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Abstract

This work studies the nonlinear fractional dynamics of asymmetric harmonic oscillators. The classical description of the physical system is generalized using the principles of fractional variational analysis. As a system of two-coupled fractional differential equations with a quadratic nonlinear component, the fractional Euler–Lagrange equations of the motion of the corresponding system are obtained. The Adams–Bashforth predictor-corrector numerical approach is used to approximate the system’s outcomes, which are then simulated comparatively with respect to various model parameter values, including mass, linear and quadratic nonlinear stiffness, and the order of the fractional derivative. The simulations provided the possibility of investigating various dynamical behaviours within the same physical model that is generalized by the use of fractional operators.

Keywords

Fractional calculus Euler–Lagrange formulation quadratic nonlinear harmonic oscillator numerical approximation

Introduction

The subject of nonlinear dynamics has a lengthy history deeply connected with mathematics and physics. It is recommended that interested readers to look at two important review papers.^1,2 In Ref. 3, the author was the first who approach the problem of a nonlinear oscillator methodically, starting with the linear oscillator and looking at the impact of the stiffness nonlinearities with quadratic and cubic stiffness. In addition to taking damping into account, he emphasized the distinctions between linear and nonlinear oscillators for both forced and free vibration.³

The asymmetric oscillator model is assumed to take the following form:

\ddot{x} + d_{1} x + d_{2} x^{2} = 0 .

(1)

Here,

d_{1}, d_{2}

are positive parameters. In the asymmetric case, the amplitude in the positive x-axis differs from that along the negative direction.

In literature, many efforts have been made to study nonlinear oscillators^4–12 where advanced methods and numerical solutions are developed in Refs. 12–17. In Refs. 14–17, the generalized formulations for some interesting models of nonlinear dynamics are performed via the use of fractional variational analysis. The main principles of fractional variations of calculus are derived in Ref. 18 with applications to fractional optimal control problems (Refs. 18 and 19 and see the references therein). Most commonly analytical solutions are not probable to get for nonlinear differential equations due to the nonlinear terms (quadratic, cubic or even higher). Therefore, efficient numerical techniques are necessary to be performed to overcome this difficulty and to get an approximate result.

The dynamical models in the literature are commonly constructed based on integer-order derivative and integral operators which may cause large errors to be embedded into the system. The generalization of these operators to non-integer order, so-called fractional calculus, gives the advantage of reducing the error and contribution of memory to the evolvement of the dynamical systems (see Refs. 20 and 21 and more inside). Therefore, fractional calculus provides an important tool for the modelling of complex phenomena appearing naturally in diverse problems of science and engineering, see Refs. 22–27 and references therein for an overview of possible application areas. The range of numerical methods for the solutions of fractional models are reviewed in the book²⁸ and higher order ones are initiated in papers.^29,30

Preliminaries

The dynamics of the physical system investigated throughout this manuscript are constructed based on the Caputo fractional derivative and integral operators whose definitions are briefly stated below.^20,21

Definition 1

The Caputo fractional derivatives (CFDs) of order $α \in C (R (α) > 0)$ of a differentiable function $f (t)$ are defined by

\begin{array}{l} ({}_{a}^{C}{D_{t}^{α} f}) (t) ≔ \frac{1}{Γ (1 - α)} \int_{a}^{t} \frac{f^{'} (τ) d τ}{{(t - τ)}^{α}}, & (left - sided CFD) \end{array}

\begin{array}{l} ({}_{t}^{C}D_{b}^{α} f) (t) ≔ \frac{- 1}{Γ (1 - α)} \int_{t}^{b} \frac{f^{'} (τ) d τ}{{(τ - t)}^{α}}, & (right - sided CFD) \end{array}

where

- \infty < a < t < b < \infty

Definition 2

The Caputo fractional integrals (CFIs) of order $α \in C (R (α) > 0)$ of a function $f (t)$ are given by

\begin{array}{l} ({}_{a}^{C}{I_{t}^{α} f}) (t) ≔ \frac{1}{Γ (α)} \int_{a}^{t} \frac{f (τ) d τ}{{(τ - t)}^{1 - α}}, & (left - sided CFI) \end{array}

\begin{array}{l} ({}_{t}^{C}I_{b}^{α} f) (t) ≔ \frac{- 1}{Γ (α)} \int_{t}^{b} \frac{f (τ) d τ}{{(t - τ)}^{1 - α}} . & (right - sided CFI) \end{array}

