Periodic solution to nonlinear oscillators without artificial noises by the homotopy perturbation method

Abstract

This paper elucidates a wrong trend in solving nonlinear oscillators. The homotopy perturbation method leads to a period solution to a nonlinear oscillator with high nonlinearity, while the Parker–Sochacki method results in a chaotic one and it is invalid for solving nonlinear oscillators.

Introduction

Nonlinear vibrations arise everywhere in engineering, and it is of great importance to accurately predict its frequency. There are many analytical methods for this purpose, for example, the Parker–Sochacki method,^1,2 the homotopy perturbation method,^3–7 the variational iteration method,^8–13 the variational principle^14–17 and the fixed point theory,^18–21 just to name a few, among which the homotopy perturbation method is the most powerful tool to nonlinear oscillators, while the Parker–Sochacki method always ends in a wrong result. In this paper, we will study a nonlinear oscillator with high nonlinearity and point out some misunderstandings in the analytical approach to nonlinear oscillators.

Multiple frequencies and their lowest common multiple

In order to approximately find the frequency or period of a nonlinear oscillator, we will use multiple frequencies to approach to the exact solution. To elucidate this, we consider a general oscillator in the form

$$y^{″}\left(t\right)+y\left(t\right)+N\left(y\left(t\right)\right)=0,y\left(0\right)=A,y^{\prime }\left(0\right)=0$$

(1)where N is a nonlinear operator and A is the amplitude.

The general approach to approximate identification of its frequency is to choose the following trial solution

$$y\left(t\right)=A\cos \left(\omega t\right)$$

(2)where $\omega$ is the frequency to be determined.

In order to improve the accuracy, a multiple frequency assumption should be considered, that is we assume the solution can be expressed in the form

$$y\left(t\right)={\displaystyle \sum _{i=0}^N{a}_i\cos \left(\left(2i+1\right)\omega t\right)}$$

(3)where $a_i$ (i = 0 ∼ N) are constants to de determined late.

Recently there was a wrong trend to replace equation (3) by the following one ²

$$y\left(t\right)={\displaystyle \sum _{i=0}^N{a}_i\cos \left({\omega }_{i+1}t\right)}$$

where ${\omega }_i$ (i = 1 ∼ N − 1) are multiple frequencies.

The assumption given in equation (3) should be avoided completely in solving approximately nonlinear oscillators.

We consider the following equations

$$y\left(t\right)=0.999\cos t+0.0001\cos 3t$$

(4)

$$y\left(t\right)=0.999\cos t+0.0001\cos 0.9999t$$

(5)

The period of equation (4) is $2\pi$ , while the period of equation (5) is the lowest common multiple of $2\pi$ and $\frac{2\pi }{0.9999}=1.0001\times 2\pi$ , and its value is as large as $10001\times 2\pi$ instead of $2\pi$ . This can be understood by considering two planets with periods of $2\pi$ and $1.0001\times 2\pi$ , and it requires 10,001 circles to meet each other again.

Approximate period of a nonlinear oscillator by the homotopy perturbation method

Consider a nonlinear oscillator with fifth-order nonlinearity

$$y^{″}\left(t\right)+y\left(t\right)+\epsilon y^5\left(t\right)=0,y\left(0\right)=A,y^{\prime }\left(0\right)=0$$

(6)

By the homotopy perturbation method, we construct a homotopy equation in the form^3–7

$$y^{″}\left(t\right)+1\cdot y\left(t\right)+\epsilon p{y}^5\left(t\right)=0$$

(7)where $p\in \left[0,1\right]$ .

We assume the solution can be expressed in a power series of p

$$y\left(t\right)=y_0\left(t\right)+p{y}_1\left(t\right)+p^2{y}_2\left(t\right)+\text{L}$$

(8)

By the parameter expansion technology,^22,23 we expand the coefficient 1 of the linear term in the form

$$1={\omega }^2+p{\omega }_1+p^2{\omega }_2+\text{L}$$

(9)

Submitting equations (8) and (9) into equation (7), we have

$$\begin{array}{l}{\left[y_0\left(t\right)+p{y}_1\left(t\right)+p^2{y}_2\left(t\right)+\text{L}\right]}^{″}+\left[{\omega }^2+p{\omega }_1+p^2{\omega }_2+\text{L}\right]\\ \times \left[y_0\left(t\right)+p{y}_1\left(t\right)+p^2{y}_2\left(t\right)+\text{L}\right]y\left(t\right)+\epsilon p\left[y_0\left(t\right)+p{y}_1\left(t\right)+p^2{y}_2\left(t\right)+\text{L}\right]=0\end{array}$$

(10)

Proceeding as what is required by the homotopy perturbation method,^3–7 we have

$$y_0^{″}\left(t\right)+{\omega }^2{y}_0\left(t\right)=0,y_0\left(0\right)=A,y_0^{\prime }\left(0\right)=0$$

(11)

$$y_1^{″}\left(t\right)+{\omega }^2{y}_1\left(t\right)+{\omega }_1{y}_0+\epsilon y_0^5\left(t\right)=0,y_1\left(0\right)=0,y_1^{\prime }\left(0\right)=0$$

(12)

The other components of $y_i$ can be easily obtained and it is not necessary to write them down for simplicity. The solution of equation (11) is

$$y_0\left(t\right)=A\cos \left(\omega t\right)$$

(13)

