Modified Ji-Huan He’s Frequency Formulation for Large-Amplitude Electron Plasma Oscillations

Abstract

This paper examines oscillations governed by the generic nonlinear differential equation ${u^{″}=\omega }_{p0}^2\left(1-u\mp {\displaystyle \frac{2{\beta }^2}{u^{\gamma }}}\right),$ where ${\omega }_{p0},\beta$ and $\gamma$ are positive constants. The aforementioned differential equation is of particular importance, as it describes electron plasma oscillations influenced by temperature effects and large oscillation amplitudes. Since no analytical solution exists for the oscillation period in terms of ${\omega }_{p0},\beta ,\gamma$ and the oscillation amplitude, accurate approximations are derived. A modified He’s approach is used to account for the non-symmetrical oscillation around the equilibrium position. The motion is divided into two parts: $u_{min}\le u<u_{eq}$ and $u_{eq}<u\le u_{max}$, where $u_{min}$ and $u_{max}$ are the minimum and maximum values of $u$, and $u_{eq}$ is its equilibrium value. The time intervals for each part are calculated and summed to find the oscillation period. The proposed method shows remarkable accuracy compared to numerical results. The most significant result of this paper is that He’s approach can be readily extended to strongly non-symmetrical nonlinear oscillations. It is also demonstrated that the same approach can be extended to any case where each segment of the function $f\left(u\right)$ in the differential equation $u^{″}+f\left(u\right)=0$ (for $u_{min}\le u<u_{eq}$ and for $u_{eq}<u\le u_{max}$) can be approximated by a fifth-degree polynomial containing only odd powers.

1. Introduction

Nonlinear oscillations appear across a broad range of scientific fields, including mechanics and engineering [1,2,3,4,5,6], physics [7,8,9,10,11], and mathematics [12,13,14,15]. These oscillations are generally modeled using nonlinear differential equations (NDEs). Due to their significant nonlinearity, NDEs pose considerable challenges, prompting substantial interest in developing accurate approximation methods. Over recent decades, various approaches have been proposed to solve NDEs. Traditional perturbation methods, however, have proven inadequate for strongly nonlinear oscillators, as they fail to provide sufficiently precise approximations. To address these limitations, numerous analytical methods have been investigated. Techniques applied include the harmonic balance method [16,17], variational iteration method [18,19], homotopy analysis method [20,21], homotopy perturbation method [22,23], Li-He’s modified homotopy perturbation method [24], enhanced homotopy perturbation method [25], asymptotic method [26,27], energy balance method [28], differential transformation method [29], parameter expansion method [30,31], variational principle [32,33,34], global residue harmonic balance method [35,36], spreading residue harmonic balance method [37], and frequency–amplitude formulation [38], among others. Additionally, significant efforts have been devoted to developing general equations that can address a broad spectrum of nonlinear oscillation cases. For example, some researchers have proposed approximate solutions for nonlinear oscillations involving cubic and harmonic restoring forces [39,40]. This case is highly significant, as it can describe a wide range of phenomena in mathematics, physics, and engineering [40,41]. It serves as a general model for many specific cases, including the Duffing equation [42], the simple pendulum [43], the cubic–quintic Duffing oscillator [44], and the capillary oscillator [45,46,47]. The central idea behind the above solutions was first introduced by He [44].

More specifically, consider a nonlinear differential equation in the following general form:

$${\displaystyle \frac{d^{2}x}{d{t}^2}}+f\left(x\right)=0.$$

(1)

He proposed that the angular frequency could be approximated with the following equation [44]:

$${\omega }_{He\left(1\right)}={\left({\left.{\displaystyle \frac{df}{dx}}\right|}_{x={\displaystyle \frac{A}{2}}}\right)}^{1/2},$$

(2)

where A is the oscillation’s amplitude. This approach has gained significant popularity due to its simplicity. However, it has been shown to be accurate only for small amplitudes or in cases where the function f(x) is described by a third-degree polynomial (or can be approximated by a third-degree polynomial using Taylor series expansions) [41,42]. For other cases (e.g., the case of large amplitudes of the cubic–quintic Duffing oscillator), He proposed the alternative equation below [44]:

$${\omega }_{He\left(2\right)}={\left[{\displaystyle \frac{1}{3}}\left({\left.{\displaystyle \frac{df}{dx}}\right|}_{x=0.3A}+{\left.{\displaystyle \frac{df}{dx}}\right|}_{x=0.5A}+{\left.{\displaystyle \frac{df}{dx}}\right|}_{x=0.7A}\right)\right]}^{1/2}.$$

(3)

Equation (3) has been shown to be accurate for any amplitude in the case of the cubic–quintic oscillator, resulting in a small error of approximately 3% [44]. However, it is important to note that Equations (2) and (3) were derived for symmetrical oscillators, meaning oscillators whose equilibrium position lies at the center of the oscillatory motion.

However, this is not always the case. In this paper, a generic differential equation presented in plasma physics, which describes a non-symmetrical oscillation around an equilibrium position, will be considered. Consider the scenario of an electrically neutral plasma in equilibrium, composed of a gas of positively charged ions and negatively charged electrons. When a small displacement occurs among a group of electrons (relative to the ions), the Coulomb force acts as a restoring force, pulling the electrons back. In this situation, the charge density oscillates at the plasma frequency [48]:

$${\omega }_{p0}=\sqrt{{\displaystyle \frac{n_0{e}^2}{m{\epsilon }_0}}},$$

(4)

where $n_0$ is the density of the electrons at the equilibrium state, $e$ and $m$ are the electron charge and mass, respectively, and ${\epsilon }_0$ is the vacuum permeability. Assuming perturbations in one direction and denoting the electron density at an arbitrary state as n, we can define the dimensionless variable $u=n_0/n$. The oscillation of charge density can then be described by the following differential equation [48]:

$${\displaystyle \frac{d^{2}u}{{dt}^2}}={\omega }_{p0}^2\left(1-u\right).$$

(5)

Equation (5) is applicable only when thermal effects are neglected. To offer a straightforward model that incorporates thermal effects for large amplitude undamped oscillations, the following equation has been derived [48,49]:

$${\displaystyle \frac{d^{2}u}{{dt}^2}}={\omega }_{p0}^2\left(1-u\mp {\displaystyle \frac{2{\beta }^2}{u^{\gamma }}}\right)$$

(6)

In Equation (6), $\gamma$ is a positive constant, while $\beta$is a dimensionless parameter that accounts for thermal effects. Specifically, $={\lambda }_D/L$, where $L$ represents a characteristic distance and ${\lambda }_D$ denotes the Debye length [48]. The parameter $\beta$ must always satisfy $\beta <1$ [48,49]. There is no analytical solution for Equation (6) and, additionally, there is no analytical equation for the oscillation period. The goal of this paper is to use He’s idea to derive a simple equation for accurately calculating the period of the oscillation described by Equation (6). Two cases will be considered due to their physical significance: the case where $\gamma =1$ and the case where $\gamma =3$ (collisionless adiabatic case) [48]. In addition, only the values of the parameter β that lead to oscillatory motion will be considered.

