Analytical approximation and dynamic response of the driven cubic

Abstract

This study investigates the dynamic behavior of a periodically driven cubic–quintic Duffing oscillator, an extended form of the classical Duffing system that accounts for stronger nonlinear stiffness effects. The motivation lies in understanding nonlinear oscillations at large amplitudes, where classical analytical approaches often fail. To address this, two semi-analytical techniques—HPM and Ms-DTM—are employed to derive approximate solutions expressed as infinite power series. The accuracy and convergence of these methods are validated through comparison with numerical integration results obtained using MATLAB’s ode45 solver. Excellent agreement among the three approaches confirms the reliability of both HPM and Ms-DTM in capturing the system’s nonlinear response. The comparative analysis reveals consistent displacement, velocity, and phase behavior, including frequency shifts due to nonlinear stiffness and amplitude variations under harmonic excitation. These findings demonstrate that HPM and Ms-DTM provide efficient and accurate tools for modeling complex nonlinear oscillatory systems.

Keywords

duffing oscillator cubic–quintic nonlinearity homotopy perturbation method (HPM)differential transform method (DTM)forced nonlinear oscillator analytical approximation

1. Introduction

Nonlinear oscillators are a cornerstone of theoretical and applied mechanics because of their ability to model diverse physical phenomena, ranging from electronic circuits and microelectromechanical systems to structural and optical resonators.^1–4 Among them, the Duffing oscillator, first introduced in the early 20th century, has served as a fundamental prototype for exploring nonlinear dynamical behavior, including bifurcation, chaos, resonance, and amplitude-dependent frequency shifts.^5–7 Historically, the Duffing model was limited to cubic nonlinearity, which describes systems where restoring forces deviate slightly from Hooke’s law. However, in many real systems, higher-order nonlinearities, such as quintic terms, arise due to geometric effects, material anisotropy, or strong excitation amplitudes. The cubic–quintic Duffing oscillator thus provides a more accurate framework for describing such complex dynamics.^8,9

Over the decades, several analytical and numerical techniques have been developed to study the Duffing oscillator. Early studies employed perturbation and multiple-scale methods to approximate periodic responses and analyze resonance conditions.^10–12 With advances in computational resources, modern approaches have focused on improving convergence, stability, and accuracy in solving nonlinear differential equations. In particular, semi-analytical methods such as the Homotopy Perturbation Method (HPM),^13,14 Variational Iteration Method (VIM),¹⁵ Adomian Decomposition Method (ADM),¹⁶ and Differential Transform Method (DTM)^17,18 have attracted significant attention. These techniques bridge the gap between purely analytical and purely numerical procedures, providing efficient tools for exploring the response of nonlinear systems without incurring high computational cost. Recent implementations have demonstrated nonlinear dynamic behavior in nanomechanical resonators exhibiting Duffing-type and higher-order nonlinearities.^19,20

In recent years, enhanced versions of these techniques have been introduced. For instance, Liao’s Homotopy Analysis Method (HAM) generalized the HPM framework to overcome limitations associated with small parameters.²¹ The Multi-Stage Differential Transform Method (Ms-DTM) was subsequently proposed to improve the convergence of the DTM in long-term simulations and to effectively handle highly nonlinear oscillations.^22–25 Several studies have applied these methods to forced and damped Duffing systems, exploring topics such as chaotic attractors, frequency modulation, and multi-stability.^22,26–28 Contemporary research also investigates nonlinear stiffness modulation, fractional-order Duffing oscillators, and driven cubic–quintic models with external forcing, confirming the importance of such oscillators in advanced mechanical and physical modeling.^29–33 Such systems are increasingly relevant in vibration control, MEMS sensors, and nonlinear optical cavities.^34,35