In the next section, a one-dimensional asymmetric Harmonic oscillator (HO) with quadratic nonlinearity term is described initially in the classical sense. In the next section, the dynamics of the system are reconstructed using fractional derivative and integral operators for a generalization. In this direction, the fractional Euler–Lagrange equation of the regarding system is produced being a system of two-coupled fractional differential equations with a nonlinear term. The last is devoted to the numerical solutions and the comparative simulations obtained from the classical case and fractional case of the system. The effect of the model parameters such as mass, linear and quadratic nonlinear stiffness and order of the fractional derivative are investigated through these simulations. Finally, we close the paper with a conclusion.

The physical description of the system

Consider a one-dimensional asymmetric Harmonic oscillator (HO) with quadratic nonlinearity term. The classical Hamiltonian $(C_{H})$ of this oscillator reads:

C_{H} = \frac{p^{2}}{2 m} + \frac{1}{2} k_{1} x^{2} + \frac{1}{2} k_{2} x^{3} .

(2)

Upon substituting equation (2) into the relations

\frac{\partial C_{H}}{\partial x} = - \dot{p}

, and

\dot{x} = \frac{\partial C_{H}}{\partial p}

one gets:

\frac{\partial C_{H}}{\partial x} = - \dot{p} \Rightarrow k_{1} x + \frac{3}{2} k_{2} x^{2} = - \dot{p},

(3)

\dot{x} = \frac{\partial C_{H}}{\partial p} \Rightarrow \dot{x} = \frac{p}{m} .

(4)

Combining equations (3) and (4), we got the following classical Hamilton’s equation (CHE)

\ddot{x} + ω_{o}^{2} x + \frac{3}{2} ω_{1}^{2} x^{2} = 0 .

(5)

The classical Lagrangian $(C_{L})$ is related to the classical Hamiltonian $(C_{H})$ as $C_{H} = \dot{x} p - C_{L}$ . Thus, using equations (2) and (4) we have

C_{L} = \frac{1}{2} m {\dot{x}}^{2} - \frac{1}{2} k_{1} x^{2} - \frac{1}{2} k_{2} x^{3} .

(6)

Now, to obtain the Classical Euler–Lagrange Equation (CELE) known also as equations of motion, substituting (6) into the formula $\frac{d}{d t} (\frac{\partial C_{L}}{\partial \dot{x}}) - \frac{\partial C_{L}}{\partial x} = 0$ produce that

\ddot{x} + ω_{o}^{2} x + \frac{3}{2} ω_{1}^{2} x^{2} = 0,

(7)

where

ω_{o} = \sqrt{\frac{k_{1}}{m}}

is the angular frequency of the HO due to the linear term.

ω_{1} = \sqrt{\frac{k_{2}}{m}}

is the angular frequency of the HO due to the quadratic nonlinear term.

We note that equations (7) and (5) are identical. This means that CELE and CHE are identical to each other.

Generalization of the system dynamics using fractional operators

In this section, based on the descriptions of fractional-order differentiation and integration operators stated in Section 2, we start firstly by fractionalizing the classical Lagrangian of the system given in equation (6). In fractional form it reads:

F_{L} = \frac{1}{2} m {({{}_{a}D}_{t}^{α} x)}^{2} - \frac{1}{2} k_{1} x^{2} - \frac{1}{2} k_{2} x^{3} .

(8)

Next, the fractional Euler–Lagrange equation (FELE) can be obtained by applying (8) into the equation $\frac{\partial F_{L}}{\partial q_{i}} + {{}_{t}D}_{b}^{α} \frac{\partial F_{L}}{\partial {{}_{a}D}_{t}^{α} q_{i}} + {{}_{a}D}_{t}^{β} \frac{\partial F_{L}}{\partial {{}_{t}D}_{b}^{β} q_{i}} = 0$ , and as a result we have

{{}_{t}D}_{b}^{α} {{}_{a}D}_{t}^{α} x - ω_{o}^{2} x - \frac{3}{2} ω_{1}^{2} x^{2} = 0 .

(9)

It is important to mention in deriving equation (9), we follow the formalism presented in Ref. 18 which enables us to derive the FELE’s. It is based on the fractional integration by parts formula.

Moreover, we aim to derive the fractional Hamilton’s equation (FHE) of motion. In this regard, we define the following generalized momenta $p_{α}$ , $p_{β}$ such that

p_{α} = \frac{\partial F_{L}}{\partial {{}_{a}D}_{t}^{α} x} = m {{}_{a}D}_{t}^{α} x .