The motivation behind this paper was to find elementary solutions for oscillations that are not symmetric about the equilibrium position. Equation (6) serves as an ideal case for this purpose. More specifically, it will be shown that for large values of the parameter β, there is significant asymmetry (i.e., the oscillation’s equilibrium position is not at the center of the domain $u_{min}\le u\le u_{max}$), leading to substantial differences in the time intervals required for the u variable to change from $u_{min}$ to the equilibrium position, and then from the equilibrium position to $u_{max}$. Finding simple equations to determine the oscillation period in these cases is significant, as it represents a step forward in the search for global elementary solutions to strong nonlinear oscillations.

This paper is organized as follows: In Section 2, the exact numerical solutions for the oscillation period when γ = 1 are presented. Subsequently, in Section 3, a modified version of He’s approach is introduced to calculate the oscillation period, demonstrating excellent accuracy in all cases. In Section 4, the numerical and analytical results derived from the proposed method are also presented for the case where γ = 3. In Section 5 (Discussion), further considerations regarding the rationale behind the proposed method, as well as additional applications of the method, are presented.

2. Exact Numerical Solutions for γ = 1

First, the procedure for obtaining the exact numerical solutions for the oscillation period will be presented. The first case to be examined is the one with $\gamma =1$. Let $v\left(u\right)={\displaystyle \frac{du}{dt}},$ then by the chain rule, ${\displaystyle \frac{d^{2}u}{d{t}^2}}={\displaystyle \frac{dv}{dt}}={\displaystyle \frac{dv}{du}}{\displaystyle \frac{du}{dt}}$. Therefore, the differential Equation (6) is written as follows [48]:

$$v{\displaystyle \frac{dv}{du}}={\omega }_{p0}^2\left(1-u\mp {\displaystyle \frac{2{\beta }^2}{u}}\right)\Rightarrow vdv={\omega }_{p0}^2\left(1-u\mp {\displaystyle \frac{2{\beta }^2}{u}}\right)du\Rightarrow$$

$$\int vdv=\int {\omega }_{p0}^2\left(1-u\mp {\displaystyle \frac{2{\beta }^2}{u}}\right)du\Rightarrow {\displaystyle \frac{v^2}{2}}=-{\omega }_{p0}^2\left[{\displaystyle \frac{{\left(u-1\right)}^2}{2}}\pm 2{\beta }^{2}lnu\right]+c\Rightarrow$$

$${\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{du}{dt}}\right)}^2+{\omega }_{p0}^2\left[{\displaystyle \frac{{\left(u-1\right)}^2}{2}}\pm 2{\beta }^{2}lnu\right]=c,$$

(7)

where $c$ is the integration constant. Equation (7) represents the conservation of energy. The potential energy is defined as below [48]:

$${V\left(u\right)=\omega }_{p0}^2\left[{\displaystyle \frac{{\left(u-1\right)}^2}{2}}\pm 2{\beta }^{2}lnu\right].$$

(8)

By considering as initial conditions $u\left(0\right)=u_{max}$ and ${\displaystyle \frac{du\left(0\right)}{dt}}=0$, $c=V\left(u_{max}\right)$ and by solving with respect to time, the oscillation’s period is calculated as follows:

$$T={\displaystyle \frac{2}{{\omega }_{p0}}}\underset{u_{min}}{\overset{u_{max}}{\int }}{\displaystyle \frac{1}{\sqrt{2V\left(u_{max}\right)-2\left[\frac{{\left(u-1\right)}^2}{2}\pm 2{\beta }^{2}lnu\right]}}}du.$$

(9)

It is important to note that the oscillation period is defined only under the following condition:

$$V\left(u_{max}\right)-\left[{\displaystyle \frac{{\left(u-1\right)}^2}{2}}\pm 2{\beta }^{2}lnu\right]\ge 0.$$

(10)

If Equation (10) does not hold, then Equation (6) does not have periodic solutions. By using Equation (10), the limits of the oscillation can be easily calculated. The procedure is to consider Equation (10) as an equality. For example, consider $u_{max}=1.35$ and $\beta =0.26$ as presented in [48] for the upper positive sign in Equations (7)–(10). By considering Equation (10) as an equality we conclude that $u_{max}=1.35$ and $u_{min}=0.204487$. In addition, when considering $\beta =0.27$, the function $u=f\left(t\right)$ is not periodic. The lower limits of the oscillation when considering $u_{max}=1.35$ and $0\le \beta \le 0.25$ are presented in

Table 1. (γ = 1) The case of the upper negative sign in Equation (6). Comparison between the accurate numerical results (Equation (9)) and proposed approximations (1st approximation, Equation (17) and 2nd approximations (Equations (19)–(21)).

