x d ) and all the coefficients are constants, comprise the simplest but extremely important class of dynamical systems. This class includes, for example, Lotka Volterra models [14] as well as Lorentz [1,15] and Rössler [16] systems. The interest in quadratic systems is also due to the tenth Hilbert problem [5]. For d 2 system (0.2) cannot be integrated apart from few exceptions. Thereby, its properties are often formulated on the basis of numerical solutions. (Moreover, a rigorous proof is not presented in many cases because the authors suppose that the conclusions based on numerical experiments are quite sufficient). This circumstance explains the particular importance both of the convenience of the method used for the approximate solution and of the order of its accuracy. In most cases, the one-step Runge Kutta method is preferred; it gives a numerical solution up to h N for N =2 5 (very rarely, for N = 6) within one step [2,8]. In principle, the Runge Kutta method is applicable 1 Ulugbek National University of Uzbekistan, Tashkent, Uzbekistan abdulla.azamov@gmail.com and mansu@mail.ru S17 j=1

2 S18 AZAMOV, BEKIMOV for obtaining a solution with a higher degree of accuracy; however, in practice, this is avoided, since the corresponding formulas (even for N>6) become overly cumbersome. In this paper we discuss a scheme for the approximate solution of the Cauchy problem for quadratic systems based on Taylor s formula x(t + h) = n k=0 x (k) (t) h k + R n+1 (t, h). (0.3) k! Formula (0.3) is not applied in practice either, since, for d 2, the expressions for x (n) in terms of the function f and its derivatives are even more cumbersome as compared to the Runge Kutta method. Nevertheless, it turns out that the situation is greatly simplified in the case of quadratic systems. In this case, it is possible to get an explicit formula for estimating the remainder and propose a relatively simple algorithm for the Taylor expansion. The algorithm is based on the fact that the terms in the formula for x (n) (which are here called Taylor terms) form a language over the three-letter alphabet {0, 1, 2} with a context-free Chomsky grammar [3, 9]. Let us first consider the case d = 1. Although equation (0.1) is solved explicitly in this case, the reasoning used in the calculation of the Taylor terms can be easily transferred to the estimation of the remainder in the general case. In the case under consideration, for any x, thevaluesf(x), f (x), and f (x) are numbers and, moreover, f (x) = const. In what follows, we will call these expressions factors and will write them briefly in the form f, f,andf except for cases where this can lead to misunderstanding. [ n 1 ] Let numbers Dn k,wheren 1, k =0, 1, 2,..., ([a] denotes the integer part of a), be 2 defined by the recurrence relations D 0 n =1, Dk n+1 =(n 2k +1)Dk 1 n +(n +1)D k n. (0.4) It is easy to establish that Dn 1 =2 n 1 n; in addition, 2 n k Dn k 2 n+k for n 4. Proposition. We have the formula x (n) (t) = [(n 1)/2] k=0 D k nf k f n 2k 1 f k+1. (0.5) Proof. The proposition is proved similarly to Newton s binomial formula, i.e., by induction on n based on relations (0.4). Note that it is convenient to consider the cases of even and odd n separately, since, in the transition from x (n) to x (n+1), the number of Taylor terms increases by one in the first case and remains the same in the second case. For small values of n, we obtain the following formulas: ẋ = f, ẍ = f f, x III = f 2 f, x IV = f 3 f +4f f f 2, x V = f 4 f +11f f 2 f 2 +4f 2 f 3,.... (0.6) Note that the powers of f and f in each term are uniquely defined by n and the power of f. Now, let d 2. In this case, f is a vector, f is a Jacobian matrix (i.e., the tensor f i(x) of x j rank (1, 1)), and f is a vector of quadratic forms (i.e., the tensor 2 f i (x) x j of rank (1, 2); see [4,6]). xk In the multidimensional case, we have f = const. However, formula (0.5), in general, does not hold here. For example, the expression for x IV has the form x IV = f 3 f +2f f ff + f ff f + f f ff,