The Non-Perturbative Approach (NPA), a novel analytical technique was developed to improve on limitations associated with the use of classical perturbation theory based methodologies for the analysis of nonlinear oscillators with strong nonlinearities. The NPA does not require the assumption of “small” parameter values which is a requirement of classical perturbative techniques; therefore, it can provide an exact frequency equation for highly nonlinear systems.^36–38 In addition to its ability to yield exact frequency formulations for highly nonlinear systems, the NPA has been applied to the Duffing Oscillator to examine a variety of nonlinear behavior including resonance, bifurcations, and stability in both single and coupled configurations.^39,40 These studies have shown that the NPA not only facilitates the mathematical treatment of nonlinear systems, but that it provides improved convergence characteristics and increased accuracy than homotopy and variational iteration techniques. As a result of the improvements that have been achieved using the NPA, this study will extend the application of the NPA to the driven cubic-quintic Duffing Oscillator, and will develop a reliable, non-perturbative analytical framework that may be used to complement and enhance existing semi-analytical methodologies.

Motivated by these developments, the present work aims to study the dynamics of a periodically driven cubic–quintic Duffing oscillator using two powerful semi-analytical techniques, namely HPM and Ms-DTM. Both approaches are validated through comparison with a standard numerical integration (Runge–Kutta ode45) to demonstrate their efficiency, accuracy, and convergence in capturing the nonlinear response. To emphasize practical relevance, we also note that the cubic–quintic Duffing oscillator appears in various engineering and physical contexts, including nonlinear vibration absorbers, MEMS/NEMS devices, bistable mechanical switches, and nonlinear optical waveguides.^19,20,34,35

The paper is organized as follows: Section 2 introduces the governing equations and physical model; Sections 3 and 4 describe the HPM and Ms-DTM formulations, respectively; Section 5 presents the comparative results and discussion; and Section 6 concludes with key remarks and future perspectives.

2. The driven cubic-quintic Duffing oscillator

In the study of nonlinear oscillatory systems, the Hamiltonian and Lagrangian formulations serve as foundational tools for understanding energy distributions and dynamic behavior. For the cubic–quintic Duffing oscillator, the Hamiltonian captures the system’s total energy, encompassing both kinetic and potential energy. At the same time, the Lagrangian formalism provides insights into the system’s path optimization and conservation laws.^13,14 In this section, we derive the governing equations of motion for the driven cubic–quintic Duffing oscillator, starting from its Lagrangian representation and transitioning to the Hamiltonian framework.^13,15 This approach not only establishes the fundamental dynamics but also sets the stage for the application of approximate analytical methods, such as HPM and Ms-DTM, in subsequent sections. An external periodic force is incorporated into the model, reflecting real-world excitations that drive the system’s nonlinear response.^14,16 The main objective of this section is to construct the mathematical foundation required for the subsequent analytical and numerical analysis of the oscillator’s behavior.

We start this section by considering the following Hamiltonian ( $H$ ):

H = \frac{p^{2}}{2 m} + \frac{1}{2} k_{1} x^{2} + \frac{1}{4} k_{3} x^{4} + \frac{1}{6} k_{5} x^{6} .

(1)

where:

m

= mass of the oscillator,

p

=canonical momentum,

k_{1}

= linear stiffness coefficient,

k_{3}

= cubic nonlinear stiffness coefficient,

k_{5}

= quintic nonlinear stiffness coefficient,

x

=displacement from equilibrium.

In the subsequent analysis, we assume $k_{1} = k_{3} = k_{5} = k$ to simplify the mathematical formulation. This normalization does not alter the qualitative behavior of the system but enables a more direct comparison between the contributions of the cubic and quintic nonlinearities. The single stiffness parameter $k$ thus represents a common scaling constant that governs both the linear and nonlinear restoring forces.

Using the relation $\frac{d q_{i}}{d t} = \frac{\partial H}{\partial p_{i}}$ . So, for $q_{i} = x$ we have

\frac{d x}{d t} = \frac{\partial H}{\partial p} \to \dot{x} = \frac{p}{m} .

(2)

The Lagrangian is related to the Hamiltonian as:

H = \sum p_{i} {\dot{q}}_{i} - L .

