Sage Journals: Discover world-class research

Abstract

This paper uses the two-scale fractal dimension transform and He’s formula derived from the ancient Chinese algorithm Ying Bu Zu Shu to find the approximate frequency–amplitude expression of the fractal and forced anharmonic oscillator that can be used to study the nonlinear oscillations produced by the plasma physics fractal structures. The results show how the electron frequency and wavelength change as a function of the plasma physics fractal structure. In fact, if the value of the fractal parameter is decreased, the wavelength increases, and consequently, the system frequency decreases. The introduced solution procedure sheds a bright light on the easy-to-follow steps to obtain an accurate steady-state analytical solution of fractal anharmonic nonlinear oscillators.

Keywords

Anharmonic oscillator two-scale fractal transform ancient Chinese algorithm He’s formulation plasma physics analytical frequency-amplitude response curve

Introduction

Fractal calculus application to model several physical and engineering phenomena has gained popularity since introducing the two-scale fractal transform that unveils hidden phenomena beyond the conventional continuum mechanics.^1–6 The resulting equations of motion are usually described by nonlinear ordinary differential equations whose frequency-amplitude relationships are determined using, for instance, homotopy perturbation methods,^7–12 parameter expansion method (PEM),¹³ He’s frequency formulation,^14–16 iteration perturbation method,^17,18 the fractal variational principle,^19–23 energy balance method,²⁴ energy method,²⁵ He’s frequency formulation based on energy conservation,²⁶ frequency-amplitude’s formula for non-conservative oscillators,²⁷ the enhanced homotopy perturbation method,²⁸ Taylor series,^29,30 the equivalent power-form representation,³¹ and an ancient Chinese algorithm³² to name a few.

The use of small and large scales to describe phenomena that occur in nature has attracted the attention of several researchers since Ji-Huan He introduced the two-scale dimension transform. To illustrate its applicability, let us consider lotus leaves when water drops fall on their surface. Studies show super-hydrophobic properties of the lotus leaves’ surface due to their rough texture visible for the human eye (macro scale). However, when observing the surface morphology at the nano/micro scale using scanning electron microscope images (SEM), countless protrusions are evident on the leaf surface that can change the super-hydrophobic surface property to hydrophilic when interacting with water.^33,34 This physical behavior of the water drops on the lotus leaf surface can be captured if the mathematical model considers small- and large-scale dimensions as studied in Refs. 35–38.

Furthermore, this two-scale transform has been applied to study the mechanism of snow’s insulation properties,³⁹ to investigate one-dimensional microgravity flows,²⁰ to find the frequency-amplitude expression of nonlinear oscillators in fractal space,^40,41 to study convection-diffusion processes,⁴² for electrochemical applications such as sensors,⁴³ to model solvent evaporation during fabrication of porous fibers by electrospinning,^29,30,44,45 in porous electrodes,⁴⁶ lasers,³¹ and on water collector from air^47–49 to name a few.

Therefore, this article focuses on deriving the primary resonance frequency-amplitude expression of the following anharmonic oscillator (also known as the fractal Helmhotz-Duffing equation)

\frac{d}{d S^{α}} (\frac{d y}{d S^{α}}) + α_{0} y + α_{22} y^{2} + α_{1} y^{3} = Q \cos ω_{f} S, y (0^{α}) = A, \frac{d y (0^{α})}{d S} = 0

(1)

that can be used to investigate how the resonance frequency and wavelength change as a function of the fractal parameter values of α. Here, y represents the oscillations displacement; S is the time; α is the fractal dimensions of time; α₀ represents the system natural frequency; α₁ and α₂₂ are system parameters; Q and ω_f are the external excitation force and frequency, respectively; and A is the initial displacement.

Experimental evidence suggests that equation (1) can be used to study the nonlinear oscillation produced by the plasma physics fractal structures that arise when a metal target is hit with a sufficiently intense laser pulse producing aerosol particles, which are sources of fractal aggregates from laser plasma.^50–55 In other words, equation (1) can be used to characterize the dynamics phenomena that occur when high-density plasma interacts with high-frequency electromagnetic waves.^56–59

We shall next use the two-scale fractal transform to write equation (1) in equivalent form.