In view of equation (13), we obtain

$$y_1^{″}\left(t\right)+{\omega }^2{y}_1\left(t\right)+{\omega }_{1}A\cos \left(\omega t\right)+\epsilon A^5{\cos }^5\left(\omega t\right)=0$$

(14)

Using the following identity

$${\cos }^5\left(\omega t\right)=\frac{10}{16}\cos \left(\omega t\right)+\frac{5}{16}\cos \left(3\omega t\right)+\frac{1}{16}\cos \left(5\omega t\right)$$

(15)

Equation (14) becomes

$$y_1^{″}\left(t\right)+{\omega }^2{y}_1\left(t\right)+\left({\omega }_{1}A+\frac{10}{16}\epsilon A^5\right)\cos \left(\omega t\right)+\frac{5}{16}\epsilon A^5\cos \left(3\omega t\right)+\frac{1}{16}\epsilon A^5\cos \left(5\omega t\right)=0$$

(16)

The periodic solution should be avoided any terms of $t\cos \left(\omega t\right)$ or $t\sin \left(\omega t\right)$ in $y_1$ , this requires

$${\omega }_{1}A+\frac{5}{8}\epsilon A^5=0$$

(17)

If we search for a first-order approximate solution, by setting p = 1 and ${\omega }_i=0$ (i > 1) in equation (9), we have

$$1={\omega }^2+{\omega }_1$$

(18)

From equations (17) and (18), we have the following approximate frequency

$$\omega =\sqrt{1+\frac{5}{8}\epsilon A^4}$$

(19)

In order to compare with the result in Abdelrazik et al., ² we choose $\epsilon =0.1$ and A = 1, and the approximate period is

$$T_{app}=\frac{2\pi }{\sqrt{1+\frac{5}{8}\times 0.1y^4\left(0\right)}}=6.095585103$$

(20)

Its exact solution is illustrated in Figure 1, with the exact period of T = 6.0907.

Figure 1.

Exact solution of equation (1) with period of T = 6.0907.

Periodic solution versus chaotic solution

This section discusses the Parker–Sochacki method for nonlinear oscillators, which was originally proposed to solve a nonlinear differential equation in a power series. ¹ This method was extended to nonlinear oscillators, ² but in vain. For a nonlinear oscillator, it is of great importance to find an accurate frequency.

Abdelrazik et al. ² obtained the following solution by the Parker–Sochacki method

$$\begin{array}{l}y\left(t\right)=0.996332\cos \left(1.03096t\right)+0.00348791\cos \left(3.19861t\right)\\ +0.000179745\cos \left(5.4452t\right)+\text{1}\text{.7071}\times {\text{10}}^{-7}\cos \left(9.40308t\right)\end{array}$$

(21)

There are four primary frequencies (1.03096, 3.19861, 5.4452 and 9.40308), and their corresponding periods are

$$T_1=\frac{2\pi }{1.03096}=6.0944996$$

(22)

$$T_2=\frac{2\pi }{3.19861}=1.964348672$$

(23)

$$T_3=\frac{2\pi }{5.4452}=1.153894312$$

(24)

$$T_4=\frac{2\pi }{9.40308}=0.6682050251$$

(25)

The period of equation (21) is the least common multiple of $T_1$ , $T_2$ , $T_3$ , and $T_4$

$$T=n_1{T}_1=n_2{T}_2=n_3{T}_3=n_4{T}_4$$

(26)where $n_1$ , $n_2$ , $n_3$ , and $n_4$ are natural numbers.

It can predict that $T\gg T_i$ (i = 1∼4). For example, the least common multiple of 1 and 1.001 is 1001.

If the solution process continues by the Parker–Sochacki method, more frequencies appear in the approximate solution, and it will become chaotic, not periodic. Most frequencies become unneeded noise.

Abdelrazik et al. ² gave a wrong direction to a nonlinear oscillator to have an approximate solution, or to track its trajectory. For a nonlinear oscillator, the most important property is the period–amplitude relationship or the frequency–amplitude relationship. The approximate solution can be generally expressed as

$$y\left(t\right)={\displaystyle \sum _{n=1}^N{A}_n\cos \left(n\omega t\right)}$$

(27)where $A_n$ (n = 1∼N) are constants. When N tends to infinity, an exact solution can be obtained. Abdelrazik et al. ² actually searched for an approximate solution in the form

$$y\left(t\right)={\displaystyle \sum _{n=1}^N{A}_n\cos \left({\omega }_{n}t\right)}$$

(28)where ${\omega }_n$ (n = 1∼N) are frequencies.

Equation (28) is a chaotic solution, not a periodic one unless ${\omega }_n$ is a multiple of a primary frequency ( ${\omega }_1$ ).

Conclusions

For a nonlinear oscillator, the frequency–amplitude relationship is of utmost importance, and there is much literature on this study, such as the He's frequency formulation,^24–29 García’s frequency formulation, ³⁰ and Suárez-Antola frequency formulation. ³¹ For conservative nonlinear oscillators, there are some even simpler methods,^27,32 while the Akbari-Ganji method is a universal method for various nonlinear vibrations. ³³ We conclude that the homotopy perturbation method is the most powerful tool to nonlinear oscillators, while the Parker-Sochacki method^1,2 is totally invalid for nonlinear vibration and should be completely avoided.