Closed-Form Heteroclinic Orbits for a Three-Parameter Dynamical System Using a Modified Optimal Parametric Iteration Method

Abstract

Numerous applications from electrical engineering and mechanical structures are mathematically modeled using dynamical systems theory. Our paper concerns the behaviors of a 3D dynamic system in terms of damped or periodical oscillations and asymptotic representation, considering the dependence on three physical parameters. This system is explicitly integrated via a smooth-function solution of a third–order nonlinear differential equation, which means that the obtained exact parametric solutions describe a heteroclinical orbit. The modified Optimal Parametric Iteration Method (mOPIM) is used to study the influence of the physical parameters. The advantages of the applied method include the small number of iterations due to due to the appropriate choice of auxiliary convergence control functions. The mOPIM solutions are in good agreement with the corresponding numerical results and this aspect is highlighted qualitatively by figures and quantitatively by tables, respectively, in this work. The accuracy of the obtained solutions is assessed via a comparison with the OPIM method and the iterative solutions using 5–8 iterations, via an iterative method. A qualitative analysis of errors is performed.

1. Introduction

Dynamical systems theory is the foundation of many studies on complex phenomena in nature. It is an important branch of mathematics that is used to describe the evolution of systems by differential equations. This discipline has allowed the modeling of diverse processes, from planetary orbits and electronic circuits to complex rendezvous systems in engineering, biology, medicine, economics, secure communications, etc.

In this field of dynamical systems theory, a turning point was represented by the three-dimensional mathematical model of atmospheric circulation introduced by Lorenz [1]. This model is a system of deterministic nonlinear equations capable of exhibiting unpredictable behavior. Lorentz thus laid the foundations of chaos theory, a theory that describes and seeks to explain this type of unpredictable behavior in nonlinear systems, even over short time intervals. A defining feature of chaos in dynamical systems is its extreme sensitivity to initial conditions. A tiny difference in initial conditions can lead to completely different outcomes, often illustrated by the “butterfly effect” [2,3,4]. Thus, if trajectories that start extremely close to each other diverge exponentially in time, this is quantified by a positive Lyapunov exponent [5,6]. There are also recent studies that have shown the sensitivity of chaos to small variations in the system parameters, independent of the sensitivity to initial conditions [5].

This research was later extended by the development of other three-dimensional models, of which we mention only the Rössler [7], Chen [8], Lü [9] or Qi [10] systems, each bringing new insights into the complexity and unpredictability of the studied phenomena [11].

Three-dimensional systems of ordinary differential equations, such as the system studied in this work, provide an efficient means to capture the complicated interactions between multiple entities, exhibiting rich dynamical properties, such as equilibria, periodic orbits, heteroclinic orbits and global connections.

Heteroclinic orbits, which connect different equilibrium points, help us to understand global transitions and system stability. Their identification is used in practical applications such as chaos synchronization, network security, data encryption and secure communications. When it comes to improving encryption methods, the need to protect and secure information transmission poses significant problems, which must be addressed to further the development of computer science. Also, the dynamics of these orbits are relevant in fluid mechanics, modeling double-convection phenomena, magnetohydrodynamic flows or transport processes in porous media, with applications in soil decontamination and waste water treatment [12,13,14].

Recent advances in data-driven techniques and machine learning approaches have revolutionized the way dynamical systems are modeled and analyzed in modern times, and they are approached together with emerging techniques in data science and machine learning. The integration of these techniques with dynamical systems theory opens up opportunities to study certain phenomena that were previously unattainable in dynamical systems modeling and prediction [15]. Although data-driven prediction methods have made great progress in recent years, it is necessary to develop new techniques to improve their applicability to a wider range of complex systems in science and engineering. In [16], recent developments in the field of complex dynamical systems are presented, with an emphasis on the discovery of data-driven, data-assisted, and artificial intelligence-based dynamical systems.

The study of these dynamical systems is often limited to numerical analysis, because obtaining closed-form solutions is extremely difficult due to the high nonlinearities.

Classical numerical methods have limitations related to discretization, step size sensitivity and accumulation errors in chaotic regimes. To overcome these obstacles, semi-analytic techniques such as the Optimal Auxiliary Function Method (OAFM) [17] and the Optimal Parametric Iteration Method (OPIM) [18] have been developed. Unlike traditional numerical methods that are based on domain discretization and are sensitive to step size, OPIM works directly on the continuous problem, without requiring transformations or other discretization. The OPIM works directly on the continuous problem using some auxiliary functions depending on several real parameters that are optimally computed via the least squares method. This allows the user to control and accelerate the convergence rate of approximations to the exact solution. The method also allows high-precision results to be obtained with a low number of iterations, which is preferable in cases where traditional discretization methods converge slowly or encounter difficulties. Using this method, the obtained approximate solutions quickly converge to the exact solution, taking into consideration the stability of equilibrium points [17,18,19].

This paper proposes to determine closed-form heteroclinic orbits for one nonlinear dynamical system using mOPIM method. The mOPIM method offers high accuracy with a minimum number of iterations, making it superior to traditional iterative or variational methods. Based on the presence of a finite number of initially unknown parameters, which are optimally determined, providing a rigorous way to control the convergence of solutions, this approach not only simplifies the computational effort, but also provides a superior qualitative understanding of the impact of physical parameters on the system, for exploring a wide class of nonlinear problems with applications in science and engineering [17,18,20].

The paper is organized as follows: After Section 1, Introduction, Section 2 presents, briefly, some considerations about the dynamical analyzed system and theorems for the existence of explicitly solutions. Section 3 is dedicated to the theoretical approach of the mOPIM method. Section 4 is devoted to the results and interpretation of the physical parameters effects. Qualitative analysis of errors is presented in details in Section 5. The paper ends with conclusions.

2. Closed-Form Solutions for q 3D Dynamical System

The proposed three parameter dynamical system has the form shown in [21]:

$$\left\{\begin{array}{c}\dot{x}=-x-2y\hfill \\ \dot{y}=-xz-by-ax\hfill \\ \dot{z}=xy-cz\hfill \end{array}\right.,$$

(1)

with the initial conditions

$$\begin{array}{c}x\left(0\right)=x_{i0},y\left(0\right)=y_{i0},z\left(0\right)=z_{i0},\hfill \end{array}$$

(2)

where $a,b,c>0$ are dimensionless physical parameters.

The matrix of linear part of system (1) is:

$$J(x,y,z)=\left(\begin{array}{ccc}-1\hfill & -2\hfill & 0\hfill \\ -z-a\hfill & -b\hfill & -x\hfill \\ y\hfill & x\hfill & -c\hfill \end{array}\right).$$

(3)

This system admits three equilibrium points: $P_0(0,0,0)$, $P_1(\sqrt{c(2a-b)},-{\displaystyle \frac{1}{2}}\sqrt{c(2a-b)},{\displaystyle \frac{b-2a}{2}})$, $P_2(-\sqrt{c(2a-b)},{\displaystyle \frac{1}{2}}\sqrt{c(2a-b)},{\displaystyle \frac{b-2a}{2}})$, if $2a>b$, $c>0$.

The characteristic polynomial corresponding to the Jacobian matrix $J\left(P^{*}\right)$ from (3) around the equilibrium point $P^{*}(x^{*},y^{*},z^{*})$ is

$$p_{J\left(P^{*}\right)}\left(\lambda \right)={\lambda }^3+C_2{\lambda }^2+C_1\lambda +C_0,$$

where $C_0=bc-2x^{*}{y}^{*}-2ac-2c{z}^{*}+{\left(x^{*}\right)}^2$, $C_1=bc+{\left(x^{*}\right)}^2+b+c-2z^{*}-2a$, $C_2=b+c+1$.

Based on the Routh–Hurwitz criterion, only if $C_2>0$, $C_1>0$, $C_0>0$ and $C_2{C}_1-C_0>0$, the real parts of all the roots are negative, so the equilibrium points are asymptotically stable; thus, they are unstable.

System (1) is explicitly integrated in two cases:

Case 1. $c=2$.

The next proposition assures the existence of closed-form solutions of the System (1) with initial data (2) in this case.

Proposition 1.

For $c=2$, the System (1) is explicitly integrated via a smooth function $w:[0,\infty )\to {\mathbb{R}}^{*}$ by:

$$\left\{\begin{array}{c}x=w\hfill \\ y={\displaystyle \frac{1}{4}}\left(-2w^{\prime }-2w\right)\hfill \\ z={\displaystyle \frac{1}{4}}\left(R_1{e}^{-2t}-w^2\right)\hfill \end{array}\right.,$$

(4)

where $R_1=x_{i0}^2+z_{i0}$ and w is an unknown smooth function solution to the following problem:

$$\left\{\begin{array}{c}-2w{w}^{″}-4w^{\prime }+(2-2b)w{w}^{\prime }+(4a-2b)w^2+\left(R_1{e}^{-2t}-w^2\right)w^2=0,\hfill \\ w\left(0\right)=x_{i0},w^{\prime }\left(0\right)=-x_{i0}-2y_{i0}.\hfill \end{array}\right.$$

(5)

Proof. 

First of all, there exists only one first integral

$$x^2+4z=R_1{e}^{-2t},R_1=x_{i0}^2+z_{i0},$$

using the first and the third equation of the System (1).

Therefore, there is an unknown smooth function $w\left(t\right)$ for which:

$$\begin{array}{c}x\left(t\right)=w\left(t\right),\hfill \\ z\left(t\right)={\displaystyle \frac{1}{4}}\left(R_1{e}^{-2t}-w^2\right),\hfill \end{array}$$

(6)

with $w\left(0\right)=x_{i0}$.

By replacing $x\left(t\right)$ and $z\left(t\right)$ from Equation (6) into the third equation from System (1), we obtain the following result:

$$\begin{array}{c}y={\displaystyle \frac{1}{4}}\left(-2w^{\prime }-2w\right),\hfill \end{array}$$

(7)

whereas $w^{\prime }\left(0\right)=-x_{i0}-2y_{i0}$.

Taking into account the second equation of the System (1) using Equations (6) and (7), the results described in Equation (5) are obtained. □

Case 2. $c\ne 2$; asymptotic behaviors.

The next proposition assures the existence of the closed-form solutions of the System (1).

Without limiting generality, we assume that $c>2$.

Proposition 2.

For $c>2$, the System (1) admits closed-form solutions. It is explicitly integrated via a smooth function $w:[0,\infty )\to {\mathbb{R}}^{*}$ by:

$$\begin{array}{c}{\displaystyle x=\pm \sqrt{w-\frac{1}{2-c}(w^{\prime }+2w)},}\hfill \\ {\displaystyle y=\frac{1}{4(2-c)}\frac{w^{″}+(2+c)w^{\prime }+2cw}{\pm \sqrt{w-\frac{1}{2-c}(w^{\prime }+2w)}},}\hfill \\ {\displaystyle z=\frac{1}{4(2-c)}(w^{\prime }+2w),}\hfill \end{array}$$

(8)

where w satisfies

$$\begin{array}{c}{\displaystyle \frac{\sqrt{c-2}}{4(2-c)}}\left(w^{‴}+(c-2)w^{″}+(2c-8)w^{\prime }\right)(cw+w^{\prime })-{\displaystyle \frac{\sqrt{c-2}}{8(2-c)}}\left(w^{″}+(c-2)w^{\prime }+(2c-8)w\right)\times \hfill \\ (c{w}^{\prime }+w^{″})=-{\displaystyle \frac{1}{4(2-c)\sqrt{c-2}}}{(cw+w^{\prime })}^2(2w+w^{\prime })-\hfill \\ {\displaystyle \frac{b\sqrt{c-2}}{4(2-c)}}(cw+w^{\prime })\left(w^{″}+(c-2)w^{\prime }+(2c-8)w\right)-{\displaystyle \frac{a}{\sqrt{c-2}}}{(w^{\prime }+cw)}^2,\hfill \\ w\left(0\right)=x^2\left(0\right)+4z\left(0\right),w^{\prime }\left(0\right)=-2x^2\left(0\right)-4cz\left(0\right),\hfill \\ w^{″}\left(0\right)=4(2-c)y\left(0\right)\left|x\left(0\right)\right|+4x^2\left(0\right)+4c^{2}z\left(0\right).\hfill \end{array}$$

(9)

Proof. 

From the first and the third equations of System (1) we have, successively:

$$x\dot{x}+2\dot{z}=-x^2-2cz\Rightarrow {\displaystyle \frac{d}{dt}}(x^2+4z)=-2(x^2+4z)+z(8-4c),$$

and then

$$\begin{array}{c}{x}^2+z=e^{-2t}\left({\displaystyle C_1+(8-4c)\underset{0}{\overset{t}{\int }}z\left(s\right)e^{2s}ds}\right).\hfill \end{array}$$

(10)

Therefore, from (10) there is an unknown smooth function $\mathsf{\Psi }\left(t\right)$ for which:

$$\begin{array}{c}{x}^2\left(t\right)={\displaystyle \frac{1}{2}}{e}^{-2t}\mathsf{\Phi }\left(t\right)-\mathsf{\Psi }\left(t\right),\hfill \\ 4z\left(t\right)={\displaystyle \frac{1}{2}}{e}^{-2t}\mathsf{\Phi }\left(t\right)+\mathsf{\Psi }\left(t\right),\hfill \end{array}$$

(11)

with ${\displaystyle \mathsf{\Phi }\left(t\right)=C_1+(8-4c)\underset{0}{\overset{t}{\int }}z\left(s\right)e^{2s}ds}$, $C_1=x^2\left(0\right)+4z\left(0\right)=\mathsf{\Phi }\left(0\right)$.

