Revealing fractional effects in an asymmetric two-dimensional oscillator through variational mechanics

Abstract

The present study is an application of a generalized Lagrangian-Hamiltonian approach to the analysis of an asymmetric two-dimensional harmonic oscillator with the use of the Caputo fractional derivative. Fractional Euler-Lagrange and Hamiltonian equations were systematically developed and then solved numerically using a predictor-corrector version of the Adams-Bashforth-Moulton method. The results show that decreasing the value of the fractional order (α) generates a form of memory-based damping within the system that transforms the classical closed orbits of phase space into spiral inward trajectories and causes the total energy of the system to decrease according to a power law, regardless of whether there are any dissipative forces in the system. It was demonstrated that both the degree of coupling and the degree of asymmetry can cause the system to lose energy at different rates and to have different amounts of attenuation in each direction. It was also demonstrated that the model provides a smooth transition to classical conservative behavior as α approaches unity, which confirms the physical validity of this representation. This demonstrates that fractional variational mechanics provides a consistent and physically meaningful way to describe the transition from conservative to dissipative behavior for coupled oscillating systems that are governed by memory and non-local effects.

Keywords

fractional Euler-Lagrange formulation asymmetric harmonic oscillator numerical approximation

1. Introduction

Nonlinear differential equations are utilized to describe numerous oscillators found in many areas of science and technology, and as a result, there exists a vast array of mathematical tools and computational techniques available for analyzing their behaviours.^1–3 Due to the inherently nonlinear nature of these oscillators, they respond differently than linearly behaved oscillators as their amplitudes increase. For instance, sufficiently high motions can produce amplitude-dependent frequencies in certain systems. As a result, many physical systems utilize nonlinear or coupled oscillators, as do many biological, chemical and economic models of oscillating systems, thus explaining the continued interest in nonlinear and coupled oscillators.^4–8 One of the most well-known examples is the Van der Pol oscillator, introduced in the 1920’s, which includes nonlinear damping; however, due to the fact that it cannot be analyzed in closed form, all analyses of the Van der Pol oscillator rely upon either numerical or approximate analytical methods.^9–12

In addition to the Van der Pol oscillator, several other studies have examined related nonlinear systems. Ref. 13 provided an analysis of beam vibration with preload discontinuities, and proposed an equivalent function to account for this type of nonlinearity, while Ref. 14 provided energy conservation methods to obtain analytical solutions for nonlinear oscillators with specified initial velocities. Since oscillating systems will continue to provide models for various physical, chemical and biological processes, the cooperative behavior of interacting and coupled oscillators continues to generate considerable interest.^15–21

Fractional calculus has evolved into a useful tool for modeling systems that have memory and hereditary properties. The current state of a system modeled using fractional calculus depends on the entire deformation history. Fractional calculus is a generalization of classical differentiation and integration, permitting non-integer orders, thereby providing the means to model nonlocal effects in time and space. Fractional calculus-based models have been shown to be effective in modeling viscoelasticity, anomalous diffusion, and biological transport phenomena, where integer-order models fail to accurately replicate the results obtained in experiments.^22–29 The fractional derivative order α (0 < α ≤ 1), therefore, provides a continuum of possibilities ranging from classical memoryless systems (α = 1) to systems having long-term memory and power-law energy dissipation (α < 1).

Riewe³⁰ was the first to incorporate fractional derivatives into Lagrangian and Hamiltonian mechanics to describe non-conservative forces. Agrawal³¹ later derived generalized fractional Euler–Lagrange equations, while Baleanu, Diethelm, and Trujillo^28,32 proposed unified fractional models capable of modeling both conservative and dissipative dynamics. Tarasov³³ further demonstrated that non-locality naturally accommodates internal friction and memory effects in complex media. Caputo and Mainardi³⁴ used fractional derivatives for the description of viscoelastic materials.