$\frac{{\mathit{d}}^2\mathit{u}}{{\mathit{d}\mathit{t}}^2}={\mathit{\omega }}_{\mathit{p}0}^2\left(1-\mathit{u}-\frac{2{\mathit{\beta }}^2}{\mathit{u}}\right)\Rightarrow \frac{1}{2}{\left(\frac{\mathit{d}\mathit{u}}{\mathit{d}\mathit{t}}\right)}^2+{\mathit{\omega }}_{\mathit{p}0}^2\left[\frac{{\left(\mathit{u}-1\right)}^2}{2}+2{\mathit{\beta }}^2\mathit{l}\mathit{n}\mathit{u}\right]=\mathit{V}\left({\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}\right),$ $\mathbf{where}{\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}=1.35$
$\mathit{\beta }$	${\mathit{u}}_{\mathit{m}\mathit{i}\mathit{n}}$	${\mathit{u}}_{\mathit{e}\mathit{q}}$	${\mathit{T}}_{\mathit{n}\mathit{u}\mathit{m}.}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{T}}_{1\mathit{s}\mathit{t}\mathit{a}\mathit{p}\mathit{p}.}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{\epsilon }}_1(\%)$	${\mathit{T}}_{2\mathit{n}\mathit{d}\mathit{a}\mathit{p}\mathit{p}.}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{\epsilon }}_2(\%)$
0	0.65	1	2π	2π	0	2π	0
0.02	0.648328	0.99920	6.2826	6.2860	0.0541	6.2860	0.0541
0.04	0.643287	0.99679	6.2941	6.2943	0.0032	6.2946	0.0079
0.06	0.634808	0.99275	6.3088	6.3087	0.0016	6.3092	0.0063
0.08	0.622765	0.98703	6.3290	6.3295	0.0079	6.3305	0.0237
0.10	0.606966	0.97958	6.3583	6.3577	0.0094	6.3595	0.0189
0.12	0.587134	0.97032	6.3963	6.3947	0.0250	6.3977	0.0219
0.14	0.562867	0.95913	6.4407	6.4425	0.0279	6.4473	0.1025
0.16	0.533573	0.94587	6.5053	6.5042	0.0169	6.5121	0.1045
0.18	0.498349	0.93035	6.5869	6.5853	0.0243	6.5984	0.1746
0.20	0.455598	0.91231	6.7053	6.6951	0.1521	6.7182	0.1924
0.22	0.402806	0.89141	6.8731	6.8530	0.2924	6.8974	0.3536
0.24	0.332835	0.86715	7.1758	7.1098	0.9198	7.2146	0.5407
0.25	0.284787	0.85355	7.4731	7.3258	1.9711	7.5182	0.6035

. The phase diagrams for the case of the upper positive sign in Equation (7) are presented in Figure 1a(i–iii) for $\beta =0.25,0.26,\mathrm{a}\mathrm{n}\mathrm{d}0.27$. Periodic solutions occur only when the phase diagram is a closed loop. The phase diagrams for the case with the negative sign in Equations (7)–(10) are also presented in Figure 1b(i–iii). The exact numerical results regarding the oscillation’s period (with respect to ${\omega }_{p0}$) using Equation (9) are presented in Table 1 and

Table 2. (γ = 1) The case of the positive sign in Equation (6). Comparison between the accurate numerical results (Equation (9)) and proposed approximation (Equation (17)).

$\frac{{\mathit{d}}^2\mathit{u}}{{\mathit{d}\mathit{t}}^2}={\mathit{\omega }}_{\mathit{p}0}^2\left(1-\mathit{u}+\frac{2{\mathit{\beta }}^2}{\mathit{u}}\right)\Rightarrow \frac{1}{2}{\left(\frac{\mathit{d}\mathit{u}}{\mathit{d}\mathit{t}}\right)}^2+{\mathit{\omega }}_{\mathit{p}0}^2\left[\frac{{\left(\mathit{u}-1\right)}^2}{2}-2{\mathit{\beta }}^2\mathit{l}\mathit{n}\mathit{u}\right]=\mathit{V}\left({\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}\right),$ $\mathbf{where}{\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}=1.35$
$\mathit{\beta }$	${\mathit{u}}_{\mathit{m}\mathit{i}\mathit{n}}$	${\mathit{u}}_{\mathit{e}\mathit{q}}$	${\mathit{T}}_{\mathit{n}\mathit{u}\mathit{m}.}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{T}}_{1\mathit{s}\mathit{t}\mathit{a}\mathit{p}\mathit{p}.}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{\epsilon }}_1(\%)$
0	0.65	1	2π	2π	0
0.02	0.651669	1.0008	6.2779	6.2804	0.0398
0.04	0.656653	1.0032	6.2689	6.2723	0.0542
0.06	0.664886	1.0072	6.2591	6.2590	0.0016
0.08	0.676266	1.0126	6.2398	6.2411	0.0208
0.10	0.690660	1.0196	6.2156	6.2189	0.0531
0.12	0.707910	1.0280	6.1906	6.1933	0.0436
0.14	0.727848	1.0378	6.1609	6.1647	0.0617
0.16	0.750298	1.0488	6.1303	6.1338	0.0571
0.18	0.775085	1.0611	6.1004	6.1012	0.0131
0.20	0.802041	1.0745	6.0638	6.0673	0.0577
0.22	0.831003	1.0889	6.0293	6.0326	0.0547
0.24	0.861821	1.1043	5.9936	5.9975	0.0651
0.25	0.877881	1.1124	5.9796	5.9799	0.0050
0.26	0.894355	1.1206	5.9607	5.9622	0.0252
0.30	0.964073	1.1557	5.8921	5.8922	0.0017
0.35	1.058830	1.2046	5.8045	5.8077	0.0551
0.40	1.160770	1.2550	5.7246	5.7148	0.1712
0.45	1.268800	1.3093	5.6492	5.6507	0.0266

.

3. Approximate Solutions for γ = 1

In this section, a modified version of He’s approach will be used. To account for the non-symmetry of the u-values around the equilibrium position, we will proceed as follows. Firstly, the equilibrium position will be determined for different $\beta$-values (for $\beta$ $\le$ 0.26) using the equation below:

$$f\left(u\right)=0{\Rightarrow -\omega }_{p0}^2\left(1-u_{eq}\mp {\displaystyle \frac{2{\beta }^2}{u_{eq}}}\right)=0\Rightarrow 1-u_{eq}\mp {\displaystyle \frac{2{\beta }^2}{u_{eq}}}=0,$$

(11)

where

$$f\left(u\right)=-{\omega }_{p0}^2\left(1-u\mp {\displaystyle \frac{2{\beta }^2}{u}}\right).$$

(12)

The equilibrium position in each case is shown in Table 1 and Table 2. The oscillation will be divided into two parts. In the first part, the case $u_{eq}<u\le u_{max}$ will be considered. The time interval required for the u variable to change from $u_{eq}$ to $u_{max}$ and then back to $u_{eq}$ will be approximated as follows:

$$t_1=\pi /{\omega }_{1,}$$

(13)

where

$${{\omega }_1=\left({\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{eq}+A_1/2}\right)}^{1/2},$$

(14)

and $A_1=u_{max}-u_{eq}$ is the amplitude of this part of the oscillation. The second part is for $u_{min}\le u<u_{eq}$. The time interval required for the u variable to change from $u_{eq}$ to $u_{min}$ and then back to $u_{eq}$ will be approximated as follows:

$$t_2={\displaystyle \frac{\pi }{{\omega }_2}},$$

(15)

where

$${{\omega }_2=\left({\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{min}+A_2/2}\right)}^{1/2},$$

