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Normal forms, computer algebra and a problem of integrability of nonlinear ODEs

Normal forms, computer algebra and a problem of integrability of nonlinear ODEs. Victor Edneral Moscow State University Russia Joint work with Alexander Bruno. Introduction Normal Form of a Nonlinear System Euler – Poisson Equations

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Normal forms, computer algebra and a problem of integrability of nonlinear ODEs

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  1. Normal forms, computer algebra and a problem of integrability of nonlinear ODEs Victor Edneral Moscow State University Russia Joint work with Alexander Bruno

  2. Introduction • Normal Form of a Nonlinear System • Euler – Poisson Equations • Normal Form of the Euler – Poisson Equations in Resonance • Structure of Integrals of the System • Necessary Conditions for Existence of Additional Integrals • Calculation of the Normal Form of the Euler – Poisson Equations • Conclusions CADE-2007, February 20-23, Turku, Finland

  3. Introduction Here, we study the connection between coefficients of normal forms and integrability of the system. For this purpose, we compute normal forms of the Euler – Poisson equations, which describe the motion of a rigid body with a fixed point. This is an autonomous sixth-order system. A lot of books and papers are devoted to integrable systems and to methods for searching for such systems. A.D. Bruno noted that all normal forms of integrable systems are degenerated, so it is interesting to search domains with a such degeneration [Bruno, 2005]. The first attempt to calculate the normal form of the Euler – Poisson system was made in [Starzhinsky, 1977]. However, without computer algebra tools, he was unable to calculate a sufficient number of terms. We use a program for analytical computation of the normal form [V. Edneral, R. Khanin, 2003]. This is a modification of the LISP based package NORT for the MATHEMATICA system. The NORT package [Edneral, 1998] has been designed for the REDUCE system. CADE-2007, February 20-23, Turku, Finland

  4. Normal Form of a Nonlinear System Consider the system of order n (1.1) in a neighborhood of the stationary point X = 0 under the assumption that the vector function Φ(X) is analytical at the point X = 0 and its Taylor expansion contains no constant and linear terms. CADE-2007, February 20-23, Turku, Finland

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  10. Euler – Poisson Equations CADE-2007, February 20-23, Turku, Finland

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  14. Has the system additional local integrals? CADE-2007, February 20-23, Turku, Finland

  15. Local Integrals [Lunkevich, Sibirskii, 1982]. Let us see a system It has two stationary points At S1 it has a center and at S2 – a focus. It is integrable at S1 with the integral You can see that is an invariant line. The system is integrable in semi plane x > -1/2 and not integrable at x < -1/2. CADE-2007, February 20-23, Turku, Finland

  16. [Lunkevich, Sibirskii, 1982]. CADE-2007, February 20-23, Turku, Finland

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  21. C2 R + D2 D4 R – C1 C3 (x0,y0) D1 D5 D3 Fig. 1 CADE-2007, February 20-23, Turku, Finland

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  28. Normal Form of the Euler – Poisson Equations in Resonance Let the normal form be (6.1) and vector of eigenvalues of the matrix Λbe Let also introduce so called resonance variables. After z1 and z2 we have CADE-2007, February 20-23, Turku, Finland

  29. Lemm 1[Bruno, 2005]. At the resonance in the normal form where are power series in At this start from free terms but – from linear terms. CADE-2007, February 20-23, Turku, Finland

  30. In resonance variables we will have the system in the form for odd values of (6.2) and for even values of . G0, Hi,k and Fi,k above are linear combination of gr,m, hr.m and fr.m. CADE-2007, February 20-23, Turku, Finland

  31. Structure of Integrals of the System As it was shown in [Bruno, 1995] an expansion of the first integral of normal form contains only resonance variables with the property Thus the first integral can be written as power series where a0, am and bm are power series in z1, z2, ρ1, ρ2. CADE-2007, February 20-23, Turku, Finland

  32. For the resonance variables, the automorphism can be rewritten as even, if odd. CADE-2007, February 20-23, Turku, Finland

  33. Then we have am = bm and the integral is (7.1) If is odd then the integral A will be CADE-2007, February 20-23, Turku, Finland

  34. Necessary Conditions for Existence of Additional Integrals From the definition, the derivation in time of any first integral along the system should be zero, i.e. CADE-2007, February 20-23, Turku, Finland