The two-scale fractal transform

In order to derive the analytical fractal frequency-amplitude relationship of equation (1) using an ancient Chinese mathematical algorithm, the fractal derivative definition

\frac{d y}{d S^{α}} (S_{0}^{α}) = Γ (1 + α) {lim}_{\frac{(S - S_{0}) \to Δ t}{Δ S \neq 0}} \frac{y (S) - y (S_{0})}{{(S - S_{0})}^{α}}

(2)

and the two-scale fractal dimension transform

t = S^{α}

(3)

are introduced.^1,6 Thus, equation (1) can be written as a nonlinear differential equation of the form

\frac{d^{2} y}{d t^{2}} + α_{0} y + α_{22} y^{2} + α_{1} y^{3} = Q \cos ω_{f} t^{1 / α}, y (0^{1 / α}) = A, \frac{d}{d t^{1 / α}} y (0^{1 / α}) = 0

(4)

Equation (4) models an anharmonic oscillator that describes nonlinear phenomena that appear in physics of plasma,^53,59 thin laminated plates,⁶⁰ acoustics,⁶¹ naval engineering,⁶² eardrum oscillations,⁶³ elasto-magnetic suspensions,⁶⁴ graded beams,⁶⁵ asymmetric oscillators,⁶⁶ and electronics⁶⁷ to name a few. Notice that equation (4) has an exact solution based on Jacobi elliptic functions when and only when Q = 0.⁶⁸ For Q ≠ 0, the exact solution of equation (4) is unknown; therefore, in order to determine the angular frequency of equation (4), we use the ancient Chinese algorithm Ying Bu Zu Shu published from an ancient Chinese mathematics monograph—The Nine Chapters on the Mathematical Art.⁶⁹ In this ancient Chinese method, the approximate solutions for equation (4) are assumed to be of the form y₁(t) and y₂(t). Substitution of y₁(t) and y₂(t) into equation (4) leads to the residuals R₁(t) and R₂(t) which can be used to obtain average trail residuals ${\tilde{R}}_{1} (t)$ and ${\tilde{R}}_{2} (t)$ needed to find the frequency-amplitude expression from Ji-Huan He’s formula. Thus, in accordance with the ancient Chinese algorithm,^69–73 the steady-state trial residual functions y₁ and y₂ are assumed to be of the form

y_{i} = A_{i} \cos (ω_{i} t^{1 / α}), i = 1,2 (no sum)

(5)

where A_i and ω_i are, respectively, oscillation amplitudes and frequencies that need to be determined. Thus, the substitution of equation (5) into equation (4) yields the following trial residual functions R_i

R_{i} = \frac{d^{2} y_{i}}{d t^{2}} + α_{0} y_{i} + α_{22} y_{i}^{2} + α_{1} y_{i}^{3} - Q \cos ω_{i} t^{1 / α}

(6)

It is also assumed that ω₁ = ω_f and ω₂ = ω_f.^69–75 Expanding equation (6) gives

\begin{array}{l} R_{i} & = & A_{i}^{2} α_{22} \cos {[t^{1 / α} ω_{i}]}^{2} + 1 / 2 \cos [t^{1 / α} ω_{i}] (2 A_{i} α_{0} - 2 Q + \\ A_{i}^{3} α_{1} - 2 A_{i} t^{2 / α - 2} ω_{i}^{2} / α^{2} + A_{i}^{3} α_{1} \cos [2 t^{1 / α} ω_{i}]) + \\ A_{i} (α - 1) t^{1 / α - 2} ω_{i} \sin [t^{1 / α} ω_{i}] / α^{2} \end{array}

(7)

The next step in the ancient Chinese algorithm consists in calculating the average trail residuals ${\tilde{R}}_{1}$ and ${\tilde{R}}_{2}$ defined as

{\tilde{R}}_{i} = \frac{4}{T} \int_{0}^{T / 4} R_{i} (t) d t

(8)

where T = 2π/ω_i.^69–75 To simplify the computation of the integral (8) first, the transformation S = t^1/α is introduced. Hence, equations (7) and (8) become

\begin{array}{l} R_{i} & = & 1 / 2 \cos [S ω_{i}] (2 A_{i} α_{0} - 2 Q + A_{i}^{3} α_{1} + 2 A_{i}^{2} α_{22} \cos [S ω_{i}] + \\ A_{i}^{3} α_{1} \cos [2 S ω_{i}]) - (A_{i} S^{1 - 2 α} ω_{i} (S ω_{i} \cos [S ω_{i}] - (α - 1) \sin [S ω_{i}])) / α^{2} \end{array}