(16)

and $A_2=u_{eq}-u_{min}$ is the amplitude of this part of the oscillation. Therefore, a first approximation of the oscillation period is expressed as follows:

$$T_{1stapp.}={\displaystyle \frac{\pi }{{\left({\left.\frac{df}{du}\right|}_{u=u_{eq}+A_1/2}\right)}^{1/2}}}+{\displaystyle \frac{\pi }{{\left({\left.\frac{df}{du}\right|}_{u=u_{min}+A_2/2}\right)}^{1/2}}}.$$

(17)

For the case of the upper negative sign in Equation (12), ${\displaystyle \frac{df}{du}}=1-{\displaystyle \frac{2{\beta }^2}{u^2}}$ and for the positive sign ${\displaystyle \frac{df}{du}}=1+{\displaystyle \frac{2{\beta }^2}{u^2}}$. The percentage differences between the accurate numerical results and those provided by the approximation proposed in this paper are calculated using the equation below:

$$\epsilon \left(\%\right)=\left|\left.{\displaystyle \frac{{\rm T}-T_{app.}}{{\rm T}}}\right|\right.100\%.$$

(18)

The results are presented in Table 1 and Table 2. For the case of the upper negative sign in Equation (12) (which is the case where the oscillation is significantly non-symmetric, as shown in Figure 1a(ii)), this approach is accurate for $\beta \le 0.25$. For $\beta =0.25$ the error is ${\epsilon }_1\cong 1.97\%$. For $\beta =0.26$, the error becomes ${\epsilon }_1\cong 8.41\%$.

To improve the accuracy of the proposed method, an alternative procedure based on Equation (3) is also presented:

$$T_{2ndapp.}={\displaystyle \frac{\pi }{{\omega }_1^{\prime }}}+{\displaystyle \frac{\pi }{{\omega }_2^{\prime }}},$$

(19)

where

$${\left.{\omega }_1^{\prime }=\left\{\left({\displaystyle \frac{1}{3}}\right)\left[{\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{eq}+{0.25A}_1}+{\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{eq}+{0.5A}_1}+{\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{eq}+{0.75A}_1}\right]\right.\right\}}^{{\displaystyle \frac{1}{2}}},$$

(20)

and

$${\left.{\omega }_2^{\prime }=\left\{\left({\displaystyle \frac{1}{3}}\right)\left[{\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{min}+{0.25A}_2}+{\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{min}+{0.5A}_2}+{\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{min}+{0.75A}_2}\right]\right.\right\}}^{{\displaystyle \frac{1}{2}}}.$$

(21)

Using Equations (19)–(21), the error is significantly reduced for the case of the upper negative sign in Equation (12). For $\beta =0.26$, ${\epsilon }_2\cong 1.63\%$ (Table 1). For the case of the positive sign in Equation (12), Equation (17) yields excellent accuracy for $0\le \beta \le 0.26$ (Table 2). For $\beta =0.26$, ${\epsilon }_2\cong 0.025\%$ therefore, there is no need for a 2nd approximation. The results presented in Table 1 and Table 2 are also presented comparatively in Figure 2 and Figure 3.

It is also important to note that for the case of the positive sign in Equation (6), there are oscillatory solutions for larger values of β compared to the negative sign in Equation (6). Therefore, the accuracy of the proposed method for very large β-values is also presented in Table 2. To ensure oscillatory motion when $u_{max}=1.35$, only cases with $\beta \le 0.45$ were considered. For example, for β = 0.5, when setting Equation (12) equal to zero, it is concluded that $u_{eq}=1.36603$, which is greater than $u_{max}$. Therefore, there is no physical significance in this case.

4. The Case of γ = 3

The procedures presented in Section 2 and Section 3 can be also applied for the case for which γ = 3. In this case, the differential Equation (6) is modified as follows:

$${\displaystyle \frac{d^{2}u}{{dt}^2}}={\omega }_{p0}^2\left(1-u\mp {\displaystyle \frac{2{\beta }^2}{u^3}}\right).$$

(22)

The conservation of energy results in [48] the following:

$${\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{du}{dt}}\right)}^2+{\omega }_{p0}^2\left[{\displaystyle \frac{{\left(u-1\right)}^2}{2}}\mp {\displaystyle \frac{{\beta }^2}{u^2}}\right]=V\left(u_{max}\right),$$

(23)

where

$$V\left(u\right)={\displaystyle \frac{{\left(u-1\right)}^2}{2}}\mp {\displaystyle \frac{{\beta }^2}{u^2}}.$$

(24)

Therefore, the oscillation’s period is calculated as

$$T={\displaystyle \frac{2}{{\omega }_{p0}}}\underset{u_{min}}{\overset{u_{max}}{\int }}{\displaystyle \frac{1}{\sqrt{2V\left(u_{max}\right)-2\left[\frac{{\left(u-1\right)}^2}{2}\mp \frac{{\beta }^2}{u^2}\right]}}}du.$$

(25)

A periodic solution exists only if

$$V\left(u_{max}\right)-\left[{\displaystyle \frac{{\left(u-1\right)}^2}{2}}\mp {\displaystyle \frac{{\beta }^2}{u^2}}\right]\ge 0.$$

(26)

For the upper negative sign in Equation (22), a periodic solution exists only for $\beta \le 0.14$. The proposed approach (Equations (17)–(21)) is also applied in this case. More specifically,

$$f\left(u\right)=0{\Rightarrow -\omega }_{p0}^2\left(1-u_{eq}\mp {\displaystyle \frac{2{\beta }^2}{u_{eq}^3}}\right)=0\Rightarrow 1-u_{eq}\mp {\displaystyle \frac{2{\beta }^2}{u_{eq}^3}}=0,$$

(27)

where

$$f\left(u\right)=-{\omega }_{p0}^2\left(1-u\mp {\displaystyle \frac{2{\beta }^2}{u^3}}\right).$$

(28)

The maximum value of the parameter u is considered to be equal to 1.35 in all cases. The u-value at the equilibrium position is calculated using Equation (27). The minimum u-value is calculated by considering Equation (26) as an equality. The results are presented in

Table 3. (γ = 3) The case of the upper negative sign in Equation (22). Comparison between the accurate numerical results (Equation (25)) and proposed approximations (1st approximation, Equation (17) and 2nd approximation (Equations (19)–(21)).