3 APPROXIMATION ALGORITHM S19 which, in view of the symmetry of f in the contravariant indices, i.e., in view of the relation f (u, v) =f (v, u), can be written as x IV = f 3 f +3f f ff + f f ff. In the general case, f f ff and f f ff do not necessary coincide. For example, for f = ( y 2,x 2 ), we have f f ff = 4(x 3 y 2,x 2 y 3 )andf ff f = 4(y 5,x 5 ). That is why the Taylor terms containing the factor f k in the formula for x (n) are not grouped into monomials with the coefficients Dn k. Nevertheless, the scheme of the proof of the proposition remains valid for the following statement. Theorem 1. The expression for x (n) is the sum of groups Δ k n of Taylor terms, k =0, 1,..., [(n 1)/2]. The group Δ k n contains Dn k terms, each of which consists of k, n 2k 1, andk +1 factors f, f,andf, respectively. Suppose that the solution x(t) to the Cauchy problem ẋ(t) =f(x), x(0) = x 0 exists on the time interval [0,T] and satisfies the condition x(t) K, wheret is a specified positive number and K is a specified compact set in R d. Define M 0 =max f(x), M 1 =max f (x), M 2 = f (x) =const x K x K (the norms of the matrix and of the quadratic form are Euclidean: f (x) =max f (x)u and u 1 f (x) = max f (x)[u, v]). Therefore (see [6; Ch. 1, inequality (1.8.2)]), we have u 1, v 1, f u M1 u, f [u, v] M2 u v. Hence, the norm of any Taylor term from the group Δ k n is bounded by M k 2 M n 2k 1 1 M k+1 0.This yields the following statement. Theorem 2. The remainder in the Taylor formula for a solution to the Cauchy problem for quadratic systems is estimated as follows: R n+1 hn+1 (n +1)! k D k n+1m k 0 M n 2k 1 M k 2, k =0, 1,...,[(n 1)/2]. As noted above, in the case d 2, one cannot obtain a compact formula similar to (0.5) for the derivatives of x (n). To overcome this difficulty, we find the structure of the Taylor terms in the groups Δ k n. It is convenient to pass to a formal language, replacing the factors f, f,andf by the digits 0, 1, and 2, respectively. Then each Taylor term turns into a word over the three-letter alphabet {0, 1, 2}. The words that are obtained from the Taylor terms after such a change will be called d-words in order to distinguish them from words in general over the alphabet {0, 1, 2}. The set of d-words forms a language (in the sense of Chomsky), which we denote by T. (Usually, the empty word Λ, not containing symbols, is also included in the language.) We will denote by σ the length of a word σ, i.e., the number of symbols in it. The language T has a simple generative grammar: it follows from the rules d dt f(x) =f (x)f(x) and d dt f (x)u = f (x)[f(x),u], where u R d,thatt is generated by the two rules 0 10, 1 20 (0.7)

4 S20 AZAMOV, BEKIMOV starting with the word 0. Rules (0.7) mean that, if we substitute 10 in a d-word σ instead of 0 or 20 instead of unity, then we obtain a d-word (of length σ + 1). Thus, the language T has a context-free grammar. In the language T, formulas (0.6) are written more clearly in the form of a tree: One of the main questions of mathematical linguistics is to give a criterion that would determine whether a word over a given alphabet belongs to a certain language. It has a simple answer in the case of the language T. Theorem 3. Awordσ over the alphabet {0, 1, 2} belongs to the language T if and only if (a) it ends with the digit 0; (b) the number of zeros is greater by one than the number of the digits 2; (c) if we enumerate the digits 2 in the order of Arabic writing (i.e., from right to left), then there will be at least k +1 digits 0 ontherightofthejth digit 2. Proof. Thefactthatalld-words have properties (a) (c) follows immediately from rules (0.7). Let us prove the sufficiency. It is obvious for words of length 1 and 2. Let a word σ of length n, n 3, over the alphabet {0, 1, 2} possess properties (a) (c). Then it must have the form σ = ρε0 k, where 0 k is the word consisting of k zeros, k 1, ε = 1 or 2, and ρ is a subword of length n k 1. In the case ε = 1, we replace the subword 10 k in the word σ by the subword 0 k.inthecaseε =2 (then necessarily k 2), we replace the subword 20 k by the subword 10 k 1. The obtained word σ has length n 1 and it still satisfies properties (a) (c). Therefore, it belongs to T. Since σ is obtained from σ by one of rules (0.7), we have σ T. We can construct a simple procedure Extract(n, σ; x) that calculates the value of the corresponding d-word σ of length n for the Taylor term at the point x. For example, Extract(1, 0; x) =f(x), Extract(2, 10; x) =f (x) f(x), Extract(7, ; x) =f (x)f (x){f(x),f (x)[f(x),f (x)f(x)]}. We apply the language T for the approximate calculation of x(t + h) using the given value x(t) with an error at one step of order h N+1. Let us introduce the following operation for the description of the corresponding algorithm. Let L be some list (array) of d-words. Let us replace each word σ L by a set of d-words of length σ + 1 by a single application of rules (0.7) to all the digits 0 and 2 in the word σ. We obtain a new list, which we denote by DL. Putting n =0,L = 0, H = h, ands = x(t), we calculate Σ = σ LExtract(n, σ; x(t)) and define S = x(t)+hσ. Then, we pass to the step n := n +1. Ifn N, the calculation ends with the value x(t + h) =S. Otherwise, we create the list L := DL and pass to calculating Σ and S, setting H := Hh/n. Remark. The language T is associated with a special fractal in the Hilbert space l 2.