  35. The identity above should be discussed at odd and even values of separately. Corresponding coefficients in formulae below will be slight different. The lowest non vanished coefficients will be different also. It is very important for an estimation of order up to which we a need to calculate the normal form. If we parameterize the identity above up to the common order in z1,2, ρ1,2variables smaller than 2( ) for the odd value and up to for the even one, we will see that the identity should be right for free and linear in the common order . (7.2) CADE-2007, February 20-23, Turku, Finland

  36. So, if we write o(z1,2,ρ1,2), + o(z1,2,ρ1,2), (7.3) o(z1,2,ρ1,2), O(z1,2,ρ1,2). CADE-2007, February 20-23, Turku, Finland

  37. then the equation for the free term has the form (A) Here the vector Ξ≡ {ξi} can be calculated from the normal form. It will be a function of parameters of the system. The vector α ≡ {αi} defines a0. If you know the first integrals of the system, you can calculate the corresponding α ≡{αi}for each integral separately. If Ξ≠ 0 then equation (A) has three dimensional set of solutions α, so only three integrals can be independent. CADE-2007, February 20-23, Turku, Finland

  38. A single possibility to have an additional integral in this case is that the tree known integrals are dependent each from other. Mathematically it can be written as the vector equality Because for checking this condition we need calculate only the lowest orders of the normal form it is possible to calculate it in analytical form in variables of the system. CADE-2007, February 20-23, Turku, Finland

  39. If Ξ≡0 thenfrom (5.2), (5.3) we will have the condition (B) η and ζare coefficients which can be calculated from coefficients of the normal form as functions of parameters. The dimension of solutions (α, β) of the system above is where M is a matrix (4 x 5) which consists from vectors CADE-2007, February 20-23, Turku, Finland

  40. Let us say that formal integral (7.1) is local independent on known integrals if its linear approximation in z, ρ, ω is linear independent from first approximations of known integrals. Main theorem [Bruno, 2005]. For existence of an additional formal integral at the family of the stationary pointFδ, it is necessary a satisfaction of one of two sets of conditions: • Ξ ≠ 0 and V = 0, • Ξ = 0 and rank(M) < 2. Note, that if the original system has five first integrals, then right hand side of normalized equation is linear, and rank(M) = 0. CADE-2007, February 20-23, Turku, Finland

  41. Calculation of the Normal Form of the Euler – Poisson Equations Near stationary points of families Sσwe computed normal forms of the System up to terms of some order m For that, we used the program [Edneral, Khanin, 2003]. All calculations were lead in rational arithmetic and float point numbers in this paper are approximations of exact results. CADE-2007, February 20-23, Turku, Finland

  42. Case of Resonance (1:2) We will use the uniformization then domain corresponds the interval and at δ = 1 we have CADE-2007, February 20-23, Turku, Finland

  43. Components of external products will be The system will have only two solutions So at δ = 1in the interval above CADE-2007, February 20-23, Turku, Finland

  44. At δ = -1 Solutions CADE-2007, February 20-23, Turku, Finland

  45. So only solution h4lies in the mechanical semi-interval But h4is a special point with all zero eigenvalues and we can conclude that With respect of the Main theorem of existence of an additional integral • Ξ ≠ 0 and V = 0, • Ξ = 0 and rank(M) < 2, we should look for points where Ξ = 0. CADE-2007, February 20-23, Turku, Finland

  46. Due to automorphism (5.1) and to Property 1, the normal form has corresponding automorphism and the sum k ≡ q3 + q4 + q5 + q6 is even for all its terms. We considered sums For the normal form, it occurs that, for m = q1 + q2 + k = 4 we will have CADE-2007, February 20-23, Turku, Finland

  47. and all lower terms cancel. Here, the quantities with a hat ĝk, k = 1, …, 6, denote the normal forms calculated up to order four. It can be demonstrated that the vector Ξ has a components So we can calculate Ξ now. Coefficients a and b depend on δ2 and c.For δ2 = 1, both coefficients a and b are pure imaginary. For δ2 = –1, they are pure imaginary if c (0, c2) and are real if c (c2, 2]. CADE-2007, February 20-23, Turku, Finland

  48. c3 c2 CADE-2007, February 20-23, Turku, Finland

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  50. c4 c5 c6 c2 CADE-2007, February 20-23, Turku, Finland

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