$\frac{{\mathit{d}}^2\mathit{u}}{{\mathit{d}\mathit{t}}^2}={\mathit{\omega }}_{\mathit{p}0}^2\left(1-\mathit{u}-\frac{2{\mathit{\beta }}^2}{{\mathit{u}}^3}\right)\Rightarrow \frac{1}{2}{\left(\frac{\mathit{d}\mathit{u}}{\mathit{d}\mathit{t}}\right)}^2+{\mathit{\omega }}_{\mathit{p}0}^2\left[\frac{{\left(\mathit{u}-1\right)}^2}{2}-\frac{{\mathit{\beta }}^2}{{\mathit{u}}^2}\right]=\mathit{V}\left({\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}\right),$ $\mathbf{where}{\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}=1.35$
$\mathit{\beta }$	${\mathit{u}}_{\mathit{m}\mathit{i}\mathit{n}}$	${\mathit{u}}_{\mathit{e}\mathit{q}}$	${\mathit{T}}_{\mathit{n}\mathit{u}\mathit{m}.}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{T}}_{1\mathit{s}\mathit{t}\mathit{a}\mathit{p}\mathit{p}.}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{\epsilon }}_1(\%)$	${\mathit{T}}_{2\mathit{s}\mathit{t}\mathit{a}\mathit{p}\mathit{p}.}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{\epsilon }}_2(\%)$
0	0.65	1	2π	2π	0	2π	0
0.02	0.647911	0.9992	6.2919	6.2934	0.023837	6.2941	0.0350
0.04	0.641503	0.9968	6.3263	6.3252	0.017389	6.3283	0.0316
0.06	0.630308	0.9926	6.3850	6.3827	0.036028	6.3908	0.0908
0.08	0.613345	0.9867	6.4830	6.4745	0.131198	6.4923	0.1433
0.10	0.588593	0.9787	6.6384	6.6201	0.276049	6.6583	0.2993
0.12	0.551076	0.9683	6.9260	6.8701	0.810374	6.9629	0.5314

and

Table 4. (γ = 3) The case of the positive sign in Equation (22). Comparison between the accurate numerical results (Equation (25)) and proposed approximation (Equation (17)).

$\frac{{\mathit{d}}^2\mathit{u}}{{\mathit{d}\mathit{t}}^2}={\mathit{\omega }}_{\mathit{p}0}^2\left(1-\mathit{u}+\frac{2{\mathit{\beta }}^2}{{\mathit{u}}^3}\right)\Rightarrow \frac{1}{2}{\left(\frac{\mathit{d}\mathit{u}}{\mathit{d}\mathit{t}}\right)}^2+{\mathit{\omega }}_{\mathit{p}0}^2\left[\frac{{\left(\mathit{u}-1\right)}^2}{2}+\frac{{\mathit{\beta }}^2}{{\mathit{u}}^2}\right]=\mathit{V}\left({\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}\right),$ $\mathbf{where}{\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}=1.35$
$\mathit{\beta }$	${\mathit{u}}_{\mathit{m}\mathit{i}\mathit{n}}$	${\mathit{u}}_{\mathit{e}\mathit{q}}$	${\mathit{T}}_{\mathit{n}\mathit{u}\mathit{m}.}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{T}}_{1\mathit{s}\mathit{t}\mathit{a}\mathit{p}\mathit{p}.}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{\epsilon }}_1(\%)$
0	0.65	1	2π	2π	0
0.02	0.652067	1.0008	6.2712	6.2732	0.0319
0.04	0.658140	1.0032	6.2439	6.2441	0.0032
0.06	0.667872	1.0071	6.1956	6.1990	0.0549
0.08	0.680775	1.0123	6.1364	6.1414	0.0814
0.10	0.696315	1.0189	6.0697	6.0754	0.0939
0.12	0.713982	1.0266	5.9965	6.0044	0.1317
0.14	0.733326	1.0353	5.9239	5.9312	0.1232
0.16	0.753975	1.0449	5.8502	5.8577	0.1281
0.18	0.775628	1.0557	5.7767	5.7857	0.1557
0.20	0.798050	1.0660	5.7035	5.7147	0.1962
0.22	0.821055	1.0774	5.6352	5.6465	0.2003
0.24	0.844501	1.0892	5.5697	5.5811	0.2045
0.25	0.856353	1.0952	5.5409	5.5494	0.1533
0.26	0.868277	1.1012	5.5098	5.5183	0.1542
0.30	0.916499	1.1261	5.3928	5.4011	0.1539
0.35	0.977472	1.1578	5.2647	5.2689	0.0798
0.40	1.038730	1.1899	5.1398	5.1512	0.2218
0.45	1.099970	1.2220	5.0405	5.0461	0.1111

.

Subsequently, the first approximation for the oscillation period is obtained using Equation (17) for the upper negative sign in Equation (28). In this case, ${\displaystyle \frac{df}{du}}=1-{\displaystyle \frac{6{\beta }^2}{u^4}}$. In addition, for the case of the positive sign in Equation (28), ${\displaystyle \frac{df}{du}}=1+{\displaystyle \frac{6{\beta }^2}{u^4}}$. The results between the accurate numerical solutions (Equation (25)) and those obtained using the approximate methods presented in this paper are shown comparatively in Table 3 and Table 4 and Figure 4 and Figure 5 along with the percentage differences. For the case of the upper negative sign in Equation (22), the first approximation yields an error of approximately 4.5% for $\beta =0.14$. The error is minimized when using the second approximation (~0.76%). For the positive sign in Equation (22) the first approximation yields small errors.