Fractional calculus has also been extensively applied to nonlinear oscillators. Studies have been conducted on the fractional Duffing, Duffing-van der Pol, Rayleigh oscillators, and fractional harmonic oscillators to examine how the fractional order affects the relationships between amplitude and frequency, bifurcation behavior, and resonance characteristics.^{22–24,26,35–39} These studies demonstrate that fractional systems are different from classical oscillators, and exhibit changed phase trajectories, persistent memory, non-exponential relaxation, and changed energy dissipation rates.²³

The Caputo derivative is especially suitable for physical modeling since it allows the use of standard initial conditions defined by integer-order derivatives.^26,28 More recent developments, such as nonsingular kernels introduced by Atangana and Baleanu,⁴⁰ improve the physical interpretability and numerical efficiency of fractional models. Experimental validation of fractional approaches have also been reported in viscoelastic vibrations, beam deflection problems, biological oscillatory systems,²⁷ and fractional control of chaotic dynamics.²² Although several important contributions^{28,29,32,41–49} exist in literature, the derivations of new methodologies to obtain analytical and numerical solutions for fractional models are still an open area of fractional calculus.

Despite these advances, most fractional oscillator studies focus on one-dimensional or symmetric systems or employ phenomenological equations lacking a rigorous variational basis. Consequently, a complete physical description of energy dissipation and interaction between degrees of freedom in asymmetric, anisotropically stiff, multidimensional fractional systems remains unavailable.

Although fractional variational mechanics has progressed considerably, asymmetric two-dimensional oscillators have received limited attention within a unified fractional Euler–Lagrange and Hamiltonian framework. While earlier studies addressed dissipative systems^30–33 and coupled multidimensional oscillators with directional asymmetry,^22–24,50 none have simultaneously incorporated memory effects, coupling, and anisotropy within a single consistent model, motivating the present investigation. Multidimensional asymmetric oscillators with intrinsic memory-induced dissipation are still largely unexplored, while existing studies on fractional coupled oscillators mainly address symmetric or phenomenological formulations, such as coupled fractional Van der Pol systems and variational-based models.^51–53

The objective of this study is to develop a fractional variational formulation for an asymmetric two-dimensional harmonic oscillator and to analyze the combined influence of fractional order and system parameters.

The primary contributions of this work are summarized as follows:

(1) A consistent fractional Lagrangian–Hamiltonian formulation based on the Caputo derivative is developed, enabling physically meaningful initial conditions and a smooth transition to the classical limit.

(2) The proposed model explicitly incorporates both directional asymmetry and coupling, allowing a detailed investigation of their joint influence on phase-space structures and energy transfer mechanisms.

(3) Numerical simulations demonstrate that fractional memory induces effective dissipation, transforming classical closed orbits into inward-spiralling trajectories and producing power-law decay of mechanical energy without introducing ad hoc damping terms.

(4) The fractional variational framework provides a unified description of conservative and dissipative behavior, extending classical harmonic oscillator theory to systems governed by memory and nonlocal interactions.

The majority of current research focuses on symmetric or one-dimensional systems or uses phenomenological formulations, despite notable advancements in fractional oscillator modeling.^{22–24,35–38} There are still few fully variational Lagrangian–Hamiltonian treatments of asymmetric multidimensional oscillators with memory effects.^{30–34,50–53} By creating a consistent fractional variational framework that demonstrates anisotropic energy exchange, intrinsic memory-induced dissipation, and a seamless transition to classical conservative dynamics, this work fills this gap.

In conclusion, this study establishes a fundamentally new physical framework that generalizes classical mechanics rather than merely applying fractional operators to conventional oscillator models. By integrating fractional derivatives into both Lagrangian and Hamiltonian formulations, the proposed approach offers deeper insight into reversible–irreversible transitions, anisotropic energy exchange, and memory-induced dissipation within a single coherent theoretical structure.^{22–24,26–28,30–34,40}

2. Fractional preliminaries

Fractional calculus broadens traditional differentiation and integration to non-integer orders, facilitating the modelling of systems characterised by memory and hereditary behaviour. It accurately delineates processes where the present state is contingent upon the complete historical context, including viscoelasticity, anomalous diffusion, and biological relaxation. The Riemann–Liouville and Caputo derivatives, as well as the fractional integral, are the most common ways to describe how physical systems change over time in a way that is not local.