(3)

So, with $p_{i} = p,$ and ${\dot{q}}_{i} = \dot{x}$ , we can write:

L = \dot{x} p - H = \frac{1}{2} m {\dot{x}}^{2} - \frac{1}{2} k x^{2} - \frac{1}{4} k x^{4} - \frac{1}{6} k x^{6} .

(4)

Now, substituting (4) into the formula $\frac{\partial L}{\partial q_{i}} - \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{i}} = 0$ with $q_{i} = x$ we have:

\frac{d}{d t} \frac{\partial L}{\partial \dot{x}} = m \ddot{x};

and \frac{\partial L}{\partial x} = - k x - k x^{3} - k x^{5};

Thus, the equation of motion reads:

\ddot{x} + ω_{o}^{2} (x + x^{3} + x^{5}) = 0 .

(5)

where,

ω_{o}^{2} = \frac{k}{m}

is the natural angular frequency

Equation (5) above is known as cubic-quintic Duffing oscillator.

Assuming the Duffing oscillator is driven by the force $q_{o} \cos (w t)$ . Then Eq. (5) becomes:

\ddot{x} + ω_{o}^{2} (x + x^{3} + x^{5}) = q_{o} \cos (w t) .

(6)

3. Existence and uniqueness of the solution of (6)

In this section, we give a main theorem related to the theory of initial value problems which is the existence and uniqueness theorem. To study the existence and uniqueness of the solution of (6) we can easily relabel the variables and convert the second order differential equation into system of two first order differential equations.

Let $x (t) = u (t), \dot{x} (t) = v (t) .$ This produces the first-order system

\frac{d u}{d t} = F (t, u, v), \frac{d v}{d t} = G (t, u, v),

(7)

with initial conditions

u (t_{0}) = u_{0}, v (t_{0}) = v_{0},

(8)

Theorem 1¹²: Consider the system of two differential equations (7), with initial conditions (8), if the functions F and G are continuous and have continuous partial derivatives in a region D of $t u v$ -space defined by $α_{1} < t < β_{1}, α_{2} < u < β_{2}, α_{3} < v < β_{3}$ , and let $(u_{0}, v_{0})$ be in D. Then there is an interval $| t - t_{0} | < h$ in which there exists a unique solution $u = φ (t), v = ψ (t)$ of the system of differential equations that also satisfies the initial conditions.

Now, transform (6) to a system of first order equations in the same manner we obtain

\frac{d u}{d t} = v, \frac{d v}{d t} = {- ω}_{o}^{2} ({u + u}^{3} + u^{5}) + q_{0} \cos (ω t) .

Note that, $F (t, u, v) = v, and G (t, u, v) = {- ω}_{o}^{2} ({u + u}^{3} + u^{5}) + q_{0} \cos (ω t)$ satisfy the hypothesis of theorem 1 on any region D containing the initial values $(u_{0}, v_{0})$ , so (6) has unique solution in interval containing $t_{0}$ .

4. Homotopy perturbation method

Simulating complex events such as fluid dynamics, wave propagation, and mechanical vibrations, nonlinear differential equations have numerous physical and technological applications. Exact solutions are typically not feasible due to their inherent nonlinearity, so strong approximative analytical techniques are often employed.^17,18 When merging homotopy theory with conventional perturbation methods, the HPM is beneficial in building approximate solutions through a series expansion.^17,21 This approach accelerates convergence and avoids the need for modest perturbation values, thereby extending its application to highly nonlinear systems.^17,22 By use of approximate analytical solutions for the nonlinear differential equation controlling the cubic-quintic Duffing oscillator, this part emphasises the efficiency of HPM in capturing the system’s nonlinear dynamic behavior under external excitation.

This part uses HPM to provide approximate analytical solutions for the nonlinear differential equation controlling the cubic-quintic Duffing oscillator, thereby capturing the oscillatory behaviour of the system under external driving forces:

\ddot{x} + ω_{o}^{2} (x + x^{3} + x^{5}) = q_{0} \cos (ω t) .