By replacing $x\left(t\right)$ and $z\left(t\right)$ from Equation (11) in the third equation from System (1), we obtain:

$$\begin{array}{c}xy={\displaystyle \frac{1}{4}}\left[{\displaystyle \frac{1}{2}}(c-2)e^{-2t}\mathsf{\Phi }\left(t\right)+{\displaystyle \frac{1}{2}}{e}^{-2t}{\mathsf{\Phi }}^{\prime }\left(t\right)+{\mathsf{\Psi }}^{\prime }\left(t\right)+c\mathsf{\Psi }\left(t\right)\right],\hfill \end{array}$$

(12)

where ${\mathsf{\Phi }}^{\prime }\left(t\right)={\displaystyle \frac{d\mathsf{\Phi }}{dt}}$.

From the first equation of the System (1), using the Equations (11) and (12) yields the relation

$$\begin{array}{c}{\displaystyle \frac{2-c}{2}}\mathsf{\Psi }\left(t\right)={\displaystyle \frac{1}{4}}(c-2)e^{-2t}\mathsf{\Phi }\left(t\right)+{\displaystyle \frac{1}{2}}{e}^{-2t}{\mathsf{\Phi }}^{\prime }\left(t\right),\hfill \end{array}$$

(13)

whence, it follows that

$$\begin{array}{c}\mathsf{\Psi }\left(t\right)=-{\displaystyle \frac{1}{2}}{e}^{-2t}\mathsf{\Phi }\left(t\right)+{\displaystyle \frac{1}{2-c}}{e}^{-2t}{\mathsf{\Phi }}^{\prime }\left(t\right).\hfill \end{array}$$

(14)

Combining Equations (11) and (14), it is deduced

$$\begin{array}{c}{x}^2\left(t\right)=e^{-2t}\mathsf{\Phi }\left(t\right)-{\displaystyle \frac{1}{2-c}}{e}^{-2t}{\mathsf{\Phi }}^{\prime }\left(t\right),\hfill \\ 4z\left(t\right)={\displaystyle \frac{1}{2-c}}{e}^{-2t}{\mathsf{\Phi }}^{\prime }\left(t\right),\hfill \end{array}$$

(15)

whereas ${\mathsf{\Phi }}^{\prime }\left(0\right)=4(2-c)z\left(0\right)$.

Using the third equation from System (1), the following is obtained:

$$\begin{array}{c}y\left(t\right)={\displaystyle \frac{1}{4(2-c)}}{\displaystyle \frac{e^{-2t}{\mathsf{\Phi }}^{″}\left(t\right)+(c-2)e^{-2t}{\mathsf{\Phi }}^{\prime }\left(t\right)}{\pm \sqrt{e^{-2t}\mathsf{\Phi }\left(t\right)-\frac{1}{2-c}{e}^{-2t}{\mathsf{\Phi }}^{\prime }\left(t\right)}}},\hfill \end{array}$$

(16)

which implies ${\mathsf{\Phi }}^{″}\left(0\right)=\pm y\left(0\right)(2-c)\sqrt{\mathsf{\Phi }\left(0\right)-{\displaystyle \frac{1}{2-c}}{\mathsf{\Phi }}^{\prime }\left(0\right)}+(2-c){\mathsf{\Phi }}^{\prime }\left(0\right)$.

A new function $w\left(t\right)$ is introduced by

$$\begin{array}{c}w\left(t\right)=e^{-2t}\mathsf{\Phi }\left(t\right).\hfill \end{array}$$

(17)

Hence, $e^{-2t}{\mathsf{\Phi }}^{\prime }\left(t\right)=w^{\prime }\left(t\right)+2w\left(t\right)$, $e^{-2t}{\mathsf{\Phi }}^{″}\left(t\right)=w^{″}\left(t\right)+4w^{\prime }\left(t\right)+4w\left(t\right)$.

These relations, together with Equations (15) and (16), yield to explicitly solutions given by Equation (8), via an unknown smooth function $w\left(t\right)$ defined in (17).

Substituting Equation (8) into the second equation from System (1) yields the desired results given by Equation (9). □

3. Theoretical Approach of the Modified Optimal Parametric Iteration Method (mOPIM)

3.1. Basic Ideas

Step 1. Selection of the linear operator $\mathcal{L}[t,u,u^{\prime },u^{″},u^{‴},\dots ,u^{\left(n\right)}]$ and the nonlinear operator $\mathcal{N}[t,u,u^{\prime },u^{″},u^{‴},\dots ,u^{\left(n\right)}]$, respectively. This selection is based on an eigenvalues analysis of the system’s linear part.

The OPIM method, introduced by Marinca et al. [22] described the n-th order ODE in the form:

$$\begin{array}{c}{\displaystyle \mathcal{L}[t,u,u^{\prime },u^{″},u^{‴},\dots ,u^{\left(n\right)}]+\mathcal{N}[t,u,u^{\prime },u^{″},u^{‴},\dots ,u^{\left(n\right)}]-g\left(t\right)=0,t\in (0,\infty ),}\hfill \end{array}$$

(18)

subject to

$$\begin{array}{c}{\displaystyle \mathcal{B}[u\left(0\right),u^{\prime }\left(0\right),u^{″}\left(0\right),\dots ,u^{(n-1)}\left(0\right)]=0,}\hfill \end{array}$$

(19)

with $\mathcal{B}$ a boundary operator, g and u known/unknown functions, with u depending on the independent variable t and $u^{\prime }\left(t\right)={\displaystyle \frac{du}{dt}}$.

In this section, we propose a modified OPIM version (mOPIM) writing the initial Equation (18) in the form

$$\begin{array}{c}{\displaystyle \mathcal{L}[t,u,u^{\prime },u^{″},u^{‴},\dots ,u^{\left(n\right)}]-\phi \left(t\right)+\mathcal{N}[t,u,u^{\prime },u^{″},u^{‴},\dots ,u^{\left(n\right)}]-\tilde{g}\left(t\right)=0,}\hfill \end{array}$$

(20)

with $\phi \left(t\right)$ as a known arbitrary smooth function, $\tilde{g}\left(t\right)=g\left(t\right)-\phi \left(t\right)$.

Next, we present the idea of the OPIM technique.

Let ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, and … be real values. The idea of OPIM is to apply the Taylor formula for an analytic function F by:

$$\begin{array}{c}F(t,u+{\alpha }_0,u^{\prime }+{\alpha }_1,u^{″}+{\alpha }_2,\dots ,u^{\left(n\right)}+{\alpha }_n)=F(t,u,u^{\prime },u^{″},\dots )+{\displaystyle \frac{{\alpha }_0}{1!}}{F}_u(t,u,u^{\prime },u^{″},\dots )+\hfill \\ {\displaystyle \frac{{\alpha }_1}{1!}}{F}_{u^{\prime }}(t,u,u^{\prime },u^{″},\dots )+{\displaystyle \frac{{\alpha }_2}{1!}}{F}_{u^{″}}(t,u,u^{\prime },u^{″},\dots )+\dots ,\hfill \end{array}$$

(21)

with $F_u={\displaystyle \frac{\partial F}{\partial u}}$.

Step 2. Evaluation of expressions ${\mathcal{N}}_{u^{\left(m\right)}}={\displaystyle \frac{\partial \mathcal{N}}{\partial u^{\left(m\right)}}}$, $m=\overline{1,n}$.

Instead of solving the Equation (20), one can solve another equation, recursing to Equation (21) and to the following OPIM scheme

$$\begin{array}{c}{\displaystyle \mathcal{L}[t,u_{k+1},u_{k+1}^{\prime },u_{k+1}^{″},\dots ,u_{k+1}^{\left(n\right)}]-\phi \left(t\right)+\mathcal{N}[t,u_k,u_k^{\prime },u_k^{″},\dots ]+{\theta }_{k,0}(t,C_{i_0}){\mathcal{N}}_u[t,u_k,u_k^{\prime },u_k^{″},\dots ]+}\hfill \\ {\theta }_{k,1}(t,C_{i_1}){\mathcal{N}}_{u^{\prime }}[t,u_k,u_k^{\prime },u_k^{″},\dots ]+{\theta }_{k,2}(t,C_{i_2}){\mathcal{N}}_{u^{″}}[t,u_k,u_k^{\prime },u_k^{″},\dots ]+\cdots -\tilde{g}\left(t\right)=0,\hfill \\ {\displaystyle \mathcal{B}[u_{k+1}\left(0\right),u_{k+1}^{\prime }\left(0\right),\dots ,u_{k-1}^{(n-1)}\left(0\right)]=0,k\ge 0,}\hfill \end{array}$$

(22)

where ${\theta }_{k,m}(t,C_{i_m})$, $m\ge 0$ are auxiliary continuous functions; $\left\{i_m\right\}$ is the index set; $C_{i_m}$ is a set of arbitrary real parameters; ${\mathcal{N}}_F={\displaystyle \frac{\partial \mathcal{N}}{\partial F}}$.

A function $u_{k+1}\left(t\right)$, solution of Equation (22), called the mOPIM solution, is ($k+1$)-th-order semi-analytical solution of Equations (20) and (19) will be denoted by $\overline{u}\left(t\right)$ or ${\overline{u}}_{mOPIM}\left(t\right)$.

For known convergence–control parameters $C_{i_m}$ mOPIM solution ${\overline{u}}_{mOPIM}\left(t\right)$ of Equations (20) and (19) is built.

Step 3. Finding the initial approximation $u_0\left(t\right)$.

For $k=0$, $u_0\left(t\right)$ is the solution to the problem:

$$\begin{array}{c}{\displaystyle \mathcal{L}[t,u_0,u_0^{\prime },\dots ]-\phi \left(t\right)-g\left(t\right)=0}\hfill \\ {\displaystyle \mathcal{B}[u_0\left(0\right),u_0^{\prime }\left(0\right),\dots ]=0.}\hfill \end{array}$$

(23)

Step 4. Evaluation of expressions $\mathcal{N}[t,u_0,u_0^{\prime },\dots ]$, ${\mathcal{N}}_{u^{\left(m\right)}}[t,u_0,u_0^{\prime },\dots ]$, $m=\overline{1,n}$.

For $u_0\left(t\right)$ a solution of Equation (23), the operators $\mathcal{N}[t,u_0,u_0^{\prime },\dots ]$, ${\mathcal{N}}_u[t,u_0,u_0^{\prime },\dots ]$, ${\mathcal{N}}_{u^{\prime }}[t,u_0,u_0^{\prime },\dots ]$, ${\mathcal{N}}_{u^{″}}[t,u_0,u_0^{\prime },\dots ]$ and so on, from Equation (22), have the form

$$\begin{array}{c}{\displaystyle \sum _{i=1}^{n_{max}}{h}_i\left(t\right)g_i\left(t\right),}\hfill \end{array}$$

(24)

where $n_{max}$ is a positive integer and $h_i\left(t\right)$ are linearly independent functions and $g_i\left(t\right)$ are known functions that depend on $u_0\left(t\right)$.

Step 5. Identifying independent functions $h_i$ from above expressions.

Step 6. The choice of auxiliary functions ${\mathcal{A}}_i(t,C_{j_i})$ depending on $u_0\left(t\right)$, $g_i\left(t\right)$ such that ${\mathcal{A}}_i(t,C_{j_i})h_i\left(t\right)$ have same form with $h_i\left(t\right)g_i\left(t\right)$, $i=\overline{1,n_{max}}$.

Step 7. Calculus of the $k+1$-order approximate solution $u_{k+1}$ by integration of Equation (25).

Thus, the Equation (22) becomes:

$$\begin{array}{c}{\displaystyle \mathcal{L}[t,u_{k+1},u_{k+1}^{\prime },u_{k+1}^{″},\dots u_{k+1}^{\left(n\right)}]=\sum _{i=1}^{n_{max}}{\mathcal{A}}_i(t,C_{j_i})h_i\left(t\right)}\hfill \\ {\displaystyle \mathcal{B}[u_{k+1}\left(0\right),u_{k+1}^{\prime }\left(0\right),\dots ,u_{k+1}^{\left(n\right)}\left(0\right)]=0}\hfill \end{array},k\ge 0.$$

(25)

For $u_{k+1}$ an $(k+1)$-order approximate solution of Equations (20) and (19), the validation of this procedure is highlighted by computing the residual function given by

$$\begin{array}{c}{\displaystyle \mathcal{R}(t,u_{k+1}\left(t\right))=\mathcal{L}[t,u_{k+1},u_{k+1}^{\prime },u_{k+1}^{″},\dots u_{k+1}^{\left(n\right)}]+\mathcal{N}[t,u_{k+1},u_{k+1}^{\prime },u_{k+1}^{″},\dots u_{k+1}^{\left(n\right)}]-g\left(t\right),}\hfill \end{array}$$

(26)

such that ${\displaystyle \mathcal{R}(t,u_{k+1}\left(t\right))<<1}$, for all $t\in (0,\infty )$.