(9)

and

{\tilde{R}}_{i} = \frac{4 α}{T} \int_{0}^{{(T / 4)}^{1 / α}} R_{i} S^{(α - 1)} d S

(10)

Substitution of equation (9) into equation (10), and evaluating the integrals, we obtain the following expression for ${\tilde{R}}_{i}$

\begin{array}{l} {\tilde{R}}_{i} = 1 / T_{i}^{2} 4^{- (1 + 2 / α)} (1 / ((α - 2) α) 64 A_{i} T_{i}^{2 / α} ω_{i p}^{2} F_{q} [{1 - α / 2}, {1 / 2, 2 - α / 2}, - 4^{- 1 - 2 / α} T_{i}^{2 / α} ω_{i}^{2}] \\ - (1 / ((α - 2) α)) 64 A_{i} T_{i}^{2 / α} (α - 1) ω_{i p}^{2} F_{q} [{1 - α / 2}, {3 / 2, 2 - α / 2}, - 4^{- 1 - 2 / α} T_{i}^{2 / α} ω_{i}^{2}] \\ + 16^{1 + 1 / α} 4^{- 2} T_{i}^{2} (A_{i}^{3} α_{1 p} F_{q} [{α / 2}, {1 / 2, 1 + α / 2}, (- 9) 4^{- 1 - 2 / α} T_{i}^{2 / α} ω_{i}^{2}] \\ + {(4 A_{i} α_{0} - 4 Q + 3 A_{i}^{3} α_{1})}_{p} F_{q} [{α / 2}, {1 / 2, 1 + α / 2}, - 4^{- 1 - 2 / α} T_{i}^{2 / α} ω_{i}^{2}] \\ + 2 A_{i}^{2} α_{22} (1 +_{p} F_{q} [{α / 2}, {1 / 2, 1 + α / 2}, - 16^{- 1 / α} T_{i}^{2 / α} ω_{i}^{2}])) \end{array}

(11)

where _pF_q is the generalized hypergeometric functions.⁷⁶

The final step in the ancient Chinese algorithm consists in using He’s formula

ω_{A E}^{2} = \frac{ω_{1}^{2} {\tilde{R}}_{2} - ω_{2}^{2} {\tilde{R}}_{1}}{{\tilde{R}}_{2} - {\tilde{R}}_{1}}

(12)

to obtain an analytical fractal frequency-amplitude expression. As usual, ω₁ = 1 and ω₂ = 2.^69–75 Considering the nature of the Helmholtz-Duffing anharmonic oscillator, it is further assumed that A₁ = A and A₂ = βA where β is determined fitting the approximate backbone curve

ω_{A E} = \sqrt{\frac{12 (4 + β (α_{0} - 4) - 4 α_{0}) + 8 A^{2} (β^{3} - 4) α_{1} + 3 A π (β^{2} - 4) α_{22}}{12 (1 + β (α_{0} - 4) - α_{0}) + 8 A^{2} (β^{3} - 1) α_{1} + 3 A π (β^{2} - 1) α_{22}}}

(13)

derived from (12) with α = 1 and Q = 0, with the exact solution of the homogeneous equation (4) found in Ref. 68.

Results

This section elucidates the accuracy attained using the analytical expression of ω_AE derived using the ancient Chinese algorithm when compared to the exact solution (solid blue line) of equation (4) when Q = 0.⁶⁸

Figure 1 shows the backbone curve obtained from the exact solution of equation (4) (solid blue line)⁶⁸ and the one computed from equation (13) (dashed red line) for the system parameter values of α₀ = 1.014 82, α₁ = 0.02, and α₂₂ = 0.029 8 with β = 0.88. It is observed from Figure 1 that the theoretical backbone curve computed from equation (13) closely follows the exact backbone curve, which indicates the accuracy provided by our derived expression (13). Figure 2 illustrates the amplitude-time response curves obtained by using the exact solution of equation (4) derived in Ref. 68 and those computed from equation (13) considering the fractal parameter values of α = 0.9, 1, and 1.1. One can see from Figure 2 that the maximum oscillation amplitude is the same for all values of α; however, when α < 1 or α > 1, the oscillation frequency becomes lower or higher than that of α = 1, respectively. From Figures 1 and 2, one can conclude that the approximate frequency-amplitude and amplitude-time relationships derived using the ancient Chinese algorithm describe well the system dynamics when Q = 0.

Figure 1.