(9)

with initial conditions:

x (0) = α, \dot{x} (0) = 0 .

(10)

Equation (9) can be re-written as:

\ddot{x} + ω_{o}^{2} x + ω_{o}^{2} (x^{3} + x^{5}) = q_{0} \cos (ω t) .

(11)

Relation (11) is equivqlent to $L (x) + N (x) = f (t)$ , where $L (x) = \ddot{x} + ω_{o}^{2} x$ represents the linear term, $N (x) = ω_{o}^{2} (x^{3} + x^{5})$ represents the nonlinear term, and $f (t) = q_{0} \cos (ω t)$ .

To apply HPM, we define the homotopy function:

H (t, λ) = \ddot{x} + ω_{o}^{2} x + λ (ω_{o}^{2} (x^{3} + x^{5}) - q_{0} \cos (ω t)) = 0 .

(12)

where

λ \in [0, 1]

is an embedding parameter. The solution is expanded in terms of

λ

as:

x (t, λ) = x_{o} (t) + λ x_{1} (t) + λ^{2} x_{2} (t) + \dots .

(13)

Substituting the series expansion (13) into Eq. (12) and equating the coefficients of like powers of $λ$ , we obtain a sequence of differential equations for $x_{0} (t), x_{1} (t), x_{2} (t), \dots$ .

({\ddot{x}}_{o} (t) + λ {\ddot{x}}_{1} (t) + \dots) + ω_{o}^{2} (x_{o} (t) + λ x_{1} (t) + \dots) + λ (ω_{o}^{2} (x_{o}^{3} (t) + λ^{3} x_{1}^{3} (t) + λ^{6} x_{2}^{3} (t) + 3 x_{o}^{2} (t) λ x_{1} (t) + 3 x_{o}^{2} (t) λ^{2} x_{2} (t) + 3 x_{o} (t) λ^{2} x_{1}^{2} (t) + 3 λ^{2} x_{1}^{2} (t) λ^{2} x_{2}^{2} (t) + 3 x_{o} (t) λ^{4} x_{2}^{2} (t) + 3 λ x_{1}^{2} (t) λ^{4} x_{2}^{2} (t) + 6 x_{o} (t) λ^{3} x_{1} (t) x_{2} (t) + \dots) - q_{0} \cos (ω t)) = 0 .

(14)

To ensure consistency with the original initial conditions (10), we impose

\sum_{n = 0}^{\infty} λ^{n} x_{n} (0) = α, \sum_{n = 0}^{\infty} λ^{n} {\dot{x}}_{n} (0) = 0,

which are satisfied by

x_{0} (0) = α, {\dot{x}}_{0} (0) = 0, x_{n} (0) = 0, {\dot{x}}_{n} (0) = 0 (n \geq 1) . (10 ’)

Collecting terms of equal powers of $λ$ , we obtain the following initial–value problems:

λ^{0} : {\ddot{x}}_{0} (t) + ω_{0}^{2} x_{0} (t) = 0, x_{0} (0) = α, {\dot{x}}_{0} (0) = 0,

(15)

λ^{1} : {\ddot{x}}_{1} (t) + ω_{0}^{2} x_{1} (t) = q_{0} \cos (ω t) - ω_{0}^{2} [x_{0}^{3} (t) + x_{0}^{5} (t)], x_{1} (0) = 0, {\dot{x}}_{1} (0) = 0 .

(16)

Higher–order equations $(n \geq 2)$ can be derived in a similar way, each satisfying the homogeneous initial conditions $x_{n} (0) = {\dot{x}}_{n} (0) = 0$ .

The general solution for (15) is found as:

x_{0} (t) = C_{1} \cos (ω_{o} t) + C_{2} \sin (ω_{o} t) .

(17)

Using (10), we got $C_{1} = α$ and $C_{2} = 0$ . So, Eq (17) becomes:

x_{0} (t) = α \cos (ω_{o} t) .