Reals parameters $C_{j_i}$ are optimally identified, using one of the known methods such as: the collocation method [23,24], numerically solving the system of equations $\mathcal{R}(t_1,u_{k+1}\left(t_1\right))=0,\mathcal{R}(t_2,u_{k+1}\left(t_2\right))=0,\dots ,\mathcal{R}(t_{N_1},u_{k+1}\left(t_{N_1}\right))=0,$ using (26), or the classical least squares method by minimizing the functional

$${\displaystyle \mathcal{E}=\sum _{i_1=1}^{N_1}{\left(u_{k+1}\left(t_{i_1}\right)-u_{numerical}\left(t_{i_1}\right)\right)}^2,}$$

(27)

if numerical solution $u_{numerical}$ (computed, for example, via the fourth-order Runge–Kutta method) is known. The set $\{t_{i_1}\in [0,T_1],i_1=1,2,\dots ,N_1\}$, of real numbers is arbitrary, $N_1$ is a given number greater than the number of optimally parameters $C_{j_i}$.

Using linearly independent functions $h_1,h_2,\cdots ,h_m$, we introduce some types of approximate solutions of Equation (18).

Some types of semi-analytical solutions of Equation (18) are defined in [25].

The existence of weak $\epsilon$-approximate mOPIM solutions is evidenced by the theorem presented above.

Theorem 1.

Equation (18) admits a sequence of weak ε-approximate mOPIM solutions.

Proof. 

It is similar to the Theorem from [25]. □

3.2. Semi-Analytical Solutions via mOPIM Technique

Case 1: $c=2$

The nonlinear problem given by Equation (5) can be semi-analytically solved by the mOPIM technique as follows:

Step 1. Select operators $\mathcal{L}[w\left(t\right),w^{\prime }\left(t\right),w^{″}\left(t\right),w^{‴}\left(t\right)]$ and $\mathcal{N}[w\left(t\right),w^{\prime }\left(t\right),w^{″}\left(t\right),w^{‴}\left(t\right)]$ from Equation (18) ($g\left(t\right)=0$) as:

$$\begin{array}{c}{\displaystyle \mathcal{L}[w,w^{\prime },w^{″},w^{‴}]=w^{″}+K{w}^{\prime }-{\phi }_1,}\hfill \\ {\displaystyle \mathcal{N}[w\left(t\right),w^{\prime }\left(t\right),w^{″}\left(t\right),w^{‴}\left(t\right)]=-2w{w}^{″}-4w^{\prime }+(2-2b)w{w}^{\prime }+(4a-2b)w^2+}\hfill \\ \left(R_1{e}^{-2t}-w^2\right)w^2-w^{″}-K{w}^{\prime }+{\phi }_1,\hfill \end{array}$$

(28)

with ${\phi }_1\left(t\right)=\left({\tilde{G}}_{0}cos\left({\omega }_{0}t\right)+{\tilde{G}}_{1}sin\left({\omega }_{0}t\right)\right)e^{-K_{z}t}$ given the smooth function (taking into consideration the eigenvalues analysis of the system’s linear part.), K, $K_z$, ${\omega }_0$, ${\tilde{G}}_0$, ${\tilde{G}}_1$ arbitrary real constants at this moment.

Remark 1.

The first step of the mOPIM technique highlights a first advantage, namely the choice of the operator $\mathcal{L}$ given by (28), such that an initial guess $w_0\left(t\right)$ could be an elementary function. The choice of the linear operator $\mathcal{L}$ does not depend on the existence of any small parameter.

Step 2. Evaluation of expressions ${\mathcal{N}}_w[w,w^{\prime },w^{″},w^{‴}]$, ${\mathcal{N}}_{w^{\prime }}[w,w^{\prime },w^{″},w^{‴}]$, ${\mathcal{N}}_{w^{″}}[w,w^{\prime },w^{″},w^{‴}]$ and ${\mathcal{N}}_{w^{‴}}[w,w^{\prime },w^{″},w^{‴}]$ as:

$$\begin{array}{c}{\displaystyle {\mathcal{N}}_w[w,w^{\prime },w^{″},w^{‴}]=-2\ddot{w}+(2-2b)w^{\prime }+2(4a-2b)w+2R_1{e}^{-2t}w-4w^3,}\hfill \\ {\displaystyle {\mathcal{N}}_{w^{\prime }}[w,w^{\prime },w^{″},w^{‴}]=(2-2b)w-(4+K),}\hfill \\ {\displaystyle {\mathcal{N}}_{w^{″}}[w,w^{\prime },w^{″},w^{‴}]=-2w-1,{\mathcal{N}}_{w^{‴}}[w,w^{\prime },w^{″},w^{‴}]=0.}\hfill \end{array}$$

(29)

Step 3. Finding the initial approximation $w_0\left(t\right)$, the solution to the Equation (23) is:

$$\begin{array}{c}{\displaystyle w_0\left(t\right)=A_0+B_0{e}^{-Kt}+\left(G_{0}cos\left({\omega }_{0}t\right)+G_{1}sin\left({\omega }_{0}t\right)\right)e^{-K_{z}t},}\hfill \end{array}$$

(30)

with $A_0$, $B_0$, $G_0$, $G_1$ arbitrary real constants depending on K, $K_z$, ${\omega }_0$, ${\tilde{G}}_0$, ${\tilde{G}}_1$ at this moment.

Step 4. Evaluation of expressions $\mathcal{N}[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]$, ${\mathcal{N}}_w[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]$, ${\mathcal{N}}_{w^{\prime }}[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]$, ${\mathcal{N}}_{w^{″}}[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]$ and ${\mathcal{N}}_{w^{‴}}[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]$ using (29), (30) as:

$$\begin{array}{c}{\displaystyle \mathcal{N}[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]={\tilde{S}}_0+{\tilde{S}}_1{e}^{-Kt}+{\tilde{S}}_2{e}^{-2Kt}+{\tilde{S}}_3{e}^{-3Kt}+{\tilde{P}}_4{e}^{-2t}+{\tilde{S}}_5{e}^{-2K_{z}t}+}\hfill \\ {\tilde{S}}_6{e}^{-(K+2K_z)t}+{\tilde{S}}_7{e}^{-(2+K)t}+{\tilde{S}}_8{e}^{-(2+2K)t}+{\tilde{S}}_9{e}^{-(2+2K_z)t}+{\tilde{S}}_{10}{e}^{-(2K+2K_z)t}+\hfill \\ {\tilde{S}}_{11}{e}^{-4Kt}+{\tilde{S}}_{12}{e}^{-4K_{z}t}+{\tilde{T}}_1{e}^{-K_{z}t}cos\left({\omega }_{0}t\right)+{\tilde{T}}_2{e}^{-K_{z}t}sin\left({\omega }_{0}t\right)+\hfill \\ {\tilde{T}}_3{e}^{-3K_{z}t}cos\left({\omega }_{0}t\right)+{\tilde{T}}_4{e}^{-3K_{z}t}sin\left({\omega }_{0}t\right)+{\tilde{T}}_5{e}^{-(2+K_z)t}cos\left({\omega }_{0}t\right)+\hfill \\ {\tilde{T}}_6{e}^{-(2+K_z)t}sin\left({\omega }_{0}t\right)+{\tilde{T}}_7{e}^{-(K+K_z)t}cos\left({\omega }_{0}t\right)+{\tilde{T}}_8{e}^{-(K+K_z)t}sin\left({\omega }_{0}t\right)+\hfill \\ {\tilde{T}}_9{e}^{-(2K+K_z)t}cos\left({\omega }_{0}t\right)+{\tilde{T}}_{10}{e}^{-(2K+K_z)t}sin\left({\omega }_{0}t\right)+{\tilde{T}}_{11}{e}^{-(K+3K_z)t}cos\left({\omega }_{0}t\right)+\hfill \\ {\tilde{T}}_{12}{e}^{-(K+3K_z)t}sin\left({\omega }_{0}t\right)+{\tilde{T}}_{13}{e}^{-(3K+K_z)t}cos\left({\omega }_{0}t\right)+{\tilde{T}}_{14}{e}^{-(3K+K_z)t}sin\left({\omega }_{0}t\right)+\hfill \\ {\tilde{T}}_{15}{e}^{-(2+K+K_z)t}cos\left({\omega }_{0}t\right)+{\tilde{T}}_{16}{e}^{-(2+K+K_z)t}sin\left({\omega }_{0}t\right)+{\tilde{T}}_{17}{e}^{-2K_{z}t}cos\left(2{\omega }_{0}t\right)+\hfill \\ {\tilde{T}}_{18}{e}^{-2K_{z}t}sin\left(2{\omega }_{0}t\right)+{\tilde{T}}_{19}{e}^{-(K+2K_z)t}cos\left(2{\omega }_{0}t\right)+{\tilde{T}}_{20}{e}^{-(K+2K_z)t}sin\left(2{\omega }_{0}t\right)+\hfill \\ {\tilde{T}}_{21}{e}^{-(2+2K_z)t}cos\left(2{\omega }_{0}t\right)+{\tilde{T}}_{22}{e}^{-(2+2K_z)t}sin\left(2{\omega }_{0}t\right)+{\tilde{T}}_{23}{e}^{-(2K+2K_z)t}cos\left(2{\omega }_{0}t\right)+\hfill \\ {\tilde{T}}_{24}{e}^{-(2K+2K_z)t}sin\left(2{\omega }_{0}t\right)+{\tilde{T}}_{25}{e}^{-4K_{z}t}cos\left(2{\omega }_{0}t\right)+{\tilde{T}}_{26}{e}^{-4K_{z}t}sin\left(2{\omega }_{0}t\right)+\hfill \\ {\tilde{T}}_{27}{e}^{-3K_{z}t}cos\left(3{\omega }_{0}t\right)+{\tilde{T}}_{28}{e}^{-3K_{z}t}sin\left(3{\omega }_{0}t\right)+{\tilde{T}}_{29}{e}^{-(K+3K_z)t}cos\left(3{\omega }_{0}t\right)+\hfill \\ {\tilde{T}}_{30}{e}^{-(K+3K_z)t}sin\left(3{\omega }_{0}t\right)+{\tilde{T}}_{31}{e}^{-4K_{z}t}cos\left(4{\omega }_{0}t\right)+{\tilde{T}}_{32}{e}^{-4K_{z}t}sin\left(4{\omega }_{0}t\right),\hfill \end{array}$$

(31)

where

${\tilde{S}}_0=-A_0^4+A_0^2(4a-2b),$

${\tilde{S}}_1=-4A_0^3{B}_0+2A_0(4a-2b)B_0+4B_{0}K-A_0{B}_0(2-2b)K-2A_0{B}_0{K}^2,$

${\tilde{S}}_2=-6A_0^2{B}_0^2+(4a-2b)B_0^2-B_0^2(2-2b)K-2B_0^2{K}^2,{\tilde{S}}_3=-4A_0{B}_0^3,{\tilde{S}}_4=+A_0^2{R}_1,$

${\tilde{S}}_5=-3A_0^2{G}_0^2+{\displaystyle \frac{1}{2}}(4a-2b)G_0^2-3A_0^2{G}_1^2+{\displaystyle \frac{1}{2}}(4a-2b)G_1^2-{\displaystyle \frac{1}{2}}(2-2b)G_0^2{K}_z-{\displaystyle \frac{1}{2}}(2-2b)G_1^2{K}_z-G_0^2{K}_z^2-G_1^2{K}_z^2+G_0^2{\omega }_0^2+G_1^2{\omega }_0^2,$

${\tilde{S}}_6=-6A_0{B}_0{G}_0^2-6A_0{B}_0{G}_1^2,{\tilde{S}}_7=+2A_0{B}_0{R}_1,{\tilde{S}}_8=+B_0^2{R}_1,{\tilde{S}}_9=+{\displaystyle \frac{1}{2}}{G}_0^2{R}_1+{\displaystyle \frac{1}{2}}{G}_1^2{R}_1,$

${\tilde{S}}_{10}=-3B_0^2{G}_0^2-3B_0^2{G}_1^2,{\tilde{S}}_{11}=-B_0^4,{\tilde{S}}_{12}=-{\displaystyle \frac{3}{8}}{G}_0^4-{\displaystyle \frac{3}{4}}{G}_0^2{G}_1^2-{\displaystyle \frac{3}{8}}{G}_1^4,$

${\tilde{T}}_1=+G_0-4A_0^3{G}_0+2A_0(4a-2b)G_0+4G_0{K}_z-A_0(2-2b)G_0{K}_z+G_{0}K{K}_z-G_0{K}_z^2-2A_0{G}_0{K}_z^2-4G_1{\omega }_0+A_0(2-2b)G_1{\omega }_0-G_{1}K{\omega }_0+2G_1{K}_z{\omega }_0+4A_0{G}_1{K}_z{\omega }_0+G_0{\omega }_0^2+2A_0{G}_0{\omega }_0^2,$