(10)

p_{β} = \frac{\partial F_{L}}{\partial {{}_{t}D}_{b}^{α} x} = 0 .

(11)

The fractional Hamiltonian, which is defined as $F_{H} = p_{α} {{}_{a}D}_{t}^{α} x + p_{β} {{}_{t}D}_{b}^{α} x - F_{L}$ reads

F_{H} = \frac{1}{2} m {({{}_{a}D}_{t}^{α} x)}^{2} + \frac{1}{2} k_{1} x^{2} + \frac{1}{2} k_{2} x^{3} .

(12)

Now the fractional Hamilton’s equation (FHE) can be obtained as

\frac{\partial F_{H}}{\partial x} = {{}_{t}D}_{b}^{α} p_{α} + {{}_{a}D}_{t}^{α} p_{β} \Rightarrow {{}_{t}D}_{b}^{α} {{}_{a}D}_{t}^{α} x - ω_{o}^{2} x - \frac{3}{2} ω_{1}^{2} x^{2} = 0 .

(13)

It is clear that (13) and (9) are identical, and the FHE and the FELE are reduced to CELE as $α \to 1$ .

Numerical scheme and simulation results

In this part, the numerical solution of the generalized one-dimensional asymmetric Harmonic oscillator (HO) with quadratic nonlinearity term is presented. The corresponding FELE given by equation (9) is solved approximately using the scheme so-called the fractional Adams–Bashforth–Moulton predictor-corrector method^29,30 and simulations are performed concerning some important model parameters.

Consider in general, the following nonlinear fractional differential equation of order $α > 0$ within the derivative operator defined in Defn.1.:

{}_{0}^{c}D_{t}^{α} Y (t) = F (t, Y (t)), t > 0, Y^{l} (0) = Y_{0}^{l}; l = 0, 1, \dots, ([α] - 1),

(14)

which has the unique solution

Y

defined on some interval

[0, T]

as (see Ref. 30 for details)

\sum_{m = 0}^{[α] - 1} Y_{0}^{(m)} \frac{t^{m}}{m!} + \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} F (τ, Y (τ)) d τ .

(15)

In Refs. 29 and 30, equation (15) is integrated numerically using the fractional predictor-corrector scheme. The discretization over the interval

[0, T]

using

(N + 1)

nodes gives the step-size

h = \frac{T}{N}

so that at each time point

t_{n} = n h, n = 0, 1, \dots, N

the approximated version of equation (15) leads to

Y_{h} (t_{n + 1}) = \sum_{m = 0}^{[α] - 1} Y_{0}^{(m)} \frac{t_{n + 1}^{m}}{m!} + \frac{h^{α}}{Γ (α)} F (t_{n + 1}, Y_{h}^{p} (t_{n + 1})) + \frac{h^{α}}{Γ (α + 2)} \sum_{i = 0}^{n} a_{i, n + 1} F (t_{i}, Y_{h} (t_{i})),

(16) where

a_{i, n + 1} = {\begin{cases} n^{α + 1} - (n - α) {(n + α)}^{α}, i f i = 0, \\ {(n - i + 2)}^{α + 1} + {(n - i)}^{α + 1} - 2 {(n - i + 1)}^{α + 1}, i f 0 < i \leq n, \\ 1, i f i = n + 1, \end{cases}

and the predicted value

Y_{h}^{p} (t_{n + 1})

is obtained by

Y_{h}^{p} (t_{n + 1}) = \sum_{m = 0}^{[α] - 1} Y_{0}^{(m)} \frac{t_{n + 1}^{m}}{m!} + \frac{1}{Γ (α)} \sum_{i = 0}^{n} b_{i, n + 1} F (t_{i}, Y_{h} (t_{i})),

with

b_{i, n + 1} = \frac{h^{α}}{α} ({(n + 1 - i)}^{α} - {(n - i)}^{α}

In the figures presented below, namely, Figures 1–9, the approximate solution of the FELE in equation (9) is obtained within the initial conditions $x (0) = 1$ and ${}_{0}^{c}{D_{t}^{α} x (0) = 0}$ and simulated for the following scenarios:

Figure 1.

Dynamical behaviours for $ω_{1}^{2} = ω_{0}^{2} = 0.01$ .

Figure 2.