(18)

On the other hand, the solution for (16) is

{\ddot{x}}_{1} (t) + ω_{o}^{2} x_{1} (t) + ω_{o}^{2} (α^{3} \cos^{3} (ω_{o} t) + α^{5} \cos^{5} (ω_{o} t)) - q_{0} \cos (ω t) = 0 .

(19)

Re-arranging it as:

{\ddot{x}}_{1} (t) + ω_{o}^{2} x_{1} (t) = q_{0} \cos (ω t) - ω_{o}^{2} (α^{3} \cos^{3} (ω_{o} t) + α^{5} \cos^{5} (ω_{o} t)) .

(20)

As a result the solution of (16) reads:

x_{1} (t) = - \sin 3 t (\frac{363 t}{320000} - \frac{\sin 2 t}{1200} - \frac{\sin 4 t}{2400} + \frac{325 \sin 6 t}{128000} + \frac{203 \sin 12 t}{6400000} + \frac{\sin 18 t}{19200000}) - \cos 3 t (\frac{161 \cos 6 t}{1280000} + \frac{\cos 4 t}{2400} - \frac{c o s 2 t}{1200} + \frac{101 \cos 12 t}{3200000} + \frac{\cos 18 t}{19200000} + \frac{10489}{9600000}) .

(21)

Finally, for $λ^{1}$ the solution will be

x (t) = x_{o} (t) + x_{1} (t);

x (t) = α \cos (ω_{o} t) - \sin 3 t (\frac{363 t}{320000} - \frac{\sin 2 t}{1200} - \frac{\sin 4 t}{2400} + \frac{325 \sin 6 t}{128000} + \frac{203 \sin 12 t}{6400000} + \frac{\sin 18 t}{19200000}) - \cos 3 t (\frac{161 \cos 6 t}{1280000} + \frac{\cos 4 t}{2400} - \frac{\cos 2 t}{1200} + \frac{101 \cos 12 t}{3200000} + \frac{\cos 18 t}{19200000} + \frac{10489}{9600000}) .

(22)

This analytical solution provides a good approximation of the nonlinear oscillator’s response to external periodic stimulation.^18,22 The method can be created repeatedly with minimal cost and used to compute higher-order modifications systematically. By bypassing the need for powerful numerical simulations, the HPM not only streamlines the analysis but also provides insightful understanding of the system’s underlying nonlinear dynamics. In perturbation-based approximations, secular terms such as $t \sin (ω t)$ may appear, leading to unbounded growth in the series for long times. However, the HPM formulation used here eliminates such unbounded components, and the obtained solutions remain bounded within the analyzed time range.

5. Ms- DTM method

A complex analytical technique, the Multi-Step Differential Transformation Method (Ms-DTM) aims to solve nonlinear differential equations efficiently.^23,25 To obtain an approximate analytical solution for the nonlinear differential equation (9), the first step is to convert the original second-order form into a system of two first-order differential equations.

x = u .

(23a)

\dot{x} = v .

(23b)

This is accomplished by adding suitable state variables that reflect the oscillator’s displacement and velocity, which makes the equation structure simpler and easier to analyse.²⁶ This shift is primarily the result of Ms-DTM’s methodical application, which improves convergence and accuracy by repeatedly generating the solution in several phases.^23,25 This synthetic model enables the capture of complex nonlinear dynamics and the systematic computation of higher-order components, eliminating the need for costly numerical simulations.

Following a step-by-step transformation of the governing equation’s first-order canonical form, we first demonstrate the precise solution using the multi-stage differential transform approach. The resulting analytical expressions demonstrate how well Ms-DTM captures the system’s behaviour when subjected to periodic external forcing.²⁴

Therefore, the system can be expressed as follows:

\frac{d u}{d t} = v .

(24a)

\frac{d v}{d t} = {- ω}_{o}^{2} (u + u^{3} + u^{5}) + q_{0} \cos (ω t) .