${\tilde{T}}_2=+G_1-4A_0^3{G}_1+2A_0(4a-2b)G_1+4G_1{K}_z-A_0(2-2b)G_1{K}_z+G_{1}K{K}_z-G_1{K}_z^2-2A_0{G}_1{K}_z^2+4G_0{\omega }_0-A_0(2-2b)G_0{\omega }_0+G_{0}K{\omega }_0-2G_0{K}_z{\omega }_0-4A_0{G}_0{K}_z{\omega }_0+G_1{\omega }_0^2+2A_0{G}_1{\omega }_0^2,$

${\tilde{T}}_3=-3A_0{G}_0^3-3A_0{G}_0{G}_1^2,{\tilde{T}}_4=-3A_0{G}_0^2{G}_1-3A_0{G}_1^3,{\tilde{T}}_5=+2A_0{G}_0{R}_1,{\tilde{T}}_6=+2A_0{G}_1{R}_1,$

${\tilde{T}}_7=-12A_0^2{B}_0{G}_0+2(4a-2b)B_0{G}_0-B_0(2-2b)G_{0}K-2B_0{G}_0{K}^2-B_0(2-2b)G_0{K}_z-2B_0{G}_0{K}_z^2+B_0(2-2b)G_1{\omega }_0+4B_0{G}_1{K}_z{\omega }_0+2B_0{G}_0{\omega }_0^2,$

${\tilde{T}}_8=-12A_0^2{B}_0{G}_1+2(4a-2b)B_0{G}_1-B_0(2-2b)G_{1}K-2B_0{G}_1{K}^2-B_0(2-2b)G_1{K}_z-2B_0{G}_1{K}_z^2-B_0(2-2b)G_0{\omega }_0-4B_0{G}_0{K}_z{\omega }_0+2B_0{G}_1{\omega }_0^2,$

${\tilde{T}}_9=-12A_0{B}_0^2{G}_0,{\tilde{T}}_{10}=-12A_0{B}_0^2{G}_1,{\tilde{T}}_{11}=-3B_0{G}_0^3-3B_0{G}_0{G}_1^2,$

${\tilde{T}}_{12}=-3B_0{G}_0^2{G}_1-3B_0{G}_1^3,{\tilde{T}}_{13}=-4B_0^3{G}_0,{\tilde{T}}_{14}=-4B_0^3{G}_1,$

${\tilde{T}}_{15}=+2B_0{G}_0{R}_1,{\tilde{T}}_{16}=+2B_0{G}_1{R}_1,$

${\tilde{T}}_{17}=-3A_0^2{G}_0^2+1/2(4a-2b)G_0^2+3A_0^2{G}_1^2-1/2(4a-2b)G_1^2-1/2(2-2b)G_0^2{K}_z+1/2(2-2b)G_1^2{K}_z-G_0^2{K}_z^2+G_1^2{K}_z^2+(2-2b)G_0{G}_1{\omega }_0+4G_0{G}_1{K}_z{\omega }_0+G_0^2{\omega }_0^2-G_1^2{\omega }_0^2,$

${\tilde{T}}_{18}=-6A_0^2{G}_0{G}_1+(4a-2b)G_0{G}_1-(2-2b)G_0{G}_1{K}_z-2G_0{G}_1{K}_z^2-1/2(2-2b)G_0^2{\omega }_0+1/2(2-2b)G_1^2{\omega }_0-2G_0^2{K}_z{\omega }_0+2G_1^2{K}_z{\omega }_0+2G_0{G}_1{\omega }_0^2,$

${\tilde{T}}_{19}=-6A_0{B}_0{G}_0^2+6A_0{B}_0{G}_1^2,{\tilde{T}}_{20}=-12A_0{B}_0{G}_0{G}_1,{\tilde{T}}_{21}=+1/2G_0^2{R}_1-1/2G_1^2{R}_1,$

${\tilde{T}}_{22}=+G_0{G}_1{R}_1,{\tilde{T}}_{23}=-3B_0^2{G}_0^2+3B_0^2{G}_1^2,{\tilde{T}}_{24}=-6B_0^2{G}_0{G}_1,{\tilde{T}}_{25}=-1/2G_0^4+1/2G_1^4,$

${\tilde{T}}_{26}=-G_0^3{G}_1-G_0{G}_1^3,{\tilde{T}}_{27}=-A_0{G}_0^3+3A_0{G}_0{G}_1^2,{\tilde{T}}_{28}=-3A_0{G}_0^2{G}_1+A_0{G}_1^3,$

${\tilde{T}}_{29}=-B_0{G}_0^3+3B_0{G}_0{G}_1^2,{\tilde{T}}_{30}=-3B_0{G}_0^2{G}_1+B_0{G}_1^3,$

${\tilde{T}}_{31}=-1/8G_0^4+3/4G_0^2{G}_1^2-1/8G_1^4,{\tilde{T}}_{32}=-1/2G_0^3{G}_1+1/2G_0{G}_1^3;$

$$\begin{array}{c}{\displaystyle {\mathcal{N}}_w[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]={\tilde{P}}_0+{\tilde{P}}_1{e}^{-Kt}+{\tilde{P}}_2{e}^{-2Kt}+{\tilde{P}}_3{e}^{-3Kt}+{\tilde{P}}_4{e}^{-2t}+{\tilde{P}}_5{e}^{-2K_{z}t}+}\hfill \\ {\tilde{P}}_6{e}^{-(K+2K_z)t}+{\tilde{P}}_7{e}^{-(2+K)t}+{\tilde{Q}}_1{e}^{-K_{z}t}cos\left({\omega }_{0}t\right)+{\tilde{Q}}_2{e}^{-K_{z}t}sin\left({\omega }_{0}t\right)+\hfill \\ {\tilde{Q}}_3{e}^{-3K_{z}t}cos\left({\omega }_{0}t\right)+{\tilde{Q}}_4{e}^{-3K_{z}t}sin\left({\omega }_{0}t\right)+{\tilde{Q}}_5{e}^{-(2+K_z)t}cos\left({\omega }_{0}t\right)+\hfill \\ {\tilde{Q}}_6{e}^{-(2+K_z)t}sin\left({\omega }_{0}t\right)+{\tilde{Q}}_7{e}^{-(K+K_z)t}cos\left({\omega }_{0}t\right)+{\tilde{Q}}_8{e}^{-(K+K_z)t}sin\left({\omega }_{0}t\right)+\hfill \\ {\tilde{Q}}_9{e}^{-(2K+K_z)t}cos\left({\omega }_{0}t\right)+{\tilde{Q}}_{10}{e}^{-(2K+K_z)t}sin\left({\omega }_{0}t\right)+{\tilde{Q}}_{11}{e}^{-2K_{z}t}cos\left(2{\omega }_{0}t\right)+\hfill \\ {\tilde{Q}}_{12}{e}^{-2K_{z}t}sin\left(2{\omega }_{0}t\right)+{\tilde{Q}}_{13}{e}^{-(K+2K_z)t}cos\left(2{\omega }_{0}t\right)+{\tilde{Q}}_{14}{e}^{-(K+2K_z)t}sin\left(2{\omega }_{0}t\right)+\hfill \\ {\tilde{Q}}_{15}{e}^{-3K_{z}t}cos\left(3{\omega }_{0}t\right)+{\tilde{Q}}_{16}{e}^{-3K_{z}t}sin\left(3{\omega }_{0}t\right),\hfill \end{array}$$

(32)

where

${\tilde{P}}_0=-4A_0^3+2A_0(4a-2b),{\tilde{P}}_1=-12A_0^2{B}_0+2(4a-2b)B_0-B_0(2-2b)K-2B_0{K}^2,$

${\tilde{P}}_2=-12A_0{B}_0^2,{\tilde{P}}_3=-4B_0^3,{\tilde{P}}_4=2A_0{R}_1,{\tilde{P}}_5=-6A_0{G}_0^2-6A_0{G}_1^2,$

${\tilde{P}}_6=-6B_0{G}_0^2-6B_0{G}_1^2,{\tilde{P}}_7=2B_0{R}_1,$

${\tilde{Q}}_1=-12A_0^2{G}_0+2(4a-2b)G_0-(2-2b)G_0{K}_z-2G_0{K}_z^2+(2-2b)G_1{\omega }_0+4G_1{K}_z{\omega }_0+2G_0{\omega }_0^2,$

${\tilde{Q}}_2=-12A_0^2{G}_1+2(4a-2b)G_1-(2-2b)G_1{K}_z-2G_1{K}_z^2-(2-2b)G_0{\omega }_0-4G_0{K}_z{\omega }_0+2G_1{\omega }_0^2,$

${\tilde{Q}}_3=-3G_0^3-3G_0{G}_1^2,{\tilde{Q}}_4=-3G_0^2{G}_1-3G_1^3,{\tilde{Q}}_5=2G_0{R}_1,{\tilde{Q}}_6=2G_1{R}_1,$

${\tilde{Q}}_7=-24A_0{B}_0{G}_0,{\tilde{Q}}_8=-A_0{B}_0{G}_1,{\tilde{Q}}_9=-12B_0^2{G}_0,{\tilde{Q}}_{10}=-12B_0^2{G}_1,$

${\tilde{Q}}_{11}=-6A_0{G}_0^2+6A_0{G}_1^2,{\tilde{Q}}_{12}=-12A_0{G}_0{G}_1,{\tilde{Q}}_{13}=-6B_0{G}_0^2+6B_0{G}_1^2,{\tilde{Q}}_{14}=-12B_0{G}_0{G}_1,$${\tilde{Q}}_{15}=-G_0^3+3G_0{G}_1^2,{\tilde{Q}}_{16}=-3G_0^2{G}_1+G_1^3;$

$$\begin{array}{c}{\displaystyle {\mathcal{N}}_{w^{\prime }}[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]=-4-K+(2-2b)A_0+B_0(2-2b)e^{-Kt}+}\hfill \\ (2-2b)\left(G_{0}cos\left({\omega }_{0}t\right)+G_{1}sin\left({\omega }_{0}t\right)\right)e^{-K_{z}t};\hfill \end{array}$$

(33)

$$\begin{array}{c}{\displaystyle {\mathcal{N}}_{w^{″}}[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]=-2A_0-2B_0{e}^{-Kt}-2\left(G_{0}cos\left({\omega }_{0}t\right)+G_{1}sin\left({\omega }_{0}t\right)\right)e^{-K_{z}t}.}\hfill \end{array}$$

(34)

Step 5. Identifying independent functions $h_i$ from the above expressions (31)–(34), according to Equation (24).

The above expressions are linear combinations between elementary functions $g_j\in \{1,cos\left(k{\omega }_{0}t\right),sin\left(k{\omega }_{0}t\right)\}$ and $h_j$ defined below:

$\begin{array}{cccc}{h}_1=1,\hfill & h_2=e^{-Kt},\hfill & h_3=e^{-2Kt},\hfill & h_4=e^{-3Kt},\hfill \\ h_5=e^{-2t},\hfill & h_6=e^{-2K_{z}t},\hfill & h_7=e^{-(K+2K_z)t},\hfill & h_8=e^{-(2+K)t},\hfill \\ h_9=e^{-(2+2K)t},\hfill & h_{10}=e^{-(2+2K_z)t},\hfill & h_{11}=e^{-(2K+2K_z)t},\hfill & h_{12}=e^{-4Kt},\hfill \\ h_{13}=e^{-4K_{z}t},\hfill & h_{14}=e^{-K_{z}t},\hfill & h_{15}=e^{-3K_{z}t},\hfill & h_{16}=e^{-(2+K_z)t},\hfill \\ h_{17}=e^{-(K+K_z)t},\hfill & h_{18}=e^{-(2K+K_z)t},\hfill & h_{19}=e^{-(K+3K_z)t},\hfill & h_{20}=e^{-(3K+K_z)t},\hfill \\ h_{21}=e^{-(2+K+K_z)t}.\hfill & & \end{array}$

