Once again on the Samoilenko numerical-analytic method of successive periodic approximations

2009

Nonlinear Oscill

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We propose a new numerical-analytic algorithm for the investigation of periodic solutions of nonlinear autonomous systems of ordinary differential equations in the critical case.The present paper is devoted to the investigation of the existence of periodic solutions of a nonlinear autonomous system of differential equations with separated linear part, namelyin the case where all solutions of the corresponding linear differential system have a certain common period T, and the period of a solution of the nonlinear system either coincides with T or is close to it. Analogous problems were considered in numerous works (see, e.g., [1][2][3][4][5]). For the solution of this problem, one uses a modification of the numerical-analytic method [6,7], the specific feature of which lies in the fact that the restrictions imposed on the Lipschitz matrix are applied not to the entire right-hand side but only to the nonlinearity g(x). T -Periodic Solutions of a Nonlinear SystemWe consider the nonlinear autonomous system of ordinary differential equations (1) in which J is an n × n real constant matrix such that the corresponding linear homogeneous systemhas n nontrivial solutions of period T = 2Π/ν. We investigate the problem of the existence and construction of periodic solutions of system (1) whose period coincides with the period T of solutions of the corresponding linear system (2). It is clear that the eigenvalues of the matrix J are either zero or purely imaginary. Since this matrix can always be reduced to a canonical skew-symmetric matrix by a similarity transformation, we can assume, without loss of generality, thatAssume that, in the domain

On periodic solutions of nonlinear autonomous differential systems in the critical case

2009

Nonlinear Oscill

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Numerical-analytic method for the investigation of boundary-value problems for semilinear systems of differential equations

2010

Nonlinear Oscill

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We propose a new numerical-analytic algorithm for the investigation of boundary-value problems for semilinear systems of differential equations.One of branches of the theory of ordinary differential equations that have been rapidly developed in recent years is the theory of boundary-value problems. There are many works devoted to the investigation of different aspects of this theory (see, e.g., [1-9]), among which an important place is occupied by problems of the existence and approximate construction of solutions of linear and nonlinear boundary-value problems.We consider the problem of the existence and approximate construction of solutions of systems of differential equations with separated linear partwith linear boundary conditions`x D˛;˛2 R n ;where A.t / is an n n matrix with real components, A.t/ 2 C OEa; b;`is a linear vector functional,`W C OEa; b ! R n ; and˛2 R n is a constant vector. According to the Riesz theorem [10], for any linear functional`defined on the space of functions continuous on OEa; b; there exists a matrix-valued function C.t/ of bounded variation such that this linear functional can be represented in the form of a Riemann-Stieltjes integral. Therefore, the boundary conditions (2) can be rewritten in the form b Z a dC.t/ x.t/ D˛: Remark 1. Every function C.t / of bounded variation defined on OEa; b can be represented in the formwhere C 1 .t / 2 C OEa; b and C 2 .t/ is a constant function on OEa; b except for a finite or countable set of points at which C 2 .t / has discontinuities of the first kind. In the case of a finite set of points of discontinuity, problem (1), (2) becomes a linear multipoint boundary-value problem. In this case, we have C 1 .t / D 0; and the points of discontinuity of the function C 2 .t/ are the points at which the solution takes the values appearing in the boundary conditions. We assume that the following condition is satisfied:

Investigation of the periodic solutions of nonlinear autonomous systems in the critical case

2008

Ukr Math J

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517.925We analyze the conditions of existence and the numerical-analytic method for the approximate construction of periodic solutions of nonlinear autonomous systems of differential equations in the critical case.The theory of periodic boundary-value problems is extensively used in the investigation of various engineering and natural processes. For this reason, significant attention of numerous researchers is given to the problems of existence and approximate construction of periodic solutions of differential equations (see, e.g., [1 -4]). Note that an important place in this field of investigations is occupied by the analysis of autonomous systems of differential equations [5 -7].In the present paper, we continue our investigations originated in [8,9] and propose a modification of the Samoilenko numerical-analytic method aimed at the investigation of existence and approximate construction of periodic solutions of nonlinear autonomous systemsin the critical case. In this case, the restrictions imposed on the Lipschitz matrix are valid not for the entire righthand side but only for the nonlinearity.