(24b)

The Ms-DTM may now be used to systematically transform the system of first-order differential equations into a series representation, enabling the construction of an approximate analytical solution.

u (t) = {\begin{array}{c} \begin{array}{l} \sum_{n = 0}^{K} U_{1} (n) t^{n}, t \in [0, t_{1}] \\ \sum_{n = 0}^{K} U_{2} (n) t^{n}, {(t - t_{1})}^{n}, t \in [t_{1}, t_{2}] \end{array} \\ . \\ . \\ . \\ \sum_{n = 0}^{K} U_{M} (n) {(t - (M - 1))}^{n}, t \in [t_{M - 1}, t_{M}] \end{array} .

(25)

v (t) = {\begin{array}{c} \begin{array}{l} \sum_{n = 0}^{K} V_{1} (n) t^{n}, t \in [0, t_{1}] \\ \sum_{n = 0}^{K} V_{2} (n) t^{n}, {(t - t_{1})}^{n}, t \in [t_{1}, t_{2}] \end{array} \\ . \\ . \\ . \\ \sum_{n = 0}^{K} V_{M} (n) {(t - (M - 1))}^{n}, t \in [t_{M - 1}, t_{M}] \end{array} .

(26)

The study starts with the provided beginning circumstances $U_{1} (0) = 0.1, V_{1} (0) = 0$ .

Using the Ms- DTM’s iterative framework, one generates the following recurrence relations:

U_{i} (K + 1) = \frac{1}{K + 1} V_{i} (K)

V_{i} (K + 1) = \frac{1}{K + 1} {- ω_{o}^{2} (U_{i} (K) + \sum_{K_{2} = 0}^{K} \sum_{K_{1} = 0}^{K_{2}} U_{i} (K_{1}) U_{i} (K_{2} - K_{1}) U_{i} (K - K_{2}) + \sum_{K_{4} = 0}^{K} \sum_{K_{3} = 0}^{K_{4}} \sum_{K_{2} = 0}^{K_{3}} \sum_{K_{1} = 0}^{K_{2}} U_{i} (K_{1}) U_{i} (K_{2} - K_{1}) U_{i} (K_{3} - K_{2}) U_{i} (K_{2} - K_{1}) U_{i} (K_{4} - K_{3}) U_{i} (K - K_{4}) + q_{0} \frac{ω^{K}}{K!} \cos (ω t)} .

(27)

The analytical solution developed using the Multi-Stage Differential Transform Method (Ms-DTM) is built mostly on the recurrence relations obtained in this part. We now perform a numerical study of the system’s reaction in order to confirm the dependability and correctness of the analytical approximation even more. The objective of this comparison research is to show the consistency among the analytical series and the exact numerical solutions under different forcing levels.^23,24 Compared with conventional numerical solvers for Eq. (6), the Multi-Stage Differential Transform Method (Ms-DTM) provides semi-analytical series solutions that maintain high accuracy with low computational effort. Its multi-stage formulation improves convergence for long-time responses, although the method remains limited by the convergence radius of the truncated series and may require more stages for strongly nonlinear or near-resonant cases.

6. Results and discussion

Using three different approaches—HPM, Ms-DTM, and the numerical integration scheme Runge-Kutta (ode45)—we show in Figures 1–3 the graphical representations of the solution to the cubic-quintic Duffing oscillator. Under the beginning parameters $x (0) = 0.1$ , $\dot{x} (0) = 0$ , the simulations were run with a little initial displacement and zero initial velocity. The parameters were selected as $q_{o} = 0.01$ , $ω = 1$ , and $ω_{o} = 3$ , which coincide with modest nonlinearity and periodic forcing. In nonlinear dynamic analysis, these values are usual to see both regular and perhaps chaotic oscillatory behaviour.

Figure 1.

Displacement x(t) versus Time t for the cubic-quintic Duffing oscillator using HPM, Ms-DTM, and Runge-Kutta (ode45) methods.

