Duffing equation to find periodic solutions for hi

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Duffing equation to find periodic solutions for higher order continuous Newton method

) of the Hessian matrix , V = ÷ ((P-Y0) B (P-Y0), ..., (P-Y0) B A (P-y0)). Solution of equation (6 ) and a simple approximate solution of amendments, they will have improved Newton iteration formula : P, + = P a [F (Y0)] I1F (P) a ÷ [F (Y0)], (7 ) where = ( Y said a ,[link widoczny dla zalogowanych], y). Advantage of this new iteration formula is a formula faster than the convergence rate of Newton . In summary, there is : Theorem 2 If v (e) = X (2 ~ r, P) a X (0, P) at the initial point P find a sufficient small neighborhood is continuous, then for each A , Japan (P, A) = 0 in the iterative formula ( 7 ) of at least three convergent. Newton continuity algorithm is as follows : Given the initial value of the household , the maximum number of iterations and accuracy Im = N > 0 ,: > 0. On [0,1] to do division of M> 0, and △ A = 1 / M. It = 0, P =,,:: O. (1 ) to solve the initial problem ( 2) and (3 ) by X (t, P). By equation ( 5) by H (P, A), calculated JIH (e, A) ll, when the IIH (e, A) II N, print an error message , end ; otherwise , to ( 3); (3 ) by the Newton formula P, and to make ,::,+ 1 , to ( 1); (4 ) If a JA 1f <, print the initial value of P, the end ; otherwise , to ( 5); (5 ) that A: = A + △ A, turn (1). The following gives a solving example. Periodic solutions find the following equation :''+3 x a 0.5 (sin + COS) = 3cost +2 sint +1, n = 1,3. (8 ) is clearly g () = 3x a 0.5 (sinx + cosx) satisfy the conditions of Theorem 1 , then the equation (8 ) has a unique periodic solution r 2 sting . The initial value of P randomly selected by this algorithm , obtained the required initial point P, in order to find equation (8 ) of the periodic solution. The results shown in Figure 1 and Figure 2 . Fig. 1Numericalresultsof, l = l, P = (2.1039552,1.11 2 [1] [2] [3] [4] 2O-l-221; 0-1-2 a l0l2345672O-l-2210 A I-2 a l0l23456701234567-101234567 a l01234567,,, Nume ~ cMresultsofn = 3, P = ( 2.0208221,1.1013599 , a 1.6425722 , a 1.1438553,1.6769031,1.3199460 ) Institute of Mathematics, Jilin University would like to thank the guidance of Professor Li Yong