Step 6. Choice of auxiliary functions ${\tilde{\mathcal{A}}}_i\left(t\right)$, $i=\overline{1,21}$ depending on the independent variable t and some arbitrary real parameters, as follows:

$$\begin{array}{cc}{\tilde{\mathcal{A}}}_1={\check{C}}_0,\hfill & {\tilde{\mathcal{A}}}_2={\check{G}}_2+{\check{A}}_{2}t,\hfill \\ {\tilde{\mathcal{A}}}_3={\check{G}}_3,\hfill & {\tilde{\mathcal{A}}}_4={\check{G}}_4,\hfill \\ {\tilde{\mathcal{A}}}_5={\check{G}}_{14}{e}^{-(K+2K_z)t},\hfill & {\tilde{\mathcal{A}}}_6={\check{G}}_9,\hfill \\ {\tilde{\mathcal{A}}}_7={\check{G}}_{12}+{\check{G}}_{15}{e}^{-2K_{z}t},\hfill & {\tilde{\mathcal{A}}}_8={\check{G}}_6,\hfill \\ {\tilde{\mathcal{A}}}_9={\check{G}}_7+{\check{G}}_8{e}^{-Kt},\hfill & {\tilde{\mathcal{A}}}_{10}={\check{G}}_{11},\hfill \\ {\tilde{\mathcal{A}}}_{11}={\check{G}}_{13},\hfill & {\tilde{\mathcal{A}}}_{12}={\check{G}}_5,\hfill \\ {\tilde{\mathcal{A}}}_{13}={\check{G}}_{10},\hfill & {\tilde{\mathcal{A}}}_{14}={\check{G}}_{0}cos\left({\omega }_{0}t\right)+{\check{G}}_{1}sin\left({\omega }_{0}t\right),\hfill \\ {\tilde{\mathcal{A}}}_{15}={\check{F}}_{3}cos\left({\omega }_{0}t\right)+{\check{F}}_{4}sin\left({\omega }_{0}t\right),\hfill & {\tilde{\mathcal{A}}}_{16}={\check{F}}_{11}cos\left({\omega }_{0}t\right)+{\check{F}}_{12}sin\left({\omega }_{0}t\right),\hfill \\ {\tilde{\mathcal{A}}}_{17}={\check{F}}_{5}cos\left({\omega }_{0}t\right)+{\check{F}}_{6}sin\left({\omega }_{0}t\right),\hfill & \begin{array}{c}\hfill {\tilde{\mathcal{A}}}_{18}=({\check{F}}_7+{\check{F}}_9{e}^{-Kt})cos\left({\omega }_{0}t\right)+\\ \hfill ({\check{F}}_8+{\check{F}}_{10}{e}^{-Kt})sin\left({\omega }_{0}t\right)\end{array},\hfill & \\ {\tilde{\mathcal{A}}}_{19}={\check{F}}_{15}cos\left({\omega }_{0}t\right)+{\check{F}}_{16}sin\left({\omega }_{0}t\right),\hfill & {\tilde{\mathcal{A}}}_{20}={\check{F}}_{13}cos\left({\omega }_{0}t\right)+{\check{F}}_{14}sin\left({\omega }_{0}t\right),\hfill \\ {\displaystyle {\tilde{\mathcal{A}}}_{21}=\sum _{i=1}^{N_{max}}\left[{\check{C}}_{i}cos\left(2i{\omega }_{0}t\right)+{\check{D}}_{i}sin\left(2i{\omega }_{0}t\right)\right],}\hfill & {\tilde{\mathcal{A}}}_{22}={\check{E}}_{3}cos\left(2{\omega }_{0}t\right)+{\check{E}}_{4}sin\left(2{\omega }_{0}t\right),\hfill \\ {\tilde{\mathcal{A}}}_{23}={\check{E}}_{1}cos\left(2{\omega }_{0}t\right)+{\check{E}}_{2}sin\left(2{\omega }_{0}t\right),\hfill & {\tilde{\mathcal{A}}}_{24}={\check{E}}_{5}cos\left(2{\omega }_{0}t\right)+{\check{E}}_{6}sin\left(2{\omega }_{0}t\right),\hfill \\ {\tilde{\mathcal{A}}}_{25}={\check{E}}_{7}cos\left(2{\omega }_{0}t\right)+{\check{E}}_{8}sin\left(2{\omega }_{0}t\right),\hfill & {\tilde{\mathcal{A}}}_{26}={\check{E}}_{9}cos\left(3{\omega }_{0}t\right)+{\check{E}}_{10}sin\left(3{\omega }_{0}t\right),\hfill \\ {\tilde{\mathcal{A}}}_{27}={\check{E}}_{11}cos\left(3{\omega }_{0}t\right)+{\check{E}}_{12}sin\left(3{\omega }_{0}t\right),\hfill & {\tilde{\mathcal{A}}}_{28}={\check{E}}_{13}cos\left(4{\omega }_{0}t\right)+{\check{E}}_{14}sin\left(4{\omega }_{0}t\right),\hfill \end{array}$$

(35)

and so on.

Remark 2.

This stage of the mOPIM technique highlights a second advantage, namely the choice of the auxiliary functions ${\tilde{\mathcal{A}}}_i\left(t\right)$ for convergence-control of the mOPIM solution ${\overline{w}}_{mOPIM}$. The real parameters that appear in (35) will be optimally identified via the least squares method.

Step 7. Calculus of the first-order approximate solution ${\overline{w}}_{mOPIM}\left(t\right)=w_1\left(t\right)$ using Equation (25) for $n=0$ and Equation (35).

The ${\overline{w}}_{mOPIM}$ solution, denoted by $\overline{w}$, could be obtained by integration of Equation (25) for $n=0$ as:

$$\begin{array}{c}\overline{w}\left(t\right)=C_0+(G_{0}cos\left({\omega }_{0}t\right)+G_{1}sin\left({\omega }_{0}t\right))e^{-K_{z}t}+G_2{e}^{-Kt}+\hfill \\ t(A_{0}cos\left({\omega }_{0}t\right)+A_{1}sin\left({\omega }_{0}t\right))e^{-K_{z}t}+A_{2}t{e}^{-Kt}+G_3{e}^{-2Kt}+G_4{e}^{-3Kt}+G_5{e}^{-4Kt}+\hfill \\ G_6{e}^{-(2+K)t}+G_7{e}^{-(2+2K)t}+G_8{e}^{-(2+3K)t}+G_9{e}^{-2K_{z}t}+G_{10}{e}^{-4K_{z}t}+G_{11}{e}^{-(2+2K_z)t}+\hfill \\ G_{12}{e}^{-(K+2K_z)t}+G_{13}{e}^{-(2K+2K_z)t}+G_{14}{e}^{-(2+K+2K_z)t}+G_{15}{e}^{-(K+4K_{z}t)}+\hfill \\ F_1{e}^{-K_{z}t}cos\left({\omega }_{0}t\right)+F_2{e}^{-K_{z}t}sin\left({\omega }_{0}t\right)+F_3{e}^{-3K_{z}t}cos\left({\omega }_{0}t\right)+F_4{e}^{-3K_{z}t}sin\left({\omega }_{0}t\right)+\hfill \\ F_5{e}^{-(K+K_{z}t)}cos\left({\omega }_{0}t\right)+F_6{e}^{-(K+K_{z}t)}sin\left({\omega }_{0}t\right)+F_7{e}^{-(2K+K_{z}t)}cos\left({\omega }_{0}t\right)+\hfill \\ F_8{e}^{-(2K+K_{z}t)}sin\left({\omega }_{0}t\right)+F_9{e}^{-(3K+K_{z}t)}cos\left({\omega }_{0}t\right)+F_{10}{e}^{-(3K+K_{z}t)}sin\left({\omega }_{0}t\right)+\hfill \\ F_{11}{e}^{-(2+K_{z}t)}cos\left({\omega }_{0}t\right)+F_{12}{e}^{-(2+K_{z}t)}sin\left({\omega }_{0}t\right)+F_{13}{e}^{-(2+K+K_{z}t)}cos\left({\omega }_{0}t\right)+\hfill \\ F_{14}{e}^{-(2+K+K_{z}t)}sin\left({\omega }_{0}t\right)+F_{15}{e}^{-(K+3K_{z}t)}cos\left({\omega }_{0}t\right)+F_{16}{e}^{-(K+3K_{z}t)}sin\left({\omega }_{0}t\right)+\hfill \\ {\displaystyle \sum _{i=1}^{N_{max}}\left[C_{i}cos\left(2i{\omega }_{0}t\right)+D_{i}sin\left(2i{\omega }_{0}t\right)\right]e^{-2K_{z}t}+E_1{e}^{-(2+2K_{z}t)}cos\left(2{\omega }_{0}t\right)+}\hfill \\ E_2{e}^{-(2+2K_{z}t)}sin\left(2{\omega }_{0}t\right)+E_3{e}^{-(K+2K_{z}t)}cos\left(2{\omega }_{0}t\right)+E_4{e}^{-(K+2K_{z}t)}sin\left(2{\omega }_{0}t\right)+\hfill \\ E_5{e}^{-(2K+2K_{z}t)}cos\left(2{\omega }_{0}t\right)+E_6{e}^{-(2K+2K_{z}t)}sin\left(2{\omega }_{0}t\right)+E_7{e}^{-4K_{z}t}cos\left(2{\omega }_{0}t\right)+\hfill \\ E_8{e}^{-4K_{z}t}sin\left(2{\omega }_{0}t\right)+E_9{e}^{-3K_{z}t}cos\left(3{\omega }_{0}t\right)+E_{10}{e}^{-3K_{z}t}sin\left(3{\omega }_{0}t\right)+\hfill \\ E_{11}{e}^{-(K+3K_{z}t)}cos\left(3{\omega }_{0}t\right)+E_{12}{e}^{-(K+3K_{z}t)}sin\left(3{\omega }_{0}t\right)+E_{13}{e}^{-4K_{z}t}cos\left(4{\omega }_{0}t\right)+\hfill \\ E_{14}{e}^{-4K_{z}t}sin\left(4{\omega }_{0}t\right),\hfill \end{array}$$

(36)

with real parameters $C_0$, $A_0$, $A_1$, $A_2$, $G_0÷G_{15}$, $F_1÷F_{16}$, $E_1÷E_{14}$ depending on ${\check{C}}_0$, ${\check{A}}_0$, ${\check{A}}_1$, ${\check{A}}_2$, ${\check{G}}_0÷{\check{G}}_{15}$, ${\check{F}}_1÷{\check{F}}_{16}$, ${\check{E}}_1÷{\check{E}}_{14}$ and will be optimally computed via the least squares method.

Remark 3.

Another advantage of the mOPIM procedure is the mOPIM solution ${\overline{w}}_{mOPIM}$ in the effective form.

Semi-analytical solutions via the OPIM method.

The second-order nonlinear differential problem given by Equation (5) can be semi-analytically solved using Optimal Parametric Iteration Method (OPIM) considering the following operators:

$$\begin{array}{c}{\displaystyle \mathcal{L}[w,w^{\prime },w^{″},w^{‴}]=w^{″}+K_z{w}^{\prime }+(K_z^2+{\omega }_0^2)w,}\hfill \\ {\displaystyle \mathcal{N}[w\left(t\right),w^{\prime }\left(t\right),w^{″}\left(t\right)]=-2w\ddot{w}-4\dot{w}+(2-2b)w{w}^{\prime }+(4a-2b)w^2+}\hfill \\ +\left(R_1{e}^{-2t}-w^2\right)w^2-w^{″}-K_z{w}^{\prime }-(K_z^2+{\omega }_0^2)w.\hfill \end{array}$$

(37)

The initial guess is $w_0\left(t\right)=({\check{G}}_{0}cos\left({\omega }_{0}t\right)+{\check{G}}_{1}sin\left({\omega }_{0}t\right))e^{-K_{z}t}$.

Following the same steps, the semi-analytical solution denoted by ${\overline{w}}_{OPIM}$ is built in the form:

$$\begin{array}{c}{\overline{w}}_{OPIM}\left(t\right)=C_0+(G_{0}cos\left({\omega }_{0}t\right)+G_{1}sin\left({\omega }_{0}t\right))e^{-K_{z}t}+G_2{e}^{-Kt}+\hfill \\ t(A_{0}cos\left({\omega }_{0}t\right)+A_{1}sin\left({\omega }_{0}t\right))e^{-K_{z}t}+G_9{e}^{-2K_{z}t}+G_{10}{e}^{-4K_{z}t}+G_{11}{e}^{-(2+2K_z)t}+\hfill \\ F_1{e}^{-K_{z}t}cos\left({\omega }_{0}t\right)+F_2{e}^{-K_{z}t}sin\left({\omega }_{0}t\right)+F_3{e}^{-3K_{z}t}cos\left({\omega }_{0}t\right)+F_4{e}^{-3K_{z}t}sin\left({\omega }_{0}t\right)+\hfill \\ {\displaystyle F_{11}{e}^{-(2+K_{z}t)}cos\left({\omega }_{0}t\right)+F_{12}{e}^{-(2+K_{z}t)}sin\left({\omega }_{0}t\right)+\sum _{i=1}^{N_{max}}\left[C_{i}cos\left(2i{\omega }_{0}t\right)+D_{i}sin\left(2i{\omega }_{0}t\right)\right]e^{-2K_{z}t}+}\hfill \\ E_1{e}^{-(2+2K_{z}t)}cos\left(2{\omega }_{0}t\right)+E_2{e}^{-(2+2K_{z}t)}sin\left(2{\omega }_{0}t\right)+E_7{e}^{-4K_{z}t}cos\left(2{\omega }_{0}t\right)+\hfill \\ E_8{e}^{-4K_{z}t}sin\left(2{\omega }_{0}t\right)+E_9{e}^{-3K_{z}t}cos\left(3{\omega }_{0}t\right)+E_{10}{e}^{-3K_{z}t}sin\left(3{\omega }_{0}t\right)+E_{13}{e}^{-4K_{z}t}cos\left(4{\omega }_{0}t\right)+\hfill \\ E_{14}{e}^{-4K_{z}t}sin\left(4{\omega }_{0}t\right),\hfill \end{array}$$

(38)

where all real parameters that appear in Equation (38) will be optimally computed via the least squares method.