2.1. Riemann–Liouville fractional integral

For a sufficiently smooth function f(t) and order α>0, the Riemann–Liouville fractional integral is defined as⁴⁹

I_{t}^{α} f (t) = \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} f (τ) d τ .

(1)

where

Γ (.)

denotes the Gamma function.

2.2. Riemann–Liouville fractional derivative

The Riemann–Liouville fractional derivative of order $α$ ( $n - 1 < α < n$ ) is defined as:

{}^{R L}D_{t}^{α} f (t) = \frac{1}{Γ (n - α)} \frac{d^{n}}{{d t}^{n}} \int_{0}^{t} \frac{f (τ)}{{(t - τ)}^{α - n + 1}} d τ .

(2)

This definition is widely used in mathematical analysis; however, it requires initial conditions expressed in terms of fractional integrals, which are sometimes difficult to interpret physically.

2.3. Caputo fractional derivative

The Caputo fractional derivative of order α ( $n - 1 < α < n$ ) reverses the order of differentiation and integration compared with the Riemann–Liouville form and is expressed as⁴⁹

{}^{C}D_{t}^{α} f (t) = \frac{1}{Γ (n - α)} \int_{0}^{t} \frac{f^{(n)} (τ)}{{(t - τ)}^{α - n + 1}} d τ .

(3)

This formulation is particularly suitable for physical and engineering problems because it allows the use of classical-type initial conditions such as $f (0), f^{'} (0),$ and $f^{″} (0) .$

When $α = 1$ , Eq. (3) reduces to the ordinary first-order derivative.

2.4. Physical interpretation

Both the Riemann–Liouville and Caputo operators act as convolution integrals with a power-law kernel $(t - τ) - α$ . This kernel ensures that the influence of past states decays slowly, producing long-term memory and anomalous relaxation.

The parameter $α$ controls the degree of memory:

for α=1, the model becomes classical and memoryless; for $0 < α < 1$ , memory effects dominate, leading to damping-like and diffusion-like responses even in purely conservative frameworks.

2.5. Application in analytical mechanics

By employing these operators, one can extend the variational principles of mechanics to fractional Euler–Lagrange and fractional Hamiltonian formulations. The fractional Lagrangian function $L (x, {}^{C}D_{t}^{α} x, t)$ produces Euler–Lagrange equations that naturally incorporate dissipation and nonlocality:

\frac{\partial L}{\partial x_{i}} + {}^{C}D_{t}^{α} (\frac{\partial L}{\partial ({}^{C}D_{t}^{α} x_{i})}) = 0 .

(4)

With

i = 1, 2, 3, \dots, n

These equations reduce to the classical ones when $α = 1$ . In the Hamiltonian formalism, the generalized momenta are defined through fractional derivatives of the Lagrangian, leading to a consistent nonlocal generalization of Hamilton’s equations.

In the present work, the Caputo definition is adopted to construct the fractional Euler–Lagrange and fractional Hamilton equations for the asymmetric two-dimensional harmonic oscillator. This choice ensures that the initial conditions remain physically interpretable and that the fractional model smoothly recovers the classical oscillator as $α \to 1$ . The resulting formalism offers a robust mathematical foundation for characterizing nonconservative interactions, memory phenomena, and energy dissipation within a cohesive framework.

3. Description of the physical system

3.1. Classical description via Lagrangian

We start this section, by considering asymmetric two- dimensional oscillator having the following kinetic and potential energy respectively:

T = \frac{1}{2} {\dot{x}}^{2} + \frac{1}{2} {\dot{y}}^{2} .

