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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.

In da Silva Fernandes [

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. The first algorithm is applied to systems of two first-order differential equations corresponding to the second-order differential equation, and the second algorithm is applied to the equations of variation of parameters associated with the original equation. According to these simplified algorithms, the determination of the unknown functions

The noncanonical version of the Hori method [

Consider the differential equations:

Let the transformation of variables

Following Hori [

According to the algorithm of the perturbation method proposed by Hori [

The

The determination of the functions

Let

A corollary of this proposition can be stated.

Consider the same conditions of Proposition

Now, consider (

Equation (

To determine

In the next paragraphs, the general algorithm for determining

Introducing the

the system of partial differential equations (

The vector functions

Equations (

Finally, it should be noted that

In this section two simplified algorithms will be derived from the general algorithm in the case of nonlinear oscillations described by a second-order differential equation of the general form:

The first algorithm is applied to the system of first-order differential equations with

For completeness, we present now the first simplified algorithm [

Introducing the variables

Following the algorithm of the Hori method for noncanonical systems, one finds from zero-th-order equation, (

Applying Proposition

In view of (

Substituting (

On the other hand,

Therefore, from (

The second equation of the general algorithm, (

From (

On the other hand,

Thus, introducing (

Equations (

Finally, we note that (

It should be noted that (

Finally, we note that the general solution given by (

In this section, a second simplified algorithm is derived from the general one. Introducing the transformation of variables

The sets

It should be noted that a second transformation of variables involving a fast phase,

Now, introducing the variables

Applying Proposition

Substituting (

Equation (

Thus, it follows from (

Substituting (

It should be noted that (

Equations (

Finally, we note that (

In order to illustrate the application of the simplified algorithms, two examples are presented. The noncanonical version of the Hori method will be applied in determining second-order asymptotic solutions for van der Pol and Duffing equations. For the van der Pol equation, two different choices of the vector

The section is organized in two subsections: in the first subsection, the asymptotic solutions are determined through the first simplified algorithm, and, in the second subsection, they are determined through the second simplified algorithm.

Consider the well-known van der Pol equation:

As described in preceding paragraphs, two different choices of

Following the simplified algorithm I defined by (

Introducing the general solution given by (

Computing

From (

To determine

Following the algorithm of the Hori method described in Section

In order to obtain the same result presented by Hori [

From (

In view of the choice the auxiliary vector

Computing the indefinite integral

In order to get the same result presented by Hori [

The new system of differential equations and the generating function are given, up to the second-order of the small parameter, by

Following da Silva Fernandes [

The solution of the new system of differential equations can be obtained by introducing the solution of the above variational equations into (

The originalvariables

Now, let us to consider a different choice of the auxiliary vector

Since

Now, repeating the procedure described in the previous section, that is, computing the indefinite integral

So, the new system of differential equations and the generating function are given, up to the second-order of the small parameter

The Lagrange variational equations for the new system of differential equations, defined by (

As described in the preceding subsection, the solution of the new system of differential equations, defined by (

The original variables

Finally, note that (

Consider the well-known Duffing equation:

Thus,

Following the simplified algorithm I and repeating the procedure described in Section

In view of (

Calculating the indefinite integral

Multiplying this result by

Taking

In the second-order approximation, one finds after lengthy calculations using MAPLE software:

Repeating the procedure described in the above paragraphs, one finds

The new system of differential equations and the generating function are given, up to the second-order of the small parameter

The Lagrange variational equations for the new system of differential equations, defined by (

As described in Section

The original variables

For the van der Pol equation, the function

Following the simplified algorithm II defined by (

Taking the secular part of

Following the algorithm of the Hori method described in Section

In order to obtain the same averaged Lagrange variational equations given by (

In view of the choice of

The new system of differential equations is given, up to the second-order of the small parameter

The generating function is obtained from (

The original variables

Note that (

Now, let us to take

The functions

The new system of differential equations is obtained from (

The generating function is obtained from (

A second-order asymptotic solution for the original variables

As before, note that (

For the Duffing equation, the function

Following the simplified algorithm II and repeating the procedure described in Section

In the second-order approximation, one finds

The new system of differential equations is given, up to the second-order of the small parameter

The generating function is obtained from (

A second-order asymptotic solution for the original variables

In this paper, the Hori method for noncanonical systems is applied to theory of nonlinear oscillations. Two different simplified algorithms are derived from the general algorithm proposed by Sessin. It has been shown that the