Frequency-amplitude backbone curve of the anharmonic equation for which α₀ = 1.014 8, α₁ = 0.02, α₂₂ = 0.0298, and β = 0.88. Notice that the backbone curve obtained from equation (13) closely follows the exact backbone curve computed from equation (4).

Figure 2.

Amplitude-time response curves for system parameter values of α₀ = 1.0148, α₁ = 0.02, α₂₂ = 0.0298, Q = 0, and β = 0.88. Here, the blue dashed lines describe the exact numerical solution of equation (4) obtained by using the solution derived in Ref. 68., while the color solid lines are the approximate solutions computed from equations (5) and (12).

We next plot the approximate frequency-amplitude response curves using equation (11) when an electric (restoring) force |Q| = 0.1 is acting with different fractal order values of α = 0.9, 1, and 1.1 with β = 0.88. One can see from Figure 3 that the frequency-amplitude curves shifted to the right of the curves computed with α = 1 when the fractal parameter values are bigger than 1 (α > 1), or to the left when the values of α are less than one (α < 1). Also, notice that an increase in α gives rise to a decrease in the wavelength since the oscillation frequency increases. Furthermore, if the value of the fractal parameter α is decreased, the wavelength increases, and consequently, the electron frequency decreases.⁷⁷ Figure 4 illustrates the amplitude-time curves computed using equations (5) and (13) and those obtained from equation (4) using the fourth-order RungeKutta method provided by the symbolic program of Mathematica. Notice that our analytical solution predicts the qualitative and quantitative fractal dynamic behavior of the anharmonic oscillator well.

Figure 3.

Frequency-amplitude curve of the anharmonic fractal oscillator computed from equation (12) using the values of α₀ = 1.0148; α₁ = 0.02; α₂₂ = 0.0298; Q = 0.1; β = 0.88; and α = 0.9, 1, and 1.1. Notice that the fractal frequency-amplitude curves shifted to the right of the curves computed with α = 1 when the fractal dimension values are bigger than 1 (α > 1), or to the left when the values of α are less than one (α < 1).

Figure 4.

Amplitude-time response curves for system parameter values of α₀ = 1.0148; α₁ = 0.02; α₂₂ = 0.0298; Q = −0.1; β = 0.88; and α = 0.9, 1, and 1.1. Here, the blue dashed lines describe the exact numerical solution of equation (4) obtained using the fourth-order RungeKutta method provided by the symbolic program of Mathematica, while the color solid lines are the approximate solutions computed from equations (5) and (12).

Finally, when α₁ = 0 and α₂₂ = 0 in equation (1), it becomes the classical equation used to study fractional electromagnetic waves in plasma physics.⁵⁹ In this case, the exact fractal frequency–amplitude response expression was determined by Elías-Zúñiga et al. in Ref. 78 using the Vakakis and Blanchard approach jointly with the two-scale fractal dimension transform. Following this approach, these researches avoided the complexity associated with fractional calculus providing a solution that describes physical observations well.

Conclusion

In this article, we have used the two-scale fractal calculus and He’s formula, which was developed from the ancient Chinese algorithm to derive an approximate analytical frequency-amplitude expression for the fractal Helmhotz-Duffing nonlinear differential equation that describes the nonlinear oscillations that occur in plasma physics. This analytical solution provides the fractal frequency-amplitude response curves from which it is possible to determine how the frequency and wavelength change as a function of the fractal dimension. Furthermore, the results show how the electron frequency and wavelength change as a function of the plasma physics fractal structures that arise when a metal target is hit with a sufficiently intense laser pulse. An increase in α decreases the wavelength with an increase in the electron frequency and vice versa.

The results obtained in this paper elucidate the applicability of the two-scale fractal dimension transform that unveils the critical role that the fractal parameter value α has in shifting to the left or to the right, of the backbone curve, the fractal frequency-amplitude response curves of anharmonic oscillators that model electromagnetic waves in plasma physics.

Footnotes

Acknowledgments

The authors would like to thank the financial support from Tecnológico de Monterrey-Campus Monterrey through the Research Group in Nanotechnology and Devices Design.

Authors’ contributions

AE-Z: Conceptualization, formal analysis, funding acquisition, investigation, project administration, writing—original draft, review, and editing. OM-R: Formal analysis, investigation, software, and visualization. DOT: Formal analysis, investigation, software, and visualization. LMP-P: Formal analysis, investigation, software, visualization, and writing—review.