(5)

V = \frac{1}{2} x^{2} + \frac{1}{2} y^{2} + ε x s i n h μ y .

(6)

In the above equations for simplicity, we assumed that

m = ω = 1

The novelty of the proposed system lies in the introduction of the nonlinear coupling term in the potential energy, εxsinh(εy), which fundamentally transforms the model from a trivial two-dimensional harmonic oscillator into a genuinely asymmetric and coupled system. This additional term induces nonlinearity, directional asymmetry, and cross-coupling between the degrees of freedom, thereby enabling new physical behavior and variational insights that are inaccessible in conventional separable harmonic oscillators. Consequently, the proposed formulation provides a meaningful extension of classical oscillator models and establishes a richer framework for investigating energy transfer, anisotropy, and memory-driven dynamics within a unified variational setting.

The classical Lagrangian $(L_{c})$ which is defined as $L_{c} = T - V$ reads:

L_{c} = \frac{1}{2} {\dot{x}}^{2} + \frac{1}{2} {\dot{y}}^{2} - \frac{1}{2} x^{2} - \frac{1}{2} y^{2} - ε x s i n h μ y .

(7)

Now, using the formula

\frac{d}{d t} (\frac{\partial L_{c}}{\partial \dot{q_{i}}}) - \frac{\partial L_{c}}{\partial q_{i}} = 0

with

q_{1} = x

, and

q_{2} = y

. The classical Euler- Lagrange equations (CELE’s) read:

\ddot{x} + x + ε \sin h μ y = 0 .

(8)

\ddot{y} + y + ε μ x \cos h μ y = 0 .

(9)

The classical Hamiltonian $(H_{c})$ is related to classical Lagrangian $(L_{c})$ as:

H_{c} = \sum_{i} p_{i} {\dot{q}}_{i} - L_{c} .

(10)

where

p_{i} = \frac{\partial L_{c}}{\partial {\dot{q}}_{i}}

As a result, with $i = 1, 2; x, y$ we have

p_{x} = \frac{\partial L_{c}}{\partial \dot{x}} = \dot{x} .

(11)

p_{y} = \frac{\partial L_{c}}{\partial \dot{y}} = \dot{y} .

(12)

Using (7) and (10) and(12), the classical Hamiltonian reads:

H_{c} = \frac{1}{2} [p_{x}^{2} + p_{y}^{2} + x^{2} + y^{2}] + ε x s i n h μ y .

(13)

Finally, the classical Hamilton’s equations of motion (CHEM’s) read:

- {\dot{p}}_{x} = \frac{\partial H_{c}}{\partial x} = x + ε \sinh μ y ⟹ \ddot{x} + x + ε \sinh μ y = 0 .

(14)

- {\dot{p}}_{y} = \frac{\partial H_{c}}{\partial y} = x + ε μ x c o s h μ y ⟹ \ddot{y} + y + ε μ x c o s h μ y = 0 .

(15)

As, one can notice the systems (8), (9), (14) and (15) are identical to each other. These systems are a coupled non- linear second order differential equations.

3.2. Fractional description

In this section, we start by fractionalizing (7). In terms of Caputo fractional derivative, the fractional form of (7) reads:

L_{f} = \frac{1}{2} {({{}_{a}^{c}D}_{t}^{α} x)}^{2} + \frac{1}{2} {({{}_{a}^{c}D}_{t}^{α} y)}^{2} - \frac{1}{2} x^{2} - \frac{1}{2} y^{2} - ε x s i n h μ y .

(16)

One can derive the fractional Euler-Lagrange equations (FELE’s) by substituting (16) into $\frac{\partial L_{f}}{\partial q_{i}} + {{}_{t}^{C}D}_{b}^{α} \frac{\partial L_{f}}{\partial {{}_{a}^{C}D}_{t}^{α} q_{i}} + {{}_{a}^{C}D}_{t}^{β} \frac{\partial L_{f}}{\partial {{}_{t}^{C}D}_{b}^{β} q_{i}} = 0$ . Thus, we yield two equations for $q_{1} = x$ and $q_{2} = y$ respectively as:

{{}_{t}^{C}D}_{b}^{α} {{}_{a}^{C}D}_{t}^{α} x - x - ε \sin h μ y = 0 .

