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Journal of Applied Mathematics and Physics, 2013, 1, 1-4 http://dx.doi.org/10.4236/jamp.2013.13001 Published Online August 2013 (http://www.scirp.org/journal/jamp) Construction of Periodic Solutions of One Class Nonautonomous Systems of Differential Equations Alexander N. Pchelintsev Tambov State Technical University, Tambov, Russia Email: pchelintsev.an@yandex.ru Received June 15, 2013; revised July 16, 2013; accepted August 1, 2013 Copyright © 2013 Alexander N. Pchelintsev. This is an open access article distributed under the Creative Commons Attribution Li- cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this article we proposed a method for constructing approximations to periodic solutions of one class nonautono- mous system of ordinary differential equations. It is based on successive approximation scheme using parallel sym- bolic calculations to obtain solutions in analytical form. We showed the convergence of the scheme of successive ap- proximations on the period, and also considered an example of a second order system where the described scheme of calculations can be applied. Keywords: Periodic Solution; System of Ordinary Differential Equations; Scheme of Successive Approximations; Symbolic Calculations 1. Introduction Quite often (and in practice as well) there appears a prob- lem of constructing of periodic solutions of the normal sys- tem of ordinary differential equations of the form , x ftx , (1) where x t—vector function of a real variable t, —vector function equal to ),( xtf , f txx ht , (2) where vector x —multidimensional polynomial and function is a trigonometric polynomial (T-peri odi c vector function). ht Many of the theorems [1] of existence of periodic solu- tions of system (1) us e the fun damental f act that such solu- tions are completely determined by the fixed points of the shift operator along the trajectories of the system. These theorems can not be used to direct finding of the desired periodic solution. Let it be known that the system (1) has a unique T-periodic solutio n x t . Examples of systems that have a unique periodic solution, are the systems with conver- gence [2,3]. In this article one class of such systems would be considered so that for them we provide a method of constructing of approximations to the solution x t ; given the conditions imposed on the function f, it allows showing the convergence of the scheme of successive ap- proximations on the period. At that it introduces an auxil- iary system for constructing in a symbolic form of ap- proximation to some periodic function, which depends on the initial conditions for the system (1). By varying those conditions we will find an approximation to the solution x t . The parallel calculations can be used to improve efficiency of calculation process. The symbolic form of representation is convenient because it further allows you to analyze the harmonic components of the desired appro- ximation. 2. The Conditions Imposed on Original System We shall consider the class of systems (1), which obeys the following cond itions: 1) The system (1) is a system with convergence, i.e., it has a unique T-periodic solution x t , that asymptoti- cally stable in whole. 2) Closed ball of radius r, which contains the val- ues of function r S x t , is contained within a ball R S of radius R. For vectors and from R S Lipschitz inequality takes place l , (3) where the positive number l satisfies the condition 12lT. (4) 3) There can be found a positive number M, such that C opyright © 2013 SciRes. JAMP A. N. PCHELINTSEV 2 for all vectors x from R S takes place ,,2 f txM rTMR. (5) 3. The Transition to Auxiliary System of Equations To simplify the notation let’s assu me that initial time mo- ment is zero. Let’s rewrite (1) in the integral form 0 ,d t xt Cfx , where vector C defines the initial conditions. So far as x t —T-peri o di c funct i on t he n 0 x xT C 0 . Then 0 ,d T fx . (6) Therefore 0 1,d T fx T 0 . So far as function x t satisfies the system (1), then it also satisfies the system 0 1 ,, T yftyf y Td . (7) Let’s pass from Equation (7) to the integral relation: 00 1 ,, tT ytCf sysfys T dd . (8) Let’s notice that transition to the auxiliary syste m (8) is necessary because the nested integral (mean integral value over the period) in calculations allows avoiding the ap- pearance degrees of t in symbolic expressions. This re- duces the amount of memory allocated for them, and also forms a symbolic representation of the desired function as an approximation to the Fourier series of periodic solu- tions of system (1) . 4. Scheme of Successive Approximations To obtain an approximation to the solution x t on the period, first let’s construct an approximation to a func- tion that is a solution to the system (8). Then by varying the vector C let’s find the desired approximation. To do this let’s show that we are in terms of applicability of the method of successive approximations. Let —space of continuous T-periodic vector func- tions y t with values in a ball R S. The distance be- tween the functions we’ll define as ,pq 0, ,max tT pqptqt . Thus space is metric. In let’s consider an op- erator 00 0 1 ,, d. tT t ytCf sysfys T Cuss dd (9) Let’s show that function g ty t belongs to space , when . y By virtue of (9) and Cr (we will search a vector C in a ball r, because it contains values of a function S x t ) we obtain: 0, , 0 d2max, t tTy tCussrT ftyt g . Given that y t takes values from R S, and inequal- ity (5), we obtain g tR for all 0,tT. Now let’s show the T-periodicity of function g t: 0gt Tgt for any real values t. With (9) let’s consider the differ- ence 00 0 0 dd dd tT t tT t . g tT gtuss uss us suss Since the function f T-periodic in t and y t is also T-periodic then function will be T-periodic. In view of this making the change us s aT in first inte- gral we obtain 0 0 0 dd dd t Tt T T . g tTgtua aus s us sus s Let ’s assume 0 ,d T Bfy . Then 0 1 ,d 10. T g tTgtfsysB s T BBT T Copyright © 2013 SciRes. JAMP A. N. PCHELINTSEV 3 Thus if then . y g Let’s show that mapping y is contracted. Let’s es- timate 0, 0 0 ,max, , 1,,dd. t tT T p qfspsfsqs fp fqs T By virtue of inequality (3) and (2) we obtain ,2,pqlTpq . From (4) it follows that 21lT . Thus the mapping is a compression. Then, according to the method of successive approxi- mations the scheme 11 00 1 ,, tT mmm yt Cfsysfys T dd (10) converges to solution of a system (8). 