This OPIM-solution is compared with the corresponding mOPIM solution for Case 1 $c=2$ in Section 4.

Case 2. $c>2$ for asymptotic behaviors.

As in the previous case $c=2$, applying the same procedure, we can choose the linear operator $\mathcal{L}[t,w,w^{\prime },w^{″},w^{‴}]$ and the nonlinear operator $\mathcal{N}[t,w,w^{\prime },w^{″},w^{‴}]$ by $\mathcal{L}[t,w,w^{\prime },w^{″},w^{‴}]=w^{‴}+K{w}^{″}-{\phi }_1,$

$$\begin{array}{c}\mathcal{N}[t,w,w^{\prime },w^{″},w^{‴}]={\displaystyle \frac{\sqrt{c-2}}{4(2-c)}}\left(w^{‴}+(c-2)w^{″}+(2c-8)w^{\prime }\right)(cw+w^{\prime })-\hfill \\ {\displaystyle \frac{\sqrt{c-2}}{8(2-c)}}\left(w^{″}+(c-2)w^{\prime }+(2c-8)w\right)(c{w}^{\prime }+w^{″})+{\displaystyle \frac{1}{4(2-c)\sqrt{c-2}}}{(cw+w^{\prime })}^2(2w+w^{\prime })+\hfill \\ {\displaystyle \frac{b\sqrt{c-2}}{4(2-c)}}(cw+w^{\prime })\left(w^{″}+(c-2)w^{\prime }+(2c-8)w\right)+{\displaystyle \frac{a}{\sqrt{c-2}}}{(w^{\prime }+cw)}^2-w^{‴}-K{w}^{″}+{\phi }_1,\hfill \end{array}$$

where $K>0$ is an arbitrary real constant, while ${\phi }_1$ is introduced in the above case.

The above linear operator $\mathcal{L}[t,w,w^{\prime },w^{″},w^{‴}]$ yields to the initial approximation $w_0\left(t\right)=A_0+{\alpha }_{0}t+B_0{e}^{-Kt}+\left(G_{0}cos\left({\omega }_{0}t\right)+G_{1}sin\left({\omega }_{0}t\right)\right)e^{-K_{z}t}$.

The asymptotic behavior implies that ${\alpha }_0=0$, so the initial guess becomes $w_0\left(t\right)=A_0+B_0{e}^{-Kt}+\left(G_{0}cos\left({\omega }_{0}t\right)+G_{1}sin\left({\omega }_{0}t\right)\right)e^{-K_{z}t}$.

The nonlinear expressions ${\mathcal{N}}_w[w,w^{\prime },w^{″},w^{‴}]$, ${\mathcal{N}}_{w^{\prime }}[w,w^{\prime },w^{″},w^{‴}]$, ${\mathcal{N}}_{w^{″}}[w,w^{\prime },w^{″},w^{‴}]$ and ${\mathcal{N}}_{w^{‴}}[w,w^{\prime },w^{″},w^{‴}]$ become:

$$\begin{array}{c}{\mathcal{N}}_w[w,w^{\prime },w^{″},w^{‴}]={\displaystyle \frac{\sqrt{c-2}}{4(2-c)}}\left(w^{‴}+(c-2)w^{″}+(2c-8)w^{\prime }\right)-(2c-8){\displaystyle \frac{\sqrt{c-2}}{8(2-c)}}(c{w}^{\prime }+w^{″})+\hfill \\ {\displaystyle \frac{2c}{4(2-c)\sqrt{c-2}}}(cw+w^{\prime })(2w+w^{\prime })+{\displaystyle \frac{2}{4(2-c)\sqrt{c-2}}}{(cw+w^{\prime })}^2+{\displaystyle \frac{cb\sqrt{c-2}}{4(2-c)}}\left(w^{″}+(c-2)w^{\prime }+(2c-8)w\right)+\hfill \\ (2c-8){\displaystyle \frac{b\sqrt{c-2}}{4(2-c)}}(cw+w^{\prime })+{\displaystyle \frac{2ac}{\sqrt{c-2}}}(w^{\prime }+cw);\hfill \end{array}$$

$$\begin{array}{c}{\mathcal{N}}_{w^{\prime }}[w,w^{\prime },w^{″},w^{‴}]={\displaystyle \frac{\sqrt{c-2}}{4(2-c)}}\left(w^{‴}+(c-2)w^{″}+(2c-8)w^{\prime }\right)+(2c-8){\displaystyle \frac{\sqrt{c-2}}{4(2-c)}}(cw+w^{\prime })+\hfill \\ {\displaystyle \frac{\sqrt{c-2}}{8}}(cw+w^{\prime })-{\displaystyle \frac{c\sqrt{c-2}}{8(2-c)}}\left(w^{″}+(c-2)w^{\prime }+(2c-8)w\right)+{\displaystyle \frac{2}{4(2-c)\sqrt{c-2}}}(cw+w^{\prime })(2w+w^{\prime })+\hfill \\ {\displaystyle \frac{1}{4(2-c)\sqrt{c-2}}}{(cw+w^{\prime })}^2+{\displaystyle \frac{b\sqrt{c-2}}{4(2-c)}}\left(w^{″}+(c-2)w^{\prime }+(2c-8)w\right)+(c-2){\displaystyle \frac{b\sqrt{c-2}}{4(2-c)}}(cw+w^{\prime })+\hfill \\ {\displaystyle \frac{2a}{\sqrt{c-2}}}(w^{\prime }+cw);\hfill \end{array}$$

$$\begin{array}{c}{\mathcal{N}}_{w^{″}}[w,w^{\prime },w^{″},w^{‴}]=-{\displaystyle \frac{\sqrt{c-2}}{4}}(cw+w^{\prime })-{\displaystyle \frac{\sqrt{c-2}}{8(2-c)}}(c{w}^{\prime }+w^{″})-\hfill \\ {\displaystyle \frac{\sqrt{c-2}}{8(2-c)}}\left(w^{″}+(c-2)w^{\prime }+(2c-8)w\right)+{\displaystyle \frac{b\sqrt{c-2}}{4(2-c)}}(cw+w^{\prime })-K;\hfill \end{array}$$

$$\begin{array}{c}{\mathcal{N}}_{w^{‴}}[w,w^{\prime },w^{″},w^{‴}]={\displaystyle \frac{\sqrt{c-2}}{4(2-c)}}(cw+w^{\prime })-1.\hfill \end{array}$$

The same procedure as in Case 1 yields to the first-order approximate solution $w_1\left(t\right)={\overline{w}}_{mOPIM}$ given by Equation (36), with the exception of terms containing $e^{-4Kt}$, $e^{-4K_{z}t}$, $e^{-4Kt}cos\left(4{\omega }_{0}t\right)$, $e^{-4Kt}sin\left(4{\omega }_{0}t\right)$, $e^{-4K_{z}t}cos\left(4{\omega }_{0}t\right)$, $e^{-4K_{z}t}sin\left(4{\omega }_{0}t\right)$, because the above expressions contain terms which are, at most, third-order, namely ${(cw+w^{\prime })}^2(2w+w^{\prime })$. Therefore, the nonlinear expressions $\mathcal{N}[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]$, ${\mathcal{N}}_w[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]$, ${\mathcal{N}}_{w^{\prime }}[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]$, ${\mathcal{N}}_{w^{″}}[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]$ and ${\mathcal{N}}_{w^{‴}}[w_0,w_0^{\prime },w_0^{″},w_0^{‴}]$ does not contain any combination between the elementary functions $e^{-4Kt}$, $e^{-4K_{z}t}$, $e^{-4Kt}cos\left(4{\omega }_{0}t\right)$, $e^{-4Kt}sin\left(4{\omega }_{0}t\right)$, $e^{-4K_{z}t}cos\left(4{\omega }_{0}t\right)$, $e^{-4K_{z}t}sin\left(4{\omega }_{0}t\right)$.

4. Numerical Results and Discussion

In this section, the obtained mOPIM results are quantitatively and qualitatively emphasized by tables and figures.

A comparative analysis between obtained mOPIM results and the corresponding numerical results is represented in Figure 1 for physical constants $a=0.65$, $b=0.34$, $c=2$, and in Figure 2 and Figure 3 for $a=0.65$, $b=0.34$, $c=3.75$. This comparison reflects the accuracy of the obtained results. Quantitatively, this accuracy is presented in

Table 1. Comparative values for numerical results $w_{numerical}$ when: $x_{i0}=0.55$, $y_{i0}=0.35$, $z_{i0}=1.75$, physical parameters: $a=0.65$, $b=0.34$, $c=2$ and index $N_{max}=25$; ${\overline{w}}_{mOPIM}$ solution for System (1) using Equations (36) and (A1) and absolute values: ${\epsilon }_w=|w_{numerical}-{\overline{w}}_{mOPIM}|$.

$\mathit{t}\left[\mathit{s}\right]$	${\overline{\mathit{w}}}_{\mathit{mOPIM}}$	${\mathit{w}}_{\mathit{numerical}}$	${\mathit{\epsilon }}_{\mathit{w}}$
	${\mathit{N}}_{\mathit{max}}=\mathbf{25}$		for ${\mathit{N}}_{\mathit{max}}=\mathbf{25}$
0	0.5499999951	0.55	4.8231 $\times {10}^{-9}$
3	0.4773703776	0.4773703851	7.5475 $\times {10}^{-9}$
6	1.3774410902	1.3774410805	9.6353 $\times {10}^{-9}$
9	1.3757946522	1.3757946423	9.9399 $\times {10}^{-9}$
12	1.3878088268	1.3878088303	3.5324 $\times {10}^{-9}$
15	1.3853010140	1.3853010159	1.8866 $\times {10}^{-9}$
18	1.3856816387	1.3856816367	1.9998 $\times {10}^{-9}$
21	1.3856370801	1.3856370757	4.4526 $\times {10}^{-9}$
24	1.3856407573	1.3856407584	1.0533 $\times {10}^{-9}$
27	1.3856406979	1.3856406950	2.8460 $\times {10}^{-9}$
30	1.3856406327	1.3856406295	3.1503 $\times {10}^{-9}$

(for $a=0.65$, $b=0.34$, $c=2$) and

Table 2. Comparative values for numerical results $w_{numerical}$ when: $x_{i0}=0.55$, $y_{i0}=0.35$, $z_{i0}=1.75$, physical parameters: $a=0.65$, $b=0.34$, $c=3.75$ and index $N_{max}=25$; ${\overline{w}}_{mOPIM}$ solution for System (1) using Equations (36) and (A2) and absolute values: ${\epsilon }_w=|w_{numerical}-{\overline{w}}_{mOPIM}|$.

$\mathit{t}\left[\mathit{s}\right]$	${\mathit{w}}_{\mathit{numerical}}$	${\overline{\mathit{w}}}_{\mathit{mOPIM}}$	${\mathit{\epsilon }}_{\mathit{w}}$
		${\mathit{N}}_{\mathit{max}}=\mathbf{25}$	for ${\mathit{N}}_{\mathit{max}}=\mathbf{25}$
0	6.9024999993	6.9025	6.1467 $\times {10}^{-10}$
3	0.0065148716	0.0065148701	1.4414 $\times {10}^{-9}$
6	0.1288368206	0.1288368208	2.0911 $\times {10}^{-10}$
9	1.2878994227	1.2878994219	8.0715 $\times {10}^{-10}$
12	1.6959357827	1.6959357821	5.7847 $\times {10}^{-10}$
15	1.6794403021	1.6794403026	5.1767 $\times {10}^{-10}$
18	1.6800157815	1.6800157831	1.5464 $\times {10}^{-9}$
21	1.6799997839	1.6799997813	2.5877 $\times {10}^{-9}$
24	1.6799999853	1.6799999854	8.1595 $\times {10}^{-11}$
27	1.6800000005	1.6800000010	4.5572 $\times {10}^{-10}$
30	1.6799999979	1.6799999998	1.8108 $\times {10}^{-9}$

(for $a=0.65$, $b=0.34$, $c=3.75$), respectively. It is observed that the order of magnitude is ${10}^{-7}$.

In both cases the optimally parameters are given in Appendix A.

Case 1. $c=2$.

In this case, we obtained the following limit: ${\displaystyle \underset{t\to \infty }{lim}({\overline{x}}_{mOPIM},{\overline{y}}_{mOPIM},{\overline{z}}_{mOPIM})=}$ $(1.385640646123426,-0.6928203230597704,-0.4800000000473364)=$$\left(\sqrt{(2a-b)c},-{\displaystyle \frac{\sqrt{(2a-b)c}}{2}},{\displaystyle \frac{b-2a}{2}}\right)\approx (1.3856406460551018,-0.6928203230275509,-0.48),$ and, therefore the trajectory $({\overline{x}}_{mOPIM},{\overline{y}}_{mOPIM},{\overline{z}}_{mOPIM})$ describes a heteroclinic orbit of the System (1).

Case 2. $c\ne 2$.