(17)

{{}_{t}^{C}D}_{b}^{α} {{}_{a}^{C}D}_{t}^{α} y - y - ε μ x c o s h μ y = 0 .

(18)

Furthermore, to construct the fractional Hamiltonian

(H_{f})

we start by defining the following four generalized momenta

p_{α, x}, p_{β, x}, p_{α, y}, and p_{β, y}

respectively as:

p_{α, x} = \frac{\partial L_{f}}{\partial {{}_{a}^{C}D}_{t}^{α} x} = {{}_{a}^{C}D}_{t}^{α} x .

(19a)

p_{β, x} = \frac{\partial L_{f}}{\partial {{}_{t}^{C}D}_{b}^{β} x} = 0 .

(19b)

p_{α, y} = \frac{\partial L_{f}}{\partial {{}_{a}^{C}D}_{t}^{α} y} = {{}_{a}^{C}D}_{t}^{α} y .

(19c)

p_{β, y} = \frac{\partial L_{f}}{\partial {{}_{t}^{C}D}_{b}^{β} y} = 0 .

(19d)

On the other hand, the fractional Hamiltonian

(H_{f})

is related to these generalized momenta and

(L_{f})

by the following relation

H_{f} = p_{α, x} {{}_{a}^{C}D}_{t}^{α} x + p_{β, x} {{}_{t}^{C}D}_{b}^{β} x + p_{α, y} {{}_{a}^{C}D}_{t}^{α} y + p_{β, y} {{}_{t}^{C}D}_{b}^{β} y - L_{f}

As a result, we got the following equation:

H_{f} = \frac{1}{2} {({{}_{a}^{C}D}_{t}^{α} x)}^{2} + \frac{1}{2} {({{}_{a}^{C}D}_{t}^{α} y)}^{2} + \frac{1}{2} x^{2} + \frac{1}{2} y^{2} + ε x s i n h μ y .

(20)

Finally, substituting (20) into the following relation $\frac{\partial H_{f}}{\partial q_{i}} = {{}_{t}^{C}D}_{b}^{α} p_{α, q_{i}} + {{}_{a}^{C}D}_{t}^{β} p_{β, q_{i}}$ . With $q_{1} = x$ and $q_{2} = y$ respectively we obtained the following two fractional Hamilton’s equations of motion (FHEM’s)

{{}_{t}^{C}D}_{b}^{α} {{}_{a}^{C}D}_{t}^{α} x - x - ε \sin h μ y = 0 .

(21)

{{}_{t}^{C}D}_{b}^{α} {{}_{a}^{C}D}_{t}^{α} y - y - ε μ x c o s h μ y = 0 .

(22)

As easily one can notice (21) and (22) are identical to (17) and (18). Moreover, as $α \to 1$ , these equations (17) and (18) reduced to the CELE’s derived in section 2.

Within the current model, the parameters ε and µ clearly describe their physical meaning. The value of ε is used to characterize the level of stiffness asymmetry between the two orthogonal coordinate axes, as well as to define the magnitude of the restoring forces associated with each axis. On the other hand, the parameter µ defines the interaction energy between the coordinates; it therefore defines the strength of coupling between them. The range of values of ε and µ that were selected in this work were designed to illustrate representative dynamic behavior by showing how different levels of asymmetry and coupling can be combined with fractional memory (or memory) characteristics. These values of ε and µ are not intended to represent any particular physical system; instead they are presented as examples of generic and physically meaningful behaviors exhibited by asymmetric fractional oscillators.