Dynamical behaviours for $ω_{1}^{2} = ω_{0}^{2} = 0.1$ .

Figure 3.

Dynamical behaviours for $ω_{1}^{2} = ω_{0}^{2} = 1$ .

Figure 4.

Dynamical behaviours for $ω_{1}^{2} = {2 ω}_{0}^{2}$ with $ω_{0}^{2} = 0.01$ .

Figure 5.

Dynamical behaviours for $ω_{1}^{2} = {2 ω}_{0}^{2}$ with $ω_{0}^{2} = 0.1$ .

Figure 6.

Dynamical behaviours for $ω_{1}^{2} = {2 ω}_{0}^{2}$ with $ω_{0}^{2} = 1$ .

Figure 7.

Dynamical behaviours for $ω_{0}^{2} = {2 ω}_{1}^{2}$ with $ω_{1}^{2} = 0.01$ .

Figure 8.

Dynamical behaviours for $ω_{0}^{2} = {2 ω}_{1}^{2}$ with $ω_{1}^{2} = 0.1$ .

Figure 9.

Dynamical behaviours for $ω_{0}^{2} = {2 ω}_{1}^{2}$ with $ω_{1}^{2} = 1$ .

Case 1: $ω_{1}^{2} = ω_{0}^{2} = 0.01, 0.1, 1$ (presented by Figures 1–3)

Case 2: $ω_{1}^{2} = 2 ω_{0}^{2}$ with $ω_{0}^{2} = 0.01, 0.1, 1$ (presented by Figures 4–6)

Case 3: $ω_{0}^{2} = 2 ω_{1}^{2}$ with $ω_{1}^{2} = 0.01, 0.1, 1$ (presented by Figures 7–9)

Figures 1 through 9 display the numerical results of the discretized FELE via equally spaced meshing. These figures demonstrate how the fractional order has a major impact on the behavior of fractional Euler–Lagrange equations. All figures have been generated on a 64-bit computer equipped with a 12th Gen Intel(R) Core(TM) i7-1255U processor and 16 GB of RAM, using MATLAB (R2023b) for code implementation.

The figures above show that, as predicted, the fractional solution converges to the classical case as $α$ approaches 1. More importantly, they reveal distinct dynamical behaviours in the fractional case depending on the value of $α$ . This feature is absent in the classical case, offering an additional degree of freedom that improves the ability to capture and analyse different aspects of real-world phenomena. Thus, the fractional Lagrangian method provides an effective model for representing various properties of the physical system under study.

Conclusion

This paper studies the generalized asymmetric harmonic oscillator with quadratic nonlinearity based on fractional variational principles. Here, the Caputo definition is used for fractional integration and differentiation operators. The values of the order of the Caputo fractional derivatives are considered as the cases $= 0.75, 0.80, 0.85, 0.90, 0.95, 1$ , where $α = 1$ represents the classical (integer order) case. Moreover, three distinct cases are investigated for physical part of the considered problem, such as $ω_{1}^{2} = ω_{0}^{2} = 0.01, 0.1, 1$ (presented by Figures 1–3), $ω_{1}^{2} = 2 ω_{0}^{2}$ with $ω_{0}^{2} = 0.01, 0.1, 1$ (presented by Figures 4–6), and $ω_{0}^{2} = 2 ω_{1}^{2}$ with $ω_{1}^{2} = 0.01, 0.1, 1$ (presented by Figures 7–9). The numerical outcomes of the corresponding nonlinear FELE and FHE are presented via the simulations in a comparative way. It is observed that the fractional operators are seen to function well in representing various behaviours of the system within the same model representation, containing the classical one. The study can be extended by using different definitions of fractional operators as a future work.

Footnotes

Declaration of conflicting 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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Jihad Asad

References

Holmes

. Ninety plus thirty years of nonlinear dynamics: less is more and more is different. Int J Bifurcation Chaos 2005; 15: 2703–2716.

Shaw

Balachandran

. A review of nonlinear dynamics of mechanical systems in year 2008. Journal of System Design and Dynamics 2008; 2(3): 611–640.

Duffing

. Erzwungene Schwingungen bei veranderlicher Eigenfrequenz und ihre technische Bedeutung. Braunschweig: Vieweg & Sohn, 1918.

Nayfeh

. Combination tones in the response of single degree of freedom systems with quadratic and cubic nonlinearities. J Sound Vib 1984; 92(3): 379–386.