In this case, we obtained the following limit: ${\displaystyle \underset{t\to \infty }{lim}({\overline{x}}_{mOPIM},{\overline{y}}_{mOPIM},{\overline{z}}_{mOPIM})=(1.8973665961699835,-0.9486832980547296,-0.48000000003168947)=}$ $\left(\sqrt{(2a-b)c},-{\displaystyle \frac{\sqrt{(2a-b)c}}{2}},{\displaystyle \frac{b-2a}{2}}\right)\approx (1.8973665961010275,-0.9486832980505138,-0.48),$ and, therefore the trajectory $({\overline{x}}_{mOPIM},{\overline{y}}_{mOPIM},{\overline{z}}_{mOPIM})$ describes the heteroclinical orbit of the System (1).

The accuracy of the mOPIM solution is lower if the computational domain for optimizing the real parameters from Equation (36) decreases. A comparison between mOPIM-solution ${\overline{w}}_{mOPIM}$ given by Equation (36) and corresponding numerical results is shown in Figure 4 using a small interval $[0,15]$.

In

Table 3. Comparative values for numerical results $w_{numerical}$ when: $x_{i0}=0.55$, $y_{i0}=0.35$, $z_{i0}=1.75$, physical parameters: $a=0.65$, $b=0.34$, $c=2$ and index $N_{max}=25$; ${\overline{w}}_{OPIM}$ solution for System (1) using Equation (38) and absolute values: ${\epsilon }_w=|w_{numerical}-{\overline{w}}_{OPIM}|$.

$\mathit{t}\left[\mathit{s}\right]$	${\overline{\mathit{w}}}_{\mathit{OPIM}}$	${\mathit{w}}_{\mathit{numerical}}$	${\mathit{\epsilon }}_{\mathit{w}}$
	${\mathit{N}}_{\mathit{max}}=\mathbf{25}$		for ${\mathit{N}}_{\mathit{max}}=\mathbf{25}$
0	0.5500000118420334	0.55	1.1842 $\times {10}^{-8}$
3	0.4773688853	0.4773703851	1.4998 $\times {10}^{-6}$
6	1.3774456400	1.3774410805	4.5594 $\times {10}^{-6}$
9	1.3758109760	1.3757946423	1.6333 $\times {10}^{-5}$
12	1.3878322367	1.3878088303	2.3406 $\times {10}^{-5}$
15	1.3853921525	1.3853010159	9.1136 $\times {10}^{-5}$
18	1.3856390193	1.3856816367	4.2617 $\times {10}^{-5}$
21	1.3856460918	1.3856370757	9.0161 $\times {10}^{-6}$
24	1.3856408397	1.3856407584	8.1330 $\times {10}^{-8}$
27	1.3856398221	1.3856406950	8.7290 $\times {10}^{-7}$
30	1.3856410367	1.3856406295	4.0719 $\times {10}^{-7}$

, we present some quantitative values of the semi-analytical solution ${\overline{w}}_{OPIM}$ using OPIM technique. It can be seen that the magnitude of the residual function $|w_{numerical}-{\overline{w}}_{OPIM}|$ is greater than in the case of mOPIM-solution.

By evaluating the absolute values ${\epsilon }_w$ in Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, it can be observed that the magnitude of ${\epsilon }_w$ decreases over time, which justifies the convergence of the mOPIM solutions.

The comparison between the OPIM solution and the corresponding numerical results is presented in Figure 6 for residual function ${\epsilon }_w\left(t\right)=|w_{numerical}-{\overline{w}}_{OPIM}|$. From this representation, we can see that the magnitude of the residual values is of the ${10}^{-4}$ order versus the corresponding mOPIM solution of the ${10}^{-7}$ order shown in Figure 5.

Figure 7 shows a profile of the residual function ${\epsilon }_w\left(t\right)=|w_{numerical}-{\overline{w}}_{mOPIM}|$ between the mOPIM solution and corresponding numerical results on reduced domain $[0,14.05]$.

In Figure 8 is depicted a profile of the residual function ${\epsilon }_w\left(t\right)=|w_{numerical}-{\overline{w}}_{mOPIM}|$ for $N_{max}=10$.

A performance indicator to highlight the accuracy of the mOPIM solution is the rate of convergence given by:

$${\displaystyle r\left(t\right)=\underset{n\to \infty }{lim}\frac{|{\overline{w}}_n\left(t\right)-w_{exact}\left(t\right)|}{|{\overline{w}}_{n-1}\left(t\right)-w_{exact}\left(t\right){|}^p},}$$

(39)

with $p\ge 1$ order of convergence, with ${\overline{w}}_n\left(t\right)$ as the n-th-order semi-analytical solution using mOPIM.

An alternative formula instead of (39) for the rate of convergence in our case is:

$${\displaystyle r\left(t\right)=\frac{|{\overline{w}}_{mOPIM}\left(t\right)-w_{numerical}\left(t\right)|}{|{\overline{w}}_0\left(t\right)-w_{numerical}\left(t\right){|}^p},}$$

(40)

where ${\overline{w}}_1={\overline{w}}_{mOPIM}$ is the first-order semi-analytical solution obtained in Section 3.2.

The convergence rate (for $p=1$ linear convergence) in the cases $a=0.65$, $b=0.34$, $c=2$, using Equation (A1) and $a=0.65$, $b=0.34$, $c=3.75$, using Equation (A2), respectively, is shown in the Figure 10 and Figure 11. A faster convergence is obtained for lower rate r with the increasing time t.

5. Qualitative Analysis of Errors

Multiple statistical tests for the same problem—often called a “sensitivity analysis” or “robustness check”—may be done to ensure that the findings are consistent and not merely an artifact of a specific test’s assumptions or limitations. In statistical analyses, different tests are applied (e.g., parametric and non-parametric) to confirm whether the findings hold true under different assumptions, such as checking for normal distribution.

A qualitative analysis of error for approximation focuses on understanding the nature, behavior, and sources of discrepancies between an exact value (or function) and its approximation, rather than just computing a specific numerical value of the error. It provides insight into why an approximation fails or succeeds, enabling better model selection and facilitating identification of limiting factors.

The approximation error in a given data value represents the significant discrepancy that arises when an exact, true value is compared against some approximation derived for it. This inherent error in approximation can be quantified and expressed in two principal ways: as an absolute error, which denotes the direct numerical magnitude of this discrepancy irrespective of the true value’s scale, or as a relative error, which provides a scaled measure of the error by considering the absolute error in proportion to the exact data value, thus offering a context-dependent assessment of the error’s significance (see Williams et al. [26]).

The main problem in more dynamical system is that the true values (i.e., exact solution) are unknown. To obtain an approximate solution, we use some observations or data recordings from physical phenomena or some numerical solutions or numerical simulations. In this case, the “errors” are in fact the difference between the approximate solutions and some numerical values (coming from a numerical solution); therefore, it is more appropriate to use the notion of “residuals”, and a qualitative analysis of these residuals seems to be important. It is known that in statistics and optimization, the errors represent the unobservable, theoretical difference between an observed value and the true model (i.e., exact solution), whereas residuals are the observable, calculated difference between an observed value and the estimated or predicted (i.e., approximate solution) value from a sample model. Errors are random, while residuals are used to estimate these unknown errors (see Cox et al. [27]).

It is known that the numerical solutions provide approximate, non-analytical results that are computationally expensive, require significant expertise, and may lack stability or convergence, often leading to unphysical outcomes if incorrectly applied. They are limited to specific, quantifiable data, frequently requiring complex discretization that struggles to capture, for instance, sharp discontinuities or material interfaces. Also, almost all numerical methods provide some numbers which are yielded by truncation of real numbers.

Therefore, a qualitative analysis of residuals is a vital, visual, and non-numerical technique for diagnosing the adequacy of a model or approximation. In general, a qualitative analysis of residuals is mainly focused on ensuring that they are almost random, i.e., no patterns and no periodical behavior are detected in a series of residuals. Most of the techniques used are focused on three aspects: First, the residuals should have a constant mean value (almost all approximate solutions provide residuals with zero mean value) and, probably most important, constant variance, which is known as heteroscedasticity. Second, the residuals are not correlated (or no autocorrelated); obviously, this aspect is most important for verifying whether or not there is some periodicity or pattern in the series of residuals, and this indicates that the approximate solution is the best analytical expression to describe the evolution of associated dynamical system. Third, it is used to check if the empirical distribution of residuals is closed to a known theoretical probability distribution, in general, normal or Gaussian distribution.

For each of the above aspects, there exist some adequate statistical tests. Most of them are implemented in some open or free software, such as, for example, R or R-studio software.

Of course, some graphs can help to provide simple observations about the shape of the series of residuals, such as, for example, a simple 2-dimensional plot or histogram.

All computations for these statistical tests were performed using R-software version 4.5.0.

For a good understanding of this analysis, we will consider the series of residuals coming from the case $a=0.65$, $b=0.34$, $c=3.75$ on the real interval $[0,30]$.

The series of residuals is:

$$\begin{array}{c}s2=\{3.863249720126305\times {10}^{-10},5.543591585777108\times {10}^{-9},4.156660710075144\times {10}^{-9},\hfill \\ -3.876213794384853\times {10}^{-10},-6.430631249720875\times {10}^{-9},2.374038521324451\times {10}^{-9},\hfill \\ 1.4253239255523908\times {10}^{-8},1.4100530965066582\times {10}^{-8},1.697981844017704\times {10}^{-8},\hfill \\ 4.65114169401204\times {10}^{-9},-1.5051723201153777\times {10}^{-8},1.7268848573337436\times {10}^{-9},\hfill \\ 9.55236534316839\times {10}^{-9},-1.0925711002585103\times {10}^{-8},1.605326893638903\times {10}^{-8},\hfill \\ -1.4773551493263426\times {10}^{-8},5.336680208856137\times {10}^{-9},5.7625015870144125\times {10}^{-12},\hfill \\ -1.1958365186970354\times {10}^{-8},5.048772067084428\times {10}^{-9},5.0545057028728024\times {10}^{-9},\hfill \\ -5.577088568742283\times {10}^{-9},-1.2698765150531699\times {10}^{-8},2.1014985307488132\times {10}^{-9},\hfill \\ -7.050886541293266\times {10}^{-10},-9.165895153984138\times {10}^{-9},-6.615031411527639\times {10}^{-9},\hfill \\ 1.8693644410205934\times {10}^{-9},-2.0400425793098975\times {10}^{-9},3.970154871524301\times {10}^{-9},\hfill \\ 1.27333137367458\times {10}^{-8},1.3701127787868472\times {10}^{-8},1.6324324558780745\times {10}^{-8},\hfill \\ 9.528444477879816\times {10}^{-9},8.036737497718605\times {10}^{-9},6.877024949503152\times {10}^{-9},\hfill \\ 2.5971482742193075\times {10}^{-9},-7.050719563750363\times {10}^{-9},-9.484791840819184\times {10}^{-9},\hfill \\ -5.613398856851859\times {10}^{-9},2.113242913992508\times {10}^{-10},2.948348454268057\times {10}^{-9},\hfill \\ 2.4869450943043603\times {10}^{-9},2.016361522194643\times {10}^{-9},1.0343146339408804\times {10}^{-9},\hfill \\ 3.045099727927436\times {10}^{-10},1.9398149753158123\times {10}^{-10},-5.747202713735078\times {10}^{-11},\hfill \\ -7.665956758273751\times {10}^{-11},-8.79563089029034\times {10}^{-12}\}.\hfill \end{array}$$

We split our data in three groups (the length of series is 50) i.e., $[1,17]$, $[18,36]$ and $[37,50]$.

Graphical representation in the Figure 12 for series of residuals and in the Figure 13 for histogram of the residuals.

5.1. Study of Heteroscedasticity

When studying the constant mean value and constant variance, the first step is supposed to split series of residuals into more groups. As a simple requirement, each group has 15–20 values. A classical graph representation known as the boxplot can provide some simple information about the discrepancy between the groups. Next, compute the statistical mean (arithmetical mean) and statistical standard deviation for each group. In general, at this level, all groups tend to have the same mean value (around zero) because all methods which provide an approximate solution are supposed to optimize the minimum criteria. In a few cases, the classical t-test can be used for pairs. In the third step, one should verify the homoscedasticity to check the statistical null hypothesis that all groups have the same variance versus the alternative hypothesis that there are statistically significant differences in variance from one group to another. There are more statistical tests that can be used to verify this kind of hypothesis. Tee’s classical test was administered by Bartlett (see Bartlett [28]). If the p-value is less than the significant level $\alpha =0.05$, the null hypothesis must be rejected and the alternative hypothesis will be sustained, i.e., there will be no constant variance or heteroscedasticity; otherwise, it is false to reject the null hypothesis, i.e., homoscedasticity. This classical test has some limitations if the values from the series (residuals) are not normally distributed or there exist some outliers (accidental, atypical or abnormal values). In time, other tests to verify the homoscedasticity were proposed, such as, for example, Levene’s test for homogeneity of variance (see [29,30]) or the Fligner–Killeen test of homogeneity of variance (see, for example, Conover et al. [31]). For a robust analysis, it is recommended to use multiple tests.