4. Numerical technique

Although the fractional equations of motion are analytically intractable for most values of α, a viable computational method is necessary to provide an approximation of their behaviour over time. This study uses the ABM scheme developed by Diethelm et al., which provides both stability and accuracy for systems containing fractional derivatives of the form of the Caputo type^29,44,45 due to its use as a predictive-corrective process. The ABM scheme has been shown to be successful in approximating the response of various nonlinear systems that contain memory effects and have been used in several studies as a model of a variety of physical and mechanical phenomena.^46,47,49

In general, the ABM scheme can be described by the concept of transforming the fractional differential equation of order $α (0 < α \leq 1)$ into an equivalent Volterra integral equation using the integral representation of the system. Then, the Volterra equation is discretized on a uniform time-step grid (t_n = nh) and the iterative steps of the predictive-corrector scheme are performed.

4.1. Step predictor

A fractional Adams-Bashforth formula is used to find the explicit estimate of the function’s future value. This uses previous values of the function and its derivative.

4.2. Step corrector

An implicit Adams-Moulton correction is made to refine the predicted value. The refinement improves both the local accuracy and stability of the solution, since the fractional integral is recalculated at the new or corrected value.

Since it is a combination of predictor-corrector steps, the above procedure represents a semi-implicit procedure that gives a local error of order $O (h^{\min (2, 1 + α)})$ . Also, the method is stable for relatively small time-steps.²⁸

The method can capture the long-term memory effects of the fractional dynamics, due to the use of cumulative weighted sums, to approximate the convolution integral kernel ${(t - τ)}^{{α - 1}}$ .

To compute the above process, we divide the continuous interval [0,T] into $N$ equal sized intervals of width $h = T / N$ . Then, at each iteration, the Caputo derivative is approximated by discrete convolution coefficients depending on the fractional order $α$ and the previously calculated values of the function. As a result, since the method is based on recursive iterations, all previous states are included when computing the current state, preserving the non-locality of the fractional operator.

Also, the fractional order $α$ is treated as a variable parameter controlling the degree of the memory effects. When $α$ → 1, the method is reduced to the classical Adams-Moulton method for integer order derivatives.

To assess the accuracy of the numerical calculations, the computation is repeated for decreasing time steps until convergence of the trajectories x(t) and y(t) is observed. To check the stability of the implemented method, the results from different mesh sizes are compared. The numerical solutions show a smooth temporal behavior and good agreement with the analytical predictions.

Therefore, the Predictor-Corrector ABM method offers a reliable, accurate and computationally effective way to solve the fractional Euler-Lagrange equations of the two-dimensional asymmetric oscillator, allowing for a straightforward comparison of its classical $(α = 1) and fractional (0 < α < 1)$ dynamics.^48,50

5. Results and discussion

One of the important features that emerge from the numerical simulation is that classical closed phase-space trajectories are transformed into inward spirals as the fractional order α is decreased. The emergence of these spiral trajectories is due to the inherent non-local memory kernel associated with the Caputo fractional derivative and not due to explicitly imposed (damping) terms. Similar spiral-type trajectories and long-term attenuation have also been demonstrated in systems of fractional oscillators based on variational principles and memory-based dynamics.^23,27,34,50 Physically, this results in continuous incorporation of the past states of the system resulting in a gradual redistribution of energy and an eventual “effective” dissipation of the energy of the system within a variational context.

In this part of the paper, we use a number-based method described in Section 4 to compute the fractional and classical dynamics of the asymmetric 2D-HO. The predictor-corrector Adams-Bashforth-Moulton (ABM) method is used for solving the equations of motions that is well-suited to solve fractional systems with memory. We performed simulation for various fractional orders (α) from 0.6 up to 1.0 for demonstrating in what way memory effects affect the behaviour of the system.