Atadan

Huseyin

. An intrinsic method of harmonic analysis for non-linear oscillations (a perturbation technique). J Sound Vib 1984; 95(4): 525–530.

Nayfeh

. Interaction of fundamental parametric resonances with subharmonic resonances of order one-half. J Sound Vib 1984; 96(3): 333–340.

Bajkowski

Szemplinska-Stupnicka

. Internal resonances effects-simulation versus analytical methods results. J Sound Vib 1986; 104(2): 259–275.

Szemplinska-Stupnicka

Bajkowski

. The subharmonic resonance and its transition to chaotic motion in a non-linear oscillator. Int J Non Linear Mech 1986; 21(5): 401–419.

Ha Quang

Mook

Plaut

. A non-linear analysis of the interactions between parametric and external excitations. J Sound Vib 1987; 118(3): 425–439.

10.

. Solution of a mixed parity nonlinear oscillator: harmonic balance. J Sound Vib 2007; 299: 331–338.

11.

Ghanem

Amer

, et al. Analyzing the motion of a forced oscillating system on the verge of resonance. J Low Freq Noise Vib Act Control 2023; 42(2): 563–578.

12.

Basit

Tahir

Shah

, et al. An application to formable transform: novel numerical approach to study the nonlinear oscillator. Control 2024; 43(2): 729–743.

13.

Imran

Basit

Yasmin

, et al. A proceeding to numerical study of mathematical model of bioconvective Maxwell nanofluid flow through a porous stretching surface with nield/convective boundary constraints. Sci Rep 2024; 14: 1873.

14.

Baleanu

Asad

Petras

, et al. Fractional euler-Lagrange equation of caldirola-kanai oscillator. Rom Rep Phys 2012; 64: 1171–1177.

15.

Baleanu

Jajarmi

Sajjadi

, et al. The fractional features of a harmonic oscillator with position-dependent mass. Commun Theor Phys 2020; 72(5): 055002.

16.

Baleanu

Sajjadi

Jajarmi

, et al. The fractional dynamics of a linear triatomic molecule. Rom. Rep. Phys 2021; 73: 105.

17.

Baleanu

Jajarmi

Defterli

, et al. Fractional investigation of time-dependent mass pendulum. J Low Freq Noise Vib Act Control 2024; 43(1): 196–207.

18.

Agrawal

. Formulation of Euler–Lagrange equations for fractional variational problems. J Math Anal Appl 2002; 272: 368–379.

19.

Baleanu

Defterli

Agrawal

. A central difference numerical scheme for fractional optimal control problems. J Vib Control 2009; 15(4): 583–597.

20.

Samko

Kilbas

Marichev

. Fractional integrals and derivatives: theory and applications. Boca Raton, FL: CRC Press, 1993.

21.

Podlubny

. Fractional differential equations. Mathematics in science and engineering. San Diego, CA: Academic Press, 1999, vol 198, p. xxiv+340.

22.

Hilfer

. Applications of fractional calculus in physics. Singapore: World Scientific, 2000.

23.

Sabatier

Agrawal

Machado

JAT

(eds). Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Dordrecht: Springer, 2007.

24.

Valerio

Machado

Kiryakova

. Some pioneers of the applications of fractional calculus. Fract Calc Appl Anal 2014; 17(2): 552–578.

25.

Hristov

. The craft of fractional modelling in science and engineering: II and III. Fractal Fract 2021; 5(4): 281.

26.

Defterli

. Comparative analysis of fractional order dengue model with temperature effect via singular and non-singular operators. Chaos, Solit Fractals 2021; 144: 110654.

27.

Arshad

Saleem

Defterli

, et al. Simpson’s method for fractional differential equations with a non-singular kernel applied to a chaotic tumor model. Phys Scr 2021; 96(12): 124019.

28.

Baleanu

Diethelm

Scalas

, et al. Fractional calculus: models and numerical methods. 2nd ed. New York, USA: World Scientific, 2017.

29.

Diethelm

Ford

Freed

. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 2002; 29(1–4): 3–22.

30.

Diethelm

Ford

. Analysis of fractional differential equations. J Math Anal Appl 2002; 265(2): 229–248.

On generalized asymmetric harmonic oscillator with quadratic nonlinearity within fractional variational principles

Abstract

Keywords

Introduction

Preliminaries

The physical description of the system

Generalization of the system dynamics using fractional operators

Numerical scheme and simulation results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References