Figure 1 shows for the three approaches the displacement x(t) against time t. The great overlapping of the oscillatory trajectories shows that HPM, Ms-DTM, and ode45 very precisely record the temporal development of displacement. Especially, the analytical solutions produced by HPM and Ms-DTM smoothly coincide with the numerical reference given by ode45, therefore approximating the dynamics of the system in this regime both practically. The lack of detectable minor phase changes or amplitude variations emphasises the analytical approaches’ strength.

Figure 2 shows time t against velocity $\dot{x} (t)$ . Like the displacement graphs, the three techniques provide almost identical velocity profiles. This alignment reflects the capacity of HPM and Ms-DTM to follow the nonlinear acceleration and deceleration patterns caused by the cubic and quintic stiffness components, therefore suggesting not only the spatial state but also the rate of change of the system.

Figure 2.

Velocity $\dot{x} (t)$ against $T i m e$ (t) for the cubic-quintic Duffing oscillator using HPM, Ms-DTM, and Runge-Kutta (ode45) methods.

At last, Figure 3 shows the phase portrait $\dot{x} (t)$ versus x(t). The periodic and bounded character of the oscillator is clearly emphasized in this graphical form. The trajectories form smooth, closed curves that illustrate the stable periodic motion produced by the external forcing. The smooth transitions in the phase plane also indicate that both HPM and Ms-DTM accurately reproduce the underlying phase coherence observed in the numerical solution.

Figure 3.

Phase portrait of $\dot{x} (t)$ versus $x (t)$ for the cubic-quintic Duffing oscillator, illustrating bounded periodic motion for all three solution techniques.

A visual comparison of all three figures reveals a very high agreement between the analytical methods (HPM and Ms-DTM) and the numerical solution (ode45). This consistency demonstrates how reliable and accurate the proposed analytical techniques are in capturing the intricate nonlinear dynamics of the system. Additionally, it highlights their ability to accurately model oscillatory activity without the computational burden often associated with full numerical integration. Furthermore, the well-organised phase picture and the concordance in displacement and velocity responses suggest that these methods are suitable for studying increasingly intricate driven nonlinear oscillator designs.

Table 1 below shows values from three different numerical methods (Ms-DTM, Homotopy Perturbation, and Runge-Kutta (Ode45)) over 10 values of t. It also includes two sets of absolute errors: one compares Runge-Kutta to Ms-DTM, while the other compares Runge-Kutta to Homotopy Perturbation. When compared to Runge-Kutta, Homotopy Perturbation exhibits noticeably better accuracy and agreement than Ms-DTM, according to the absolute error analysis. The small and gradually growing error between Homotopy Perturbation and Runge-Kutta indicates that the two methods are extremely. Ms-DTM, on the other hand, deviates significantly from Runge-Kutta over time, as evidenced by the constantly growing and significantly higher inaccuracies between the two. If Runge-Kutta is taken as the reference, then Ms-DTM is a less accurate. The central processing unit (CPU) running times for ode45, Ms- DTM and HPM are 0.189694, 0.054448, and 7.298927 seconds respectively, whereas the total elapsed time is 7.538428 seconds.

Table 1.

Absolute errors between Runge-Kutta and Homotopy perturbation, and absolute errors between Runge-Kutta and Ms-DTM at $t = 1, 2, \dots ., 10$ .