Numerical data illustrating the response of the asymmetrical two-dimensional harmonic oscillator to various combinations of the parameters ε and μ as illustrated by the graphical representations in Figures 1–3 are also presented here in terms of the fractional framework. In the case that both parameters have relatively low values (see Figure 1), the behavior of this system is essentially identical to that of a classical oscillator; the oscillations of x(t) and y(t) behave in an almost perfectly periodic manner at amplitudes which are nearly constant, and the phase portraits of this system are elliptically closed indicating very little fractional damping and therefore a very weak memory influence.

Figure 1.

Dynamical behaviours for $ε = 0.1$ and $μ = 0.1$ .

Figure 2.

Dynamical behaviours for $ε = 0.1$ and $μ = 1$ .

Figure 3.

Dynamical behaviours for $ε = 1$ and $μ = 0.1$ .

The chosen initial conditions were symmetric and moderate in magnitude to avoid favoring one direction over another and to allow isolation of the effects of the fractional exponent (α) on the behavior of the system. The variation in the initial conditions appears to have a significant impact on the short time transient response. However, the longtime attenuation, spiral phase trajectory and the power law form of the energy decay appear to be insensitive to the chosen initial conditions and thus seem to be controlled by the fractional memory function of the model. Thus, it appears the observed fractional damping is a characteristic of the model itself and not dependent upon the specifics of the initial conditions.

When the coupling parameter μ is raised significantly (Figure 2), the trajectories show clear phase delays, stronger modulation, and noticeable amplitude decay. This is because memory-dependent damping has a stronger effect. The phase portraits that go with them turn into inward spirals, which show a slow loss of energy even without any explicit damping terms.

When ε is high and μ is low, as shown in Figure 3, the system has a clear directional asymmetry. This is because the restoring and memory effects are stronger in the x direction than in the y direction, which makes the oscillations in the x direction die out faster. The smaller values for both parameters produce asymmetrical damping and produce non uniform deformed trajectories in phase space. Increasing either parameter will make the system’s dissipative fractional behavior stronger, make the amplitudes decay faster, and change how the two coordinates are linked. These findings validate that the fractional order formulation effectively encapsulates memory-driven dissipative behavior, anisotropic dynamics, and the seamless transition from conservative to dissipative motion.

To verify the correctness of the adopted predictor-corrector ABM scheme, the system was solved for α = 1 using both the fractional numerical method and a classical fourth-order Runge-Kutta (RK4) algorithm. The resulting time histories of the state variables in Figure 4 show excellent agreement between the two methods. These results provide a benchmark validation of the numerical implementation. Next, a step-size sensitivity analysis was performed to confirm that the numerical results are not affected by the discretization parameters. Simulations were carried out for three different time step sizes, namely h = 0.5, h = 0.25, and h = 0.1. The numerical solutions obtained with decreasing step sizes (Figure 5) exhibit consistent convergence in the time-domain responses. As can be seen, the maximum deviation between solutions corresponding to successive step sizes decreases monotonically as the time step is refined, indicating numerical stability and convergence of the employed scheme.

Figure 4.

Comparison of the fourth-order Runge-Kutta (RK4) scheme and the adapted predictor-corrector Adams-Bashforth-Moulton (ABM) method for the classical case α = 1 ( $ε = 1$ and $μ = 0.1$ ).

Figure 5.

Step-size sensitivity analysis for the asymmetric two-dimensional oscillator with time steps h = 0.5, h = 0.25, and h = 0.1 (α = 0.9, $ε = 1$ , and $μ = 0.1$ ).