5 APPROXIMATION ALGORITHM S21 ACKNOWLEDGMENTS This work was supported by the Program for Fundamental Research of the Republic of Uzbekistan (project no. F4-FA-F014). REFERENCES 1. D. V. Anosov, Lorentz attractor, in Mathematical Encyclopedia, Ed. by I. M. Vinogradov (Sov. Entsikl., Moscow, 1982), Vol. 3, p. 451 [in Russian]. 2. N. S. Bakhvalov, Numerical Methods (Nauka, Moscow, 1973; Mir, Moscow, 1977). 3. A. V. Gladkii, Formal Grammars and Languages (Nauka, Moscow, 1973; Elsevier, New York, 1982). 4. V. A. Zorich, Mathematical Analysis II (Nauka, Moscow, 1984; Springer, Berlin, 2004). 5. Yu. S. Il yashenko, Selected Problems of the Theory of Dynamical Systems (Izd. MTsNMO, Moscow, 2011) [in Russian]. 6. H. Cartan, Differential Calculus (Hermann, Paris, 1971; Mir, Moscow, 1971). 7. J. C. Artes and J. Llibre, Quadratic Hamiltonian vector fields, J. Differential Equations 107, (1994). 8. J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd ed. (Wiley, New York, 2008). 9. N. Chomsky, Three models for the description of language, IRE Trans. Inform. Theory 2, (1956). 10. W. A. Coppel, A survey of quadratic systems, J. Differential Equations 2, (1966). 11. J. Guckenheimer, Computational environments for exploring dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 1 (2), (1991). 12. J. Guckenheimer, Numerical analysis of dynamical systems, in Handbook of Dynamical Systems (Elsevier, Amsterdam, 2002), Vol. 2, pp J. Guckenheimer, Phase portraits of planar vector fields: Computer proofs, Experiment. Math. 4 (2), (1995). 14. Lotka Volterra and Related Systems, Ed. by Sh. Ahmad and I. M. Stamova (De Gruyter, Berlin, 2013). 15. A. N. Pchelintsev, Numerical and physical modelling of the dynamics of Lorenz system, Numer. Anal. Appl. 7 (2), (2014). 16. H.-O. Peitgen, H. Jurgens, and D. Saupe, Rössler attractor, in Chaos and Fractals: New Frontiers of Science (Springer, New York, 2004), pp J. W. Reyn, A Bibliography of the Qualitative Theory of Quadratic Systems of Differential Equations in the Plane: Report (Delft Univ. Technol., Delft, 1987). Translated by E. Vasil eva

Chini Equations and Isochronous Centers in Three-Dimensional Differential Systems

Qual. Th. Dyn. Syst. 99 (9999), 1 1 1575-546/99-, DOI 1.17/s12346-3- c 29 Birkhäuser Verlag Basel/Switzerland Qualitative Theory of Dynamical Systems Chini Equations and Isochronous Centers in Three-Dimensional