$t$	$x_{ode}$	$x_{H T P}$	$x_{M s - D T M}$	Absolute Error		$Relat ı ve Error$
$t$	CPU=0.189694 s	CPU=7.298927s	CPU=0.054448 s	$\| x_{H T P} - x_{o d e} \|$	$\| x_{M s - D T M} - x_{o d e} \|$	$\frac{\| x_{H T P} - x_{o d e} \|}{\| x_{o d e} \|}$	$\frac{\| x_{M s - D T M} - x_{o d e} \|}{\| x_{o d e} \|}$
0	0.100000	0.100000	0.100000	0.000000	0.000000	0.000000	0.000000
1	-0.097262	-0.097244	-0.098569	0.000018	0.001307	0.000185	0.013441
2	0.094894	0.094921	0.098011	0.000027	0.003120	0.000284	0.032847
3	-0.092504	-0.092594	-0.090745	0.000090	0.001760	0.000973	0.019015
4	0.084809	0.084917	0.084486	0.000108	0.000323	0.001273	0.003809
5	-0.078138	-0.078312	-0.072111	0.000174	0.006030	0.002227	0.077133
6	0.071210	0.071471	0.061658	0.000261	0.009552	0.003665	0.134140
7	-0.059463	-0.059741	-0.046103	0.000278	0.013360	0.004675	0.224680
8	0.049564	0.049880	0.033418	0.000316	0.016146	0.006376	0.325760
9	-0.039377	-0.039718	-0.016402	0.000341	0.022980	0.008660	0.583460
10	0.025085	0.025372	0.002871	0.000287	0.022214	0.011441	0.885550

7. Conclusion

The present study demonstrates that HPM and Ms-DTM are effective analytical techniques for predicting the dynamics of the cubic-quintic Duffing oscillator. To obtain explicit solutions for the system’s temporal response, the accuracy and reliability of the approximations were evaluated against numerical integration using the Runge-Kutta (ode45) method. The graphical representations of displacement, velocity, and phase portraits demonstrated a high degree of consistency across all three techniques, confirming the resilience of Ms-DTM and HPM in capturing nonlinear oscillatory behavior.

The Ms-DTM’s piecewise nature made it a valuable tool for accelerating convergence over extended periods. Because it allows for local error control and enhanced stability, this segmented form would be perfect for long-term simulations of nonlinear oscillators. However, with minimal computational effort, the HPM generates symbolic series expressions, providing a rapid method for analysing system behaviour and determining the influence of nonlinear stiffness components. In particular, the amplitude-dependent stiffness and frequency bending of the cubic term are reflected in the perturbation series produced by HPM. Additionally, the quintic term’s stiffening effect at larger amplitudes naturally accommodates higher harmonics. Both analyses.

The ability of both analytical methods to reproduce complex waveform distortions present in the oscillator’s response was demonstrated by their successful capture of these nonlinear effects.

While both techniques have proven to be efficient and accurate, certain limitations have been observed. In long-term simulations, especially, the Ms-DTM requires strict attention to subinterval boundaries to prevent error propagation between segments. Conversely, in substantially nonlinear or chaotic environments, where the series expansion finds difficulty preserving convergence, the HPM might suffer. Furthermore, neither approach is inherently suited to manage stochastic inputs or discontinuous forcing factors, implying that additional changes are required to enhance their relevance to such complex dynamical situations.

Future studies might investigate the expansion of these techniques to incorporate chaotic regimes and damping effects, where nonlinearity generates more complex behaviours and possible bifurcations. Investigating stochastic excitations or impulsive forces may also extend the applicability of these analytical methods and enable them to address a broader spectrum of practical nonlinear systems. The present work provides the foundation for practical and effective analytical methods that address the difficulties of severely nonlinear oscillatory behavior, thereby providing insight and precision with minimal computing overhead.

Footnotes

Acknowledgment

The authors Rania Wannan, and Jihad Asad would like to thank Palestine Technical University- Kadoorie for supporting them financially during this research.

ORCID iDs

Rania Wannan

Olivia Florea

Jihad Asad

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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Analytical approximation and dynamic response of the driven cubic–quintic Duffing oscillator

Abstract

Keywords

1. Introduction

2. The driven cubic-quintic Duffing oscillator

3. Existence and uniqueness of the solution of (6)

4. Homotopy perturbation method

5. Ms- DTM method

6. Results and discussion

7. Conclusion

Footnotes

Acknowledgment

ORCID iDs

Funding

Declaration of conflicting interests

References