5. Discussion

An interesting question that may arise at this point is why this simple approach is so effective. Since He’s approach was first used for the cubic–quintic oscillator, let us consider the following nonlinear differential equation to answer this critical question:

$${\displaystyle \frac{d^{2}u}{{dt}^2}}+{\omega }_{p0}^2\left[c_1\left(u-u_{eq}\right)+c_2{\left(u-u_{eq}\right)}^3+c_3{\left(u-u_{eq}\right)}^5\right]=0,$$

(29)

where $c_1,c_2,\mathrm{a}\mathrm{n}\mathrm{d}{c}_3$ are arbitrary constants. In addition, consider a trivial solution of the following form:

$$u=u_{eq}+Acos\left(\omega t\right),$$

(30)

By substituting Equation (30) to Equation (29) it is concluded that

$$-{\omega }^{2}A\cos \left(\omega t\right)+{\omega }_{p0}^2\left[c_{1}Acos\left(\omega t\right)+c_2{A}^3{\cos }^3\left(\omega t\right)+c_3{A}^3{\cos }^5\left(\omega t\right)\right]=0.$$

(31)

Using a series expansion of the trigonometric functions and omitting terms higher than the first order, we conclude that ${\cos }^3\left(\omega t\right)\approx {\displaystyle \frac{3}{4}}\mathrm{c}\mathrm{o}\mathrm{s}(\omega t)$ and ${\cos }^5\left(\omega t\right)\approx {\displaystyle \frac{5}{8}}\mathrm{c}\mathrm{o}\mathrm{s}(\omega t)$ [50]. Therefore, Equation (31) is simplified as follows:

$$-{\omega }^{2}A\cos \left(\omega t\right)+{\omega }_{p0}^2\left[c_{1}Acos\left(\omega t\right)+{\displaystyle \frac{3}{4}}{c}_2{A}^3\cos \left(\omega t\right)+{\displaystyle \frac{5}{8}}{c_{3}A}^5\cos \left(\omega t\right)\right]=0,$$

(32)

By solving with respect to the angular frequency ω,

$$\omega ={\omega }_{p0}{\left(c_{1}A+{\displaystyle \frac{3}{4}}{c}_2{A}^2+{\displaystyle \frac{5}{8}}{c_{3}A}^4\right)}^{1/2}.$$

(33)

On the other hand, using He’s approach for Equation (29),

$${\omega }_{He}={\left({\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{eq}+{\displaystyle \frac{A}{2}}}\right)}^{{\displaystyle \frac{1}{2}}}={\omega }_{p0}{\left[{\left.c_1+3c_2{\left(u-u_{eq}\right)}^2+5c_3{\left(u-u_{eq}\right)}^4\right|}_{u=u_{eq}+{\displaystyle \frac{A}{2}}}\right]}^{{\displaystyle \frac{1}{2}}}={\omega }_{p0}{\left(c_{1}A+{\displaystyle \frac{3}{4}}{c}_2{A}^2+{\displaystyle \frac{5}{16}}{c_{3}A}^4\right)}^{1/2}$$

(34)

The only difference between Equations (33) and (34) is the coefficient of the third term: 5/8 when using the trivial solution (30) and the Fourier series approximation, and 5/16 when using He’s approach. However, for small amplitudes (i.e., A < 0.5) the term ${c_{3}A}^4$ becomes small compared to $c_2{A}^2$ (assuming that $c_3$ is not significantly larger that $c_2)$; therefore, the difference in the coefficient has a small influence on the result. Therefore, Equation (34) becomes an excellent approximate solution for small amplitudes for Equation (29).

In conclusion, He’s approach is a reliable approximation for differential equations that can be expressed in the form of Equation (29) and for small oscillation amplitudes. In Figure 6, equations of the form $g\left(u\right)=c_1\left(u-u_{eq}\right)+c_2{\left(u-u_{eq}\right)}^3+c_3{\left(u-u_{eq}\right)}^5$ are fitted to the data represented by the function $g\left(u\right)=-1+u+{\displaystyle \frac{2{\beta }^2}{u}}$ for $\beta =0.26$, which is the strongest nonlinear case (as shown in Figure 1a(ii)). For $u_{eq}\le u\le u_{max}$,

$$g\left(u\right)=0.8311\left(u-u_{eq}\right)+0.3463{\left(u-u_{eq}\right)}^3-0.6205{\left(u-u_{eq}\right)}^5.$$

(35)

The R-squared coefficient in this case resulted in 1.0000. For $u_{min}\le u\le u_{eq}$,

$$g\left(u\right)=0.7522\left(u-0.83882\right)-0.2838{\left(u-0.83882\right)}^3-2.547{\left(u-0.83882\right)}^5$$

(36)

In this case, the R-squared coefficient was 0.9982. The same considerations apply to any values of the parameters β and γ used in this paper. The fact that each part of the function $f\left(u\right)={\omega }_{p0}^2\left(-1+u\pm {\displaystyle \frac{2{\beta }^2}{u^{\gamma }}}\right)$, for $\mathrm{u}>u_{eq}$ and $u<u_{eq}$, can be approximated by a fifth-degree polynomial (with only odd powers) provides a reasonable explanation for the effectiveness of He’s modified approach in this paper.

In addition, for cases of very strong nonlinearity, Equations (19)–(21) demonstrated higher accuracy compared to Equation (17). According to Table 1, Equation (17) underestimates the oscillation period (and consequently overestimates the angular frequency) for large amplitudes. On the other hand, Equations (19)–(21) demonstrated excellent accuracy in these cases. This result can be also easily explained using the analogy with Equation (29). In particular,

$${\left.\omega =\left\{\left({\displaystyle \frac{1}{3}}\right)\left[{\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{eq}+{0.25A}_1}+{\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{eq}+{0.5A}_1}+{\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{eq}+{0.75A}_1}\right]\right.\right\}}^{{\displaystyle \frac{1}{2}}}={\omega }_{p0}{\left(c_{1}A+0.875c_2{A}^2+0.638{c_{3}A}^4\right)}^{1/2}.$$

(37)

The higher coefficients, 0.875 and 0.638, in the second and third terms of Equation (37), compared to 3/4 and 5/16 in Equation (34), lead to accurate results due to the negative signs of the $c_2$ and $c_3$ coefficients in Equation (36) ($c_2=-0.2838$, $c_3=-2.547$). In other words, by using Equation (34),

$${\omega }_{1stapp.}={\omega }_{p0}{\left(0.7522A-{\displaystyle \frac{3}{4}}0.2838A^2-{\displaystyle \frac{5}{8}}{2.547A}^4\right)}^{1/2}.$$

(38)

On the contrary, when using Equation (38),

$${\omega }_{2ndapp.}={\omega }_{p0}{\left(0.7522A-0.875·0.2838A^2-0.638·{2.547A}^4\right)}^{1/2}.$$

(39)

Therefore, ${\omega }_{2ndapp.}<{\omega }_{1stapp.}$. It is also important to note that the modified, ‘more symmetric’ Equation (37) was used instead of the initially derived Equation (3) because it provided better accuracy for large β-values. In conclusion, by using Equations (19)–(21), we successfully minimize the underestimation of the oscillation period for large amplitudes. Another interesting aspect to consider is the case of the positive sign in Equation (6). As clearly demonstrated in Table 2 and Table 4, there is no significant non-symmetry in this case (i.e., the equilibrium position is approximately at the midpoint between the maximum and minimum u values). Therefore, a reasonable question is whether the classic He’s approach can be applied directly to such cases. To evaluate the aforementioned idea, the angular frequency will be calculated using the following equation:

$${\mathit{\omega }}_{\mathit{H}\mathit{e}}={\left({\left.{\displaystyle \frac{\mathit{d}\mathit{f}}{\mathit{d}\mathit{u}}}\right|}_{{\mathit{u}}_{\mathit{e}\mathit{q}}+{\displaystyle \frac{\mathit{A}}{2}}}\right)}^{1/2},$$

(40)

where $A=u_{max}-u_{eq}$.