Declaration of conflict interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Tecnológico de Monterrey funded this research through the Research Group of Nanotechnology for Devices Design, and by the Consejo Nacional de Ciencia y Tecnología de México (Conacyt), Project Numbers 242269, 255837, and 296176.

ORCID iDs

Alex Elías-Zúñiga

Luis M Palacios-Pineda

Oscar Martínez-Romero

Daniel O Trejo

References

J-H

. A tutorial review on fractal spacetime and fractional calculus. Int J Theor Phys 2014; 53: 3698–3718.

J-H

Ain

. New promises and future challenges of fractal calculus: from two-scale thermodynamics to fractal variational principle. Therm Sci 2020; 24(2A): 659–681.

Ain

J-H

. On two-scale dimension and its applications. Therm Sci 2019; 23(3): 1707–1712.

J-H

. Fractal calculus and its geometrical explanation. Results Phys 2018; 10: 272–276.

C-H

J-H

Sedighi

. Fangzhu (方诸): an ancient Chinese nanotechnology for water collection from air: history, mathematical insight, promises, and challenges. Math Methods Appl Sci 2020. DOI: 10.1002/mma.6384

J-H

Kou

, et al. Fractal oscillation and its frequency-amplitude property. Fractals 2021; 29(4): 2150105.

Sedighi

Daneshmand

. Nonlinear transversely vibrating beams by the homotopy perturbation method with an auxiliary term. J Appl Comput Mech 2015; 1(1): 1–9.

Anjum

J-H

. Two modifications of the homotopy perturbation Method for Nonlinear Oscillators. J Appl Comput Mech 2020; 6(SI): 1420–1425.

Amer

Galal

Elnaggar

. The vibrational motion of a dynamical system using homotopy perturbation technique. Appl Math 2020; 11(11): 1081–1099.

10.

J-H

Amer

Elnaggar

, et al. Periodic property and instability of a rotating pendulum system. Axioms 2021; 10: 191.

11.

J-H

El-Dib

Mady

. Homotopy perturbation method for the fractal Toda oscillator. Fractal and Fract 2021; 5(3): 93.

12.

Nadeem

J-H

. The homotopy perturbation method for fractional differential equations: part 2, two-scale transform. Int J Numer Methods Heat Fluid Flow 2021; 11(3): 3490–3504.

13.

Sedighi

. Size-dependent dynamic pull-ininstability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory. Acta Astronaut 2014; 95: 111–123.

14.

Liu

, et al. Passive atmospheric water harvesting utilizing an ancient Chinese ink slab. Facta Univ Ser Mech Eng 2021; 19(2): 229–239.

15.

Wang

. He’s frequency formulation for fractal nonlinear oscillator arising in a microgravity space. Numer Methods Partial Differ Equ 2021; 37(2): 1374–1384.

16.

Qie

Houa

J-H

. The fastest insight into the large amplitude vibration of a string. Rep Mech Eng 2020; 2(1): 1–5.

17.

. Iteration perturbation method for strongly nonlinear oscillations. J Vib Control 2001; 7(5): 631–642.

18.

Sedighi

Daneshmand

. Static and dynamic pull-in instability of multi-walled carbon nanotube probes by He’s iteration perturbation method. J Mech Sci Technol 2014; 28(9): 3459–3469.

19.

J-H

Kong

Chen

, et al. Variational iteration method for Bratu-like equation arising in electrospinning. Carbohydr Polym 2014; 105: 229–230.

20.

J-H

. A fractal variational theory for one-dimensional compressible flow in a microgravity space. Fractals 2020; 28(02): 2050024.

21.

Wang

Yao

. Fractal variational theory for Chaplygin-He Gas in a microgravity condition. Comput Methods Appl Mech Eng 2020; 6(SI): 1606–1612.

22.

Wang

Yao

Liu

, et al. A fractal variational principle for the telegraph equation with fractal derivatives. Fractals 2020; 28(04): 2050058.

23.

Wang

. A new fractal model for the soliton motion in a microgravity space. Int J Numer Methods Heat Fluid Flow 2021; 31(1): 442–451.

24.

Goharimanesh

Koochi

. Nonlinear oscillations of CNT nano-resonator based on nonlocal elasticity: the energy balance method. Rep Mech Eng 2021; 2(1): 41–50.

25.

Elías-Zúñiga

Martínez-Romero

Energy method to obtain approximate solutions of strongly nonlinear oscillators. Math Probl Engr 2013; 2013: 620591.