The parameters ε and μ are fundamental in creating the anisotropic behavior of the fractional oscillator. Increasing the coupling parameter μ will improve energy transfer from one coordinate to another; however, when utilizing fractional dynamics, there is unequal memory-induced loss that causes phase delay and amplitude modulations.^22,23 The same can be said for the effect of increasing the asymmetry parameter ε. This creates direction dependent stiffness, therefore it will amplify fractional dissipation in one of the coordinates as opposed to the other. This ultimately produces non uniform spiral trajectories and distorted phase portraits, and thus demonstrates that fractional memory effects and structural asymmetries have a strong interdependence relationship with each other.^27,50

Another key verification of our proposed model is that it recovers classical conservative behavior smoothly when α → 1. In the case of α = 1, memory is eliminated, all phase-space orbits are closed and the total energy is preserved; in complete conformity to classical Hamiltonian dynamics. When α < 1, the computer simulations provide evidence of a progressive (power-law) decline in energy consistent with observed characteristics of fractional oscillators in viscoelastic and complex materials^23,26–28; which indicates that the fractional order α has a physical basis as a dimensionless control parameter governing memory and internal damping rather than being simply an arbitrary numerical parameter.

In addition to the above-mentioned points, note that this study is limited in its use of a linear asymmetric oscillator and relies on numerical approximations of fractional derivatives, which will add complexity and computational time to real-time applications. The development of numerical methods suitable for both non-linear implementations and identification of the fractional parameter represents an important area of potential research.

6. Conclusion

We developed and analyzed a new fractional model of the two-dimensional asymmetric harmonic oscillator using the Caputo fractional derivative. This allowed them to incorporate into the classical Lagrangian and Hamiltonian formulations memory and hereditary properties present in all physical systems. The derived fractional Euler-Lagrange and Hamiltonian equations represent a single mathematical formulation that can describe both conservative and dissipative types of dynamics.

The authors used the predictor-corrector ABM algorithm to find numerical solutions to the fractional equations of motion. Their numerical studies demonstrated how decreasing the fractional order α resulted in several distinct transitions in the dynamics of the system, specifically amplitude attenuation, phase lag, and energy loss; thus, they confirmed that the fractional derivative inherently causes a type of non-local damping (even without explicit resistive terms). They showed that the classical behavior of the system is obtained smoothly when α → 1, and therefore validated the robustness and consistency of their fractional formalism.

A comprehensive comparison of the classical and fractional cases has shown that the fractional oscillator exhibited power law relaxation of the total energy and memory dependent dissipation, while retaining the essential coupling between the two orthogonal coordinates. Also, they have demonstrated that the asymmetry of the stiffness constants $k x \neq k y$ , induces directional damping resulting in non uniform energy exchange and more rich dynamic behavior than that of the symmetric case.

Therefore, the authors conclude that the fractional mechanics are able to not only generalizing the traditional oscillation models but also to capture the slow relaxation phenomenon and the viscoelastic damping observed in experiments. Therefore, the fractional parameter α effectively controls the transition from the ideal harmonic motion to the realistic dissipative response and represents physically a measure of memory depth and internal friction.

Future work may extend this approach to nonlinear fractional oscillators, forced or parametrically excited systems and quantum analogs of fractional Hamiltonians where long range correlations play a major role. Furthermore, the connection of the fractional calculus with modern computational or machine learning techniques could lead to further insight into the non-local mechanical systems and the memory driven energy transfer processes.

In this general sense, these results show that fractional variational mechanics provides an intrinsic, physically transparent way to model coupled, asymmetric, memory-based dissipative systems as well as their conservative limits,^30–33,50 in contrast to the need for ad-hoc damping terms in classical models.

The most important new contribution of this study extends beyond the numbers it produced. The study also introduces a physically consistent fractional variational model for an asymmetric coupled oscillator as well as illustrates the relationship between memory, asymmetry and coupling; demonstrating that fractional calculus does not simply provide a mathematically generalized description of behavior, but provides a real physical description of how dissipation and redistribution of energy occur.

In conclusion, the proposed fractional extension of the asymmetric harmonic oscillator is a flexible theoretical and computational tool to explore the complex systems with memory, dissipation and coupling - closing the gap between the classical idealized mechanics and the realistic physical phenomena.

Footnotes

ORCID iDs

Jihad Asad

Dumitru Baleanu

Amin Jajarmi

Ozlem Defterli

Noorhan F. AlShaikh Mohammad

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

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