Table 5. (γ = 1) The case of the positive sign in Equation (6): A comparison between the accurate numerical results (Equation (9)) and the proposed approximations (Equations (17) and (40)). The percentage differences between the numerical results and Equation (17) (${\epsilon }_{1stapp-num}(\%)$), as well as between the numerical results and Equation (40) (${\epsilon }_{He-num}(\%)$), are also presented.

$\frac{{\mathit{d}}^2\mathit{u}}{{\mathit{d}\mathit{t}}^2}={\mathit{\omega }}_{\mathit{p}0}^2\left(1-\mathit{u}+\frac{2{\mathit{\beta }}^2}{\mathit{u}}\right)$$({\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}=1.35$)
$\mathit{\beta }$	${\mathit{T}}_{\mathit{n}\mathit{u}\mathit{m}.}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{T}}_{1\mathit{s}\mathit{t}\mathit{a}\mathit{p}\mathit{p}.}\times {\mathit{\omega }}_{\mathit{p}0}$	$\mathit{A}={\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}-{\mathit{u}}_{\mathit{e}\mathit{q}}$	${\mathit{T}}_{\mathit{H}\mathit{e}}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{\epsilon }}_{1\mathit{s}\mathit{t}\mathit{a}\mathit{p}\mathit{p}-\mathit{n}\mathit{u}\mathit{m}}(\%)$	${\mathit{\epsilon }}_{\mathit{H}\mathit{e}-\mathit{n}\mathit{u}\mathit{m}}(\%)$
0	2π	2π	0.3500	2π	0	0
0.02	6.2779	6.2804	0.3492	6.2814	0.0398	0.0558
0.04	6.2689	6.2723	0.3468	6.2759	0.0542	0.1117
0.06	6.2591	6.2590	0.3428	6.2670	0.0016	0.1262
0.08	6.2398	6.2411	0.3374	6.2546	0.0208	0.2372
0.10	6.2156	6.2189	0.3304	6.2389	0.0531	0.3749
0.12	6.1906	6.1933	0.3220	6.2201	0.0436	0.4765
0.14	6.1609	6.1647	0.3122	6.1985	0.0617	0.6103
0.16	6.1303	6.1338	0.3012	6.1743	0.0571	0.7177
0.18	6.1004	6.1012	0.2889	6.1476	0.0131	0.7737
0.20	6.0638	6.0673	0.2755	6.1188	0.0577	0.9070
0.22	6.0293	6.0326	0.2611	6.0882	0.0547	0.9769
0.24	5.9936	5.9975	0.2457	6.0558	0.0651	1.0378
0.25	5.9796	5.9799	0.2376	6.0391	0.0050	0.9950
0.26	5.9607	5.9622	0.2294	6.0221	0.0252	1.0301
0.30	5.8921	5.8922	0.1943	5.9512	0.0017	1.0030
0.35	5.8045	5.8077	0.1454	5.8587	0.0551	0.9338
0.40	5.7246	5.7148	0.0950	5.7631	0.1712	0.6725
0.45	5.6492	5.6507	0.0407	5.6675	0.0266	0.3239

presents the case of the positive sign in Equation (6) with $\gamma =1$. Equation (40) provides accurate results for any amplitude.

The same approach was also applied for the case of γ = 3 with the positive sign in Equation (6) (

Table 6. (γ = 3) The case of the positive sign in Equation (6): A comparison between the accurate numerical results (Equation (25)) and the proposed approximations (Equations (17) and (40)). The percentage differences between the numerical results and Equation (17) (${\epsilon }_{1stapp-num}(\%)$), as well as between the numerical results and Equation (40) (${\epsilon }_{He-num}(\%)$), are also presented.

$\frac{{\mathit{d}}^2\mathit{u}}{{\mathit{d}\mathit{t}}^2}={\mathit{\omega }}_{\mathit{p}0}^2\left(1-\mathit{u}+\frac{2{\mathit{\beta }}^2}{\mathit{u}}\right)$$({\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}=1.35$)
$\mathit{\beta }$	${\mathit{T}}_{\mathit{n}\mathit{u}\mathit{m}.}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{T}}_{1\mathit{s}\mathit{t}\mathit{a}\mathit{p}\mathit{p}.}\times {\mathit{\omega }}_{\mathit{p}0}$	$\mathit{A}={\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}-{\mathit{u}}_{\mathit{e}\mathit{q}}$	${\mathit{T}}_{\mathit{H}\mathit{e}}\times {\mathit{\omega }}_{\mathit{p}0}$	${\mathit{\epsilon }}_{1\mathit{s}\mathit{t}\mathit{a}\mathit{p}\mathit{p}-\mathit{n}\mathit{u}\mathit{m}}(\%)$	${\mathit{\epsilon }}_{\mathit{H}\mathit{e}-\mathit{n}\mathit{u}\mathit{m}}(\%)$
0	2π	2π	0.3500	2π	0	0
0.02	6.2712	6.2732	0.3492	6.2792	0.0319	0.1276
0.04	6.2439	6.2441	0.3468	6.2675	0.0032	0.3780
0.06	6.1956	6.1990	0.3429	6.2483	0.0549	0.8506
0.08	6.1364	6.1414	0.3377	6.2221	0.0814	1.3966
0.10	6.0697	6.0754	0.3311	6.1895	0.0939	1.9737
0.12	5.9965	6.0044	0.3234	6.1513	0.1317	2.5815
0.14	5.9239	5.9312	0.3147	6.1082	0.1232	3.1111
0.16	5.8502	5.8577	0.3051	6.0609	0.1281	3.6016
0.18	5.7767	5.7857	0.2943	6.0103	0.1557	4.0438
0.20	5.7035	5.7147	0.2840	5.9565	0.1962	4.4359
0.22	5.6352	5.6465	0.2726	5.9007	0.2003	4.7115
0.24	5.5697	5.5811	0.2608	5.8433	0.2045	4.9123
0.25	5.5409	5.5494	0.2548	5.8142	0.1533	4.9324
0.26	5.5098	5.5183	0.2488	5.7847	0.1542	4.9893
0.30	5.3928	5.4011	0.2239	5.6657	0.1539	5.0605
0.35	5.2647	5.2689	0.1922	5.5164	0.0798	4.7809
0.40	5.1398	5.1512	0.1601	5.3699	0.2218	4.4768
0.45	5.0405	5.0461	0.1280	5.2283	0.1111	3.7258