26.

J-H

Hou

Qie

, et al. Hamiltonian-based frequency-amplitude formulation for nonlinear oscillators. Facta Univ Ser Mech Eng 2021; 19(2): 199–208.

27.

J-H

García

. The simplest amplitude-period formula for non-conservative oscillators. Rep Mech Eng 2021; 2(1): 143–148.

28.

J-H

El-Dib

. The enhanced homotopy perturbation method for axial vibration of strings. Facta Univ Ser Mech Eng; 59: 1139–1150, DOI: 10.22190/FUME210125033H.

29.

Shen

, et al. Taylor series solution for fractal Bratu-type equation arising in electrospinning process. Fractals 2020; 28(01): 2050011.

, et al. Equivalent power-form transformation for fractal Bratu’s equation. Fractals 2021; 29(1): 21500019.

, et al. Equivalent power-form representation of the fractal Toda oscillator. Fractals 2021; 29(1): 21500341.

, et al. An efficient ancient Chinese algorithm to investigate the dynamics response of a fractal microgravity forced oscillator. Fractals 2021; 29(06): 2150144.

, et al. Superhydrophobic surfaces developed by mimicking hierarchical surface morphology of lotus leaf. Molecules 2014; 19: 4256–4283. DOI: 10.3390/molecules19044256.

, et al. Fractal N/Mems: from pull-in instability to pull-in stability. Fractals 2021; 29(2): 2150030.

35.

J-H

Sedighi

. Fangzhu: an ancient Chinese nanotechnology for water collection from air: history, mathematical insight, promises and challenges. Math Meth Appl Sci 2020: 1–10.

36.

Akgül

Ahmad

. Reproducing kernel method for Fangzhu’s oscillator for water collection from air. Math Methods Appl Sci 2020: 1–10. DOI: 10.1002/mma.6853.

, et al. Dynamics response of the forced Fangzhu fractal device for water collection from air. Fractals 2021; 29(7): 2150186.

, et al. Investigation of the fractal response of a nonlinear packaging system. Fractals 2021; 12(3): 231. DOI: 10.1142/S0218348X22500074.

. A fractal derivative model for snow’s thermal insulation property. Therm Sci 2019; 23(4): 2351–2354.

, et al. Determination of the frequency-amplitude response curves of undamped forced Duffing’s oscillators using an ancient Chinese algorithm. Res Phys 2021; 24: 104085.

, et al. Analytical solution of the fractal Cubic-Quintic Duffing equation. Fractals 2021; 29(04): 2150080.

42.

J-H

. A simple approach to one-dimensional convection-diffusion equation and its fractional modification for E reaction arising in rotating disk electrodes. J Electroanal Chem 2019; 854(1): 113565.

43.

J-H

. A fractal modification of the surface coverage model for an electrochemical arsenic sensor. Electrochim Acta 2019; 296: 491–493.

. Sudden solvent evaporation in bubble electrospinning for fabrication of unsmooth nanofibers. Therm Sci 2017; 21(4): 1827–1832.

. Solvent evaporation in a binary solvent system for controllable fabrication of porous fibers by electrospinning. Therm Sci 2017; 21: 1821–1825.

, et al. A fractal model for current generation in porous electrodes. J Electroanal Chem 2021; 880(1): 114883.

47.

Wang

. Effect of Fangzhu’s nanoscale surface morphology on water collection. Math Meth Appl Sci 2020: 1–10.

48.

J-H

El-Dib

. Homotopy perturbation method for Fangzhu oscillator. J Math Chem 2020; 58: 2245–2253.

, et al. Enhanced He’s frequency-amplitude formulation for nonlinear oscillators. Results Phys 2020; 19: 103626.

. Fractal aggregates from laser plasma. J Aerosol Sci 1989; 20(8): 865–870.

51.

Koch

Günther

. Review of the state-of-the-art of laser ablation inductively coupled plasma mass spectrometry. Appl Spectrosc 2011; 65(5): 155A–162A.

52.

Smolanov

. Fractal structure of low-temperature plasma of arc discharge as a consequence of the interaction of current sheets. J Phys Conf Ser 2016; 669: 012055.

53.

Haas

Ferreira Soares

. Large amplitude oscillations in a trapped dissipative electron gas. Phys Plasmas 2018; 25: 012310.

. New method for solving strong conservative odd parity nonlinear oscillators: applications to plasma physics and rigid rotator. AIP Adv 2020; 10: 085001.

