Abstract
Some remarks on the application of the Hori method in the theory of nonlinear oscillations are presented. Two simplified algorithms for determining the generating function and the new system of differential equations are derived from a general algorithm proposed by Sessin. The vector functions which define the generating function and the new system of differential equations are not uniquely determined, since the algorithms involve arbitrary functions of the constants of integration of the general solution of the new undisturbed system. Different choices of these arbitrary functions can be made in order to simplify the new system of differential equations and define appropriate near-identity transformations. These simplified algorithms are applied in determining second-order asymptotic solutions of two well-known equations in the theory of nonlinear oscillations: van der Pol equation and Duffing equation.
Highlights
In da Silva Fernandes 1, the general algorithm proposed by Sessin 2 for determining the generating function and the new system of differential equations of the Hori method for noncanonical systems has been revised considering a new approach for the integration theory which does not depend on the auxiliary parameter t∗ introduced by Hori 3, 4
For the Duffing equation, only one generating function is determined, and the second simplified algorithm gives the same generating function obtained through Krylov-Bogoliubov method
It has been shown that the mth-order terms Tj m and Zj∗ m that define the near-identity transformation and the new system of differential equations, respectively, are not uniquely determined, since the algorithms involve at each order arbitrary functions of the constants of integration of the general solution of the undisturbed system
Summary
In da Silva Fernandes 1 , the general algorithm proposed by Sessin 2 for determining the generating function and the new system of differential equations of the Hori method for noncanonical systems has been revised considering a new approach for the integration theory which does not depend on the auxiliary parameter t∗ introduced by Hori 3, 4 In this paper, this new approach is applied to the theory of nonlinear oscillations for a second-order differential equation and two simplified versions of the general algorithm are derived. Brief descriptions of the noncanonical version of the Hori method 4 and the general algorithm proposed by Sessin 2 are presented in the two sections
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