). Also in this case, Equation (40) provides reliable results in most cases. However, for large β-values, there are non-negligible errors ranging from 3.5% to 5.1%.

As an ongoing development, the modified He’s frequency formulation can be extended to address fractal or fractional oscillators. For instance, the methods presented in this paper can also be applied to other cases of non-symmetric oscillators. For example, let us consider the case of the fractal Toda oscillator [51,52,53,54,55,56]:

$${\displaystyle \frac{d^{2}u}{d{t}^2}}+e^u-1=0.$$

(41)

By integrating Equation (40) and using the initial conditions $u\left(0\right)=A$, $u^{\prime }\left(0\right)=0$ we obtain

$${\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{du}{dt}}\right)}^2+e^u-u=e^{u_{max}}-u_{max}.$$

(42)

The period of the oscillation is

$$T=2\underset{u_{min}}{\overset{u_{max}}{\int }}{\displaystyle \frac{1}{\sqrt{2}\sqrt{e^{u_{max}}-u_{max}-e^u+u}}}du.$$

(43)

Provided that

$$e^{u_{max}}-u_{max}-e^u+u\ge 0.$$

(44)

Considering $u_{max}=0.5,$ we can easily calculate $u_{min}$ by considering Equation (44) as an equality. Therefore, $u_{min}=-0.5998$. For $u_{min}=-0.5998$ and $u_{max}=0.5$, Equation (43) results in $T=6.3613$. On the other hand, the method presented in this paper can also be used alternatively. In particular, $f\prime \left(u\right)=e^u$ and $u_{eq}=0$.

Therefore,

$${\left.{\omega }_1^{\prime }=\left\{\left({\displaystyle \frac{1}{3}}\right)\left[{\left.{\displaystyle \frac{df}{du}}\right|}_{u=0.25A_1}+{\left.{\displaystyle \frac{df}{du}}\right|}_{u=0.5A_1}+{\left.{\displaystyle \frac{df}{du}}\right|}_{u=0.75A_1}\right]\right.\right\}}^{{\displaystyle \frac{1}{2}}},$$

(45)

and

$${\left.{\omega }_2^{\prime }=\left\{\left({\displaystyle \frac{1}{3}}\right)\left[{\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{min}+0.25A_2}+{\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{min}+0.5A_2}+{\left.{\displaystyle \frac{df}{du}}\right|}_{u=u_{min}+0.75A_2}\right]\right.\right\}}^{{\displaystyle \frac{1}{2}}},$$

(46)

where $A_1=u_{max}$ and $A_2={|u}_{min}|$.

Finally, by using Equation (19),

$$T_{2ndapp.}={\displaystyle \frac{\pi }{{\omega }_1^{\prime }}}+{\displaystyle \frac{\pi }{{\omega }_2^{\prime }}}=6.3324.$$

(47)

The percentage difference between the numerical solution $T=6.3606$ and the approximate solution $T_{2ndapp.}=6.3324$ is 0.44%.

It is also worth noting that the derivation of single-step frequency formulations for nonlinear oscillators has garnered significant attention within the nonlinear science research community [57,58,59,60]. In a major contribution to the field, He and Liu [61] provided a mathematical analysis of the frequency formulation, emphasizing that the proposed one-step frequency equation is not only mathematically valid but also offers valuable insights into the fundamental physical processes involved. In addition to the general Equations (2) and (3), several recent advances of the method have been presented for solving various cases of nonlinear oscillators. In particular, an interesting recent advancement of Ji-Huan He’s formulation is discussed in Reference [57]. Consider the following nonlinear oscillator with odd nonlinearities:

$${\displaystyle \frac{d^{2}u}{d{t}^2}}+\sum _{n=0}^N{a}_{2n+1}{u}^{2n+1}=0,$$

(48)

where $a_{2n+1}$ are constants. Using Ji-Huan He’s recent formulation [57] the angular frequency is provided below:

$${\omega }^2=a_1+\sum _{n=1}^N{a}_{2n+1}{b}_{2n+1}{A}^{2n},$$

(49)

where

$$b_{2n+1}={\displaystyle \frac{3}{2n+2}}.$$

(50)

The modified Equations (49) and (50) encompass a variety of important cases, such as the Duffing oscillator, the cubic–quintic Duffing oscillator, and the mathematical pendulum, among others [62,63,64,65]. The one-step frequency formulation for nonlinear oscillators offers an efficient alternative for solving such systems. This approach streamlines the analysis and enables faster acquisition of approximate solutions, removing the need for lengthy iterative calculations. By simplifying complex computations, the frequency formulation has become a gold-standard technique for approximating the frequency of most nonlinear oscillators.

6. Conclusions

This paper focused on the general differential Equation (6), which describes one-dimensional nonlinear electron plasma oscillations for large amplitudes. Although an analytical solution is not available, it was shown that by modifying He’s approach to account for the oscillation’s asymmetry, highly accurate approximate equations for the oscillation period can be derived. This is a very interesting result that shows that the simple He’s formulas (Equations (2) and (3)) have many unexplored applications in many areas of physics. In particular, He’s approach can be equally applied to non-symmetrical oscillations (i.e., oscillations where the time interval required to move from one extreme position to the equilibrium position is significantly different from the time interval required to move from the equilibrium position to the opposite extreme position). In addition, the method presented in this paper can also be applied to other intriguing cases, such as the fractal Toda oscillator. The straightforward application and simplicity of this approach could, with appropriate modifications, evolve it into a gold-standard technique for approximating the period in most cases of undamped nonlinear oscillations.