. A new approach for modelling the damped Helmholtz oscillator: applications to plasma physics and electronic circuits. Commun Theor Phys 2021; 73: 035501.

, et al. Nonlinear dynamics of plasma oscillations modeled by an anharmonic oscillator. Phys Plasmas 2008; 15: 032308.

, et al. Analysis of vibrational resonance in biharmonically driven plasma. Chaos 2016; 26: 093117.

, et al. Nonlinear dynamics of plasma oscillations modeled by a forced modified Van der Pol-Duffing oscillator. Phys Fludyn 2013; 1308: 6132vl, arXiV.

, et al. Fractional electromagnetic waves in plasma. P Rom Acad A 2016; 17(1): 31–38.

60.

Han

Petyt

. Geometrically nonlinear vibration analysis of thin, rectangular plates using the hierarchical finite element method: 1st mode of laminated plates and higher modes of isotropic and laminated plates. Comput Struct 1997; 63(2): 295–308.

61.

Thompson

JMT

. Designing against capsize in beam seas: recent advances and new insights. Appl Mech Rev 1997; 50(5): 307–325.

62.

Spirou

Cotton

. Analytical expressions of capsize boundary of a ship with roll bias in beam waves. J Ship Res 2002; 46: 167–174.

63.

Chen

Lin

. Several numerical solution techniques for nonlinear eardrum-type oscillations. J Sound Vib 2006; 296: 1059–1067.

64.

Bonisoli

Vigliani

Passive elasto-magnetic suspensions: nonlinear models and experimental outcomes. Mech Res Comm 2007; 34: 385–394.

, et al. Accurate analytical perturbation approach for large amplitude vibration of functionally graded beams. Int J Non-linear Mech 2012; 47(5): 473–480.

66.

Elías-Zúñiga

Martínez-Romero

. Transient and steady-state responses of an asymmetric nonlinear oscillator. Math Probl Engr 2013; 2013: 574696.

, et al. Dynamics, circuitry implementation and control of an autonomous Helmholtz Jerk oscillator. J Control Autom Electr Syst 2019; 30(4): 501–511.

68.

Elías-Zúñiga

Exact solution of the quadratic mixed-parity Helmholtz-Duffing oscillator. Appl Math Comput 2012; 218: 7590–7594.

69.

J-H

. Ancient Chinese algorithm: the Ying Buzu Shu (method of surplus and deficiency) vs. Newton iteration method. Appl Math Mech 2002; 23: 1407–1412.

70.

Wang

. Generalized variational principle and periodic wave solution to the modified equal width-Burgers equation in nonlinear dispersion media. Phys Lett A 2021; 419: 127723.

71.

Wang

Zhang

. Investigation of the periodic solution of the time-space fractional Sasa-Satsuma equation arising in the monomode optical fibers. EPL 2021, DOI: 10.1209/0295-5075/ac2a62 .

72.

Wang

. Periodic solution of the time-space fractional complex nonlinear Fokas-Lenells equation by an ancient Chinese algorithm. Optik 2021; 243: 167461.

, et al. Fractal equation of motion of a non-Gaussian polymer chain: investigating its dynamic fractal response using an ancient Chinese algorithm. J Math Chem. DOI: 10.1007/s10910-021-01310-x.

74.

J-H

. An improved amplitude-frequency formulation for nonlinear oscillators. Int J Nonlinear Sci Numer Simul 2008; 9: 211–212.

75.

J-H

. Amplitude-frequency relationship for conservative nonlinear oscillators with odd nonlinearities. Int J Appl Comput Math 2017; 3: 1557–1560.

76.

Byrd

Friedman

. Handbook of elliptic integrals for engineers and physicists. Berlin, Germany: Springer-Verlag, 1953.

77.

Bhangale

Kachhia

. Fractional electromagnetic waves in plasma and dielectric media with Caputo generalized fractional derivative. Rev Mex Fís 2020; 66(6): 848–855.

, et al. Exact steady-state solution of fractals damped, and forced systems. Res Phys 2021; 28: 104580.

New analytical solution of the fractal anharmonic oscillator using an ancient Chinese algorithm: Investigating how plasma frequency changes with fractal parameter values

Abstract

Keywords

Introduction

The two-scale fractal transform

Results

Conclusion

Footnotes

Acknowledgments

Authors’ contributions

Declaration of conflict interests

Funding

ORCID iDs

References