5. Finding Periodic Solution Since the right part of the system (1) on x a multidimen- sional polynomial and in t it is a trigonometric polyno- mial the initial function 0 y t it is advisable to choose equal to vector C or as 0cos y tC t , where 2T , i.e., . 0 Note that the multiplication of two (and, consequently, any number of) trigonometric functions (cos or sin) and also an extent of such functions [4], can be represented as sum of a constant vector and trigonometric polynomial, and during the integration of trigonometric polynomial is obtained the same sum. Then from the scheme (10), each iteration can be calculated symbolically. At that the transformation of trigonometric functions in symbolic form, as well as their symbolic integration, are parallel- ized. The idea of parallelism lies in the formula (10): calculations for each component of the vector y m y t might be produced independently of each other and store the obtained symbolic expressions in network database which accessible for computational process in a distrib- uted computing environment. Following the formula (10), we build a function m y t to such value m, when 1 max ,, r mm c CS yy (11) where c —accuracy for the scheme (10). Let’s consider the integral 0 ,d T Ify . When substituting a character expression for the function we obtain vector function from C. Then redenote CI . We need to find a vector C such that there takes place an Equation (6), i.e. CC 0C . (12) So far as the system (1) has a unique periodic solution then system of algebraic Equations (12) in region will also have a unique solution. r S The system (12) is equivalent to ,0CC . From where we get the optimization problem to find an approximate value of vector : C ,min, r CCC СS . (13) The transition from the system (12) of algebraic equa- tions to the problem (13) related to the fact that directly in the calculations the functio)(t is defined approxi- mately by using the criterion (11 n). y 6. An Example of a Nonlinear Second-Order System As an example of using the described method let’s consider a nonlinear oscillations equation (in this subsection, we rename some functions, as is customary in well-known literature) x qxx gxpt , (14) where 2 0.01 12qx x, 0.03 g xx, 0.01cos 24pt t , T period of right part is equal to 12. According to Theorem 8.1 [2] for an Equation (14) the convergence property is executed. From an Equation (14) let’s move to an equivalent system of second order [2], where right part has the form (2): , x yQx Ptygx , (15) where 3 0 2 d0.01 3 x Qxqx x , Copyright © 2013 SciRes. JAMP A. N. PCHELINTSEV Copyright © 2013 SciRes. JAMP 4 0 1 dsin2 2400 t4 P tp t . Let’s find the radius r of the ball r, where all solutions of system (15) are bounded. To do this, let’s first show the fulfillment of conditions 1 - 4 [2]: S 1) Functions q, g and p are continuous, g satisfies a Lipschitz condition with constant 0.03. 2) The value of numbers a, and is equal to 1/12, 73/7200 and 1/400 respectively so that qx when x a, gx when x a, gx when x a . 3) A function is bounded at all t, i.e., Pt Pt E, 1 1200E. 4) There exists such number 1 259,200 , that Qx E when x a, Qx E when x a ; when 0Gx x a, where 2 0 d0.015Gx gx x. Then any solution of system (15) is bounded in a rectan- gl e , de f i n e d by inequalitie s 01 , x ay b , where 0 so that 0 Qx E when x a. Let’s choose it as equal to 0max xa QxE . Since —everywhere increasing function then Qx 0 . The value of constant 1 is chosen as [2] 10 a , where —any positive number and max xa g x [2]. Let’s assume 3 1200 . Then 3 600b. Thus, radius r may be taken equal to 22 730rab . Now let’s establish the fulfillment of conditions 3 and 2 from section 2 of given article. Let’s rewrite the system (15) in a ve ctor form : ,zwtz , where ,ztxt yt. Let’s consider a ball R S of radius 0.5R which have inside a ball . Then from (15) r S 0, , 3 , 2max , 2 2 max0.010.03 3 . tTz xRyR rT wtzt rTyxxEx R Local implementation of the Lipschitz condition for the function z is established the following way. Let’s represent 12 121 ,zzAxxzz 2 , where the A matrix has the form 22 1122 12 2 0.01 11 ,3 0.03 0 xxxx Ax x . Hence we obtain 12 12 , 2max , xRx R TAxx 1. 7. Acknowledgements This article was supported b y the Russian Foundatio n for Basic Research (Projects No. 11-07-00098 and 13-07- 00077). REFERENCES [1] M. A. Krasnosel’skij, “The Shift Operator along Trajec- tories of Differential Equations (in Russian),” Nauka, Moscow, 1966. [2] V. A. Pliss, “Nonlocal Problems of Oscillation Theory (in Russian),” Nauka, Moscow, 1964, pp. 107-110,113. [3] B. P. Demidovich, “Lectures on the Mathematical Stabil- ity Theory (in Russian),” Nauka, Moscow, 1967. [4] I. S. Gradshtejn and I. M. Ryzhik, “Tables of Integrals, Sums, Series and Multiplications (in Russian),” Fizmatlit, Moscow, 1963, p. 39. |