Results for our example are listed below:

• First, the boxplot graph in the Figure 14:
• The statistical mean value and, respectively, the standard deviation: Mean value of groups

  $$\begin{array}{ccc}sg1& sg2& sg3\\ -7.511786\times {10}^{-10}& -3.835192\times {10}^{-10}& -1.346109\times {10}^{-10}\end{array}$$
  Standard deviation of groups

  $$\begin{array}{ccc}sg1& sg2& sg3\\ 1.600494\times {10}^{-9}& 2.255497\times {10}^{-9}& 1.590026\times {10}^{-9}\end{array}$$
• The Bartlett test in R (see details at https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/bartlett.test, accessed on 10 February 2026)
  Null Hypothesis ($H_0$): All group variances are equal.
  Alternative Hypothesis ($H_a$): At least two group variances are different.
  Results from R-software:
  Bartlett’s K-squared = 2.6764, df = 2, p-value = 0.2623
  Because the p-value = 0.2623 > 0.05, then, the decision is that “it fails to reject $H_0$” i.e., “data is consistent with equal variances” for a statistical point of view.
• The Levene’s test in R (see https://www.rdocumentation.org/packages/lawstat/versions/3.2/topics/levene.test, accessed on 10 February 2026)
  Results from the R-software:
  Df F value $Pr(>F)$
  47 0.0972 0.9076
  Because, the p-value = 0.9076 > 0.05, then the decision is “Fail to reject the null hypothesis”, i.e., “The assumption of equal variances (homoscedasticity) holds true”.
• The Fligner–Killeen test (see for detail at https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/fligner.test, accessed on 10 February 2026)
  Results from the R-software:
  Fligner–Killeen:med chi-squared = 0.6515, df = 2, p-value = 0.722
  Because the p-value = 0.722 > 0.05, the statistical decision is “Fail to reject $H_0$” and that indicates “no significant difference in variances”.

5.2. Study of Autocorrelation

The non-autocorrelation assumption is very important when an approximate solution is given, because it is desired that its analytical expression should be as accurate as possible; that is, the expression should be well determined and any difference from the numerical solution should be purely random, being the result of rounding of real numbers. The presence of a correlation between different values in the residual series suggests the presence of a periodic—and therefore deterministic—relationship or function that was not included in the analytical expression of the approximate solution.

A classical test for checking of non-autocorrelation is the test given by Durbin and Watson (see Durbin et al. [32,33]). The test verifies the null hypothesis that there is no autocorrelation with the alternative hypothesis and significant autocorrelation is presented. A p-value below (significant level $\alpha$) 0.05 indicates significant autocorrelation, allowing rejection of the null hypothesis that residuals are uncorrelated. An alternative to this classical test is the Breusch–Godfrey test for serial correlation (see Breusch [34] and Godfrey [35]).

Results for our example are listed below:

• Durbin–Watson test (details can be found at https://www.rdocumentation.org/packages/car/versions/3.1-3/topics/durbinWatsonTest, accessed on 10 February 2026)
  Null Hypothesis ($H_0$): There is no first-order autocorrelation in the residuals.
  Alternative Hypothesis ($H_a$): First-order autocorrelation exists.
  As a simple remark for the statistical decision, a p-value below 0.05 indicates significant autocorrelation, allowing rejection of the null hypothesis that residuals are uncorrelated. Also, some common interpretations in the case of the auto-correlation presence:
  $DW\approx 2$: Residuals are likely independent (no autocorrelation).
  $DW<2$: Suggests positive autocorrelation, common in time-series data.
  $DW>2$: Suggests negative autocorrelation
  Results from the R-software:
  DW = 2.0399, p-value = 0.4415
  We find that the p-value = 0.4415 > 0.05; hence, the statistical decision is “Fail to reject the null hypothesis”.
• The Breusch–Godfrey test (see https://www.rdocumentation.org/packages/lmtest/versions/0.9-40/topics/bgtest, accessed on 10 February 2026)
  Results from R-software:
  LM test = 0.070308, df = 1, p-value = 0.7909
  We see that the p-value = 0.7909 > 0.05, so the decision is “Fail to reject the null hypothesis”; that is, “there is no serial correlation of any order”.

5.3. Study of Normality

Researchers often apply tests such as the Shapiro–Wilk test to see if numbers are normally distributed. The Shapiro–Wilk test is effective in identifying various deviations from a normal pattern. The Kolmogorov–Smirnov test checks whether the data fits well into a normal distribution, while the Anderson–Darling test looks closely at the ends of the data (see, for example http://ijmas.com/upcomingissue/01.05.2025.pdf, accessed on 10 February 2026).

To verify the hypothesis that the empirical distribution of the residuals in near to normal distribution (Gauss–Laplace probability distribution) is very important for applications in the real world. A simple graph such as a histogram can give a first impression about the series of residuals, but this is not sufficient when the length of this series is not so large. There are more classical tests that can be used to check this assumption. The most used test is the Shapiro–Wilk normality test (see Shapiro et al. [36]). If the p-value is greater (significant level $\alpha$) than $0.05$, it is false to reject the null hypothesis of normality for the series of residuals. As a limitation, the test might not adequately pick up on severe outliers that render data non-normal. As alternative to this test is the one-sample Kolmogorov–Smirnov test (see, for example, Wayne [37]) or the Anderson–Darling normality test (see Anderson et al. [38]).

It is known that Shapiro–Wilk (S-W) is generally the most powerful test for normality, especially in small samples, but it is sensitive to outliers, which can cause false rejection of normality (see for example https://doi.org/10.1007/s10260-007-0046-8, or https://doi.org/10.1080/03610918.2021.1938124 or https://www.hrpub.org/download/20211130/MS15-13423966.pdf, accessed on 10 February 2026).

• Shapiro–Wilk test (see more details about this test at https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/shapiro.test, accessed on 10 February 2026)
  Results for our data:
  W = 0.87366, p-value = 7.223×10⁻⁵
  In this case, if the p-value = 0.0007223 << 0.05, then a decision can be “Reject $H_0$”, i.e., “data is significantly non-normal”.
• Komogorov–Smirnov test (see https://www.rdocumentation.org/packages/dgof/versions/1.5.1/topics/ks.test, accessed on 10 February 2026)
  Results from R-software:
  D = 0.15222, p-value = 0.1776
  The p-value = 0.1776 > 0.05, hence, the statistical decision is “Fail to reject the null hypothesis of normality”.
• Anderson–Darling normality test (see https://www.rdocumentation.org/packages/nortest/versions/1.0-4/topics/ad.test, accessed on 10 February 2026)
  Results form R-software
  A = 0.20828, p-value = 0.8572
  In this case, the p-value = 0.8572 > 0.5; therefore, the decision is “Fail to reject the null hypothesis”.

5.4. mOPIM Technique Versus the Iterative Method

A simple integration of the system (1), using the iterative method developed by Daftardar-Gejji et al. [39], yields:

$$\begin{array}{c}{\displaystyle x\left(t\right)=x\left(0\right)+\underset{0}{\overset{t}{\int }}\left(-x\left(s\right)-2y\left(s\right)\right)ds}\hfill \\ {\displaystyle y\left(t\right)=y\left(0\right)+\underset{0}{\overset{t}{\int }}\left(-x\left(s\right)z\left(s\right)-by\left(s\right)-ax\left(s\right)\right)ds}\hfill \\ {\displaystyle z\left(t\right)=z\left(0\right)+\underset{0}{\overset{t}{\int }}\left(x\left(s\right)y\left(s\right)-cz\left(s\right)\right)ds}\hfill \end{array}.$$

(41)

The iterative algorithm leads to:

$$\begin{array}{c}{\displaystyle x_0\left(t\right)=x\left(0\right),x_1\left(t\right)=N_1(x_0,y_0,z_0)=\underset{0}{\overset{t}{\int }}\left(-x\left(0\right)-2y\left(0\right)\right)ds,}\hfill \\ {\displaystyle y_0\left(t\right)=y\left(0\right),y_1\left(t\right)=N_2(x_0,y_0,z_0)=\underset{0}{\overset{t}{\int }}\left(-x\left(0\right)z\left(0\right)-by\left(0\right)-ax\left(0\right)\right)ds,}\hfill \\ {\displaystyle z_0\left(t\right)=z\left(0\right),z_1\left(t\right)=N_3(x_0,y_0,z_0)=\underset{0}{\overset{t}{\int }}\left(x\left(0\right)y\left(0\right)-cz\left(0\right)\right)ds,}\hfill \\ \cdots \hfill \\ x_m\left(t\right)=N_1\left({\displaystyle \sum _{i=0}^{m-1}{x}_i,\sum _{i=0}^{m-1}{y}_i,\sum _{i=0}^{m-1}{z}_i}\right)-N_1\left({\displaystyle \sum _{i=0}^{m-2}{x}_i,\sum _{i=0}^{m-2}{y}_i,\sum _{i=0}^{m-2}{z}_i}\right),\hfill \\ y_m\left(t\right)=N_2\left({\displaystyle \sum _{i=0}^{m-1}{x}_i,\sum _{i=0}^{m-1}{y}_i,\sum _{i=0}^{m-1}{z}_i}\right)-N_2\left({\displaystyle \sum _{i=0}^{m-2}{x}_i,\sum _{i=0}^{m-2}{y}_i,\sum _{i=0}^{m-2}{z}_i}\right),\hfill \\ z_m\left(t\right)=N_3\left({\displaystyle \sum _{i=0}^{m-1}{x}_i,\sum _{i=0}^{m-1}{y}_i,\sum _{i=0}^{m-1}{z}_i}\right)-N_3\left({\displaystyle \sum _{i=0}^{m-2}{x}_i,\sum _{i=0}^{m-2}{y}_i,\sum _{i=0}^{m-2}{z}_i}\right),\hfill \\ m\ge 2.\hfill \end{array}$$

(42)

The iterative solution generates the solution to the Equations (1) as:

$${\displaystyle x_{iter}\left(t\right)=\sum _{m=0}^{\infty }{x}_m\left(t\right),y_{iter}\left(t\right)=\sum _{m=0}^{\infty }{y}_m\left(t\right),z_{iter}\left(t\right)=\sum _{m=0}^{\infty }{z}_m\left(t\right),}$$

The iterative solutions $x_{iter}\left(t\right)$, after eight iterations using the algorithm (42), for the initial data: $x_{i0}=0.55$, $y_{i0}=0.35$, $z_{i0}=1.75$ and the physical parameters $a=0.65$, $b=0.34$, $c=2$ (presented in

Table 4. Comparative values between the mOPIM solution ${\overline{x}}_{mOPIM}\left(t\right)$ using (A1), the numerical solution, and the iterative solution $x_{iter}\left(t\right)$.

$\mathit{t}\left[\mathit{s}\right]$	${\mathit{x}}_{\mathit{numerical}}$	${\overline{\mathit{x}}}_{\mathit{mOPIM}}$	${\mathit{x}}_{\mathit{iter}}$	${\mathit{x}}_{\mathit{iter}}$
			5 Iterations	8 Iterations
0	0.55	0.5499999951	0.55	0.55
0.15	0.4017708691	0.4017708612	0.4017660802	0.4017708684
0.30	0.3104242855	0.3104243299	0.3101733342	0.3104245484
0.45	0.2544282526	0.2544282041	0.2520579351	0.2544358594
0.60	0.2212758090	0.2212758157	0.2100902278	0.2213566010
0.75	0.2034134260	0.2034136365	0.1671499114	0.2039027309
0.90	0.1961178916	0.1961180752	0.1031772101	0.1982128316
1.05	0.1963496108	0.1963496888	−0.0063040127	0.2034692125
1.20	0.2021105486	0.2021105613	−0.1901168373	0.2227403396
1.35	0.2120704721	0.2120705248	−0.4794393043	0.2657949353
1.5	0.2253404724	0.2253406402	−0.9031510107	0.3578749377

), the iterative solutions $x_{iter}\left(t\right)$, after eight iterations using the algorithm (42) are built:

$$\begin{array}{c}{\displaystyle x_{iter}\left(t\right)=\sum _{m=0}^8{x}_m\left(t\right)=0.55-1.25t+2.064t^2-2.4574616666t^3+2.5262927833t^4-}\hfill \\ 2.2701166015t^5+1.7578143914t^6-1.1015286488t^7+0.4186353844t^8+0.2194535569t^9-\hfill \\ 0.7224845695t^{10}+1.0343864324t^{11}-1.1381575372t^{12}+1.0617940546t^{13}-0.8618132505t^{14}+\hfill \\ 0.6036134838t^{15}-0.3450661900t^{16}+0.1273296168t^{17}+0.0281831371t^{18}-0.1177692187t^{19}+\hfill \\ 0.1506774717t^{20}-0.1431944614t^{21}+0.1131554063t^{22}-0.0758511714t^{23}+0.0418218915t^{24}-\hfill \\ 0.0164498159t^{25}+0.0008716307t^{26}+0.0064682908t^{27}-0.0082471358t^{28}+0.0070899240t^{29}-\hfill \\ 0.0049416059t^{30}+0.0029190629t^{31}-0.0014628496t^{32}+0.0005983703t^{33}-0.0001724173t^{34}+\hfill \\ 6.2581\times {10}^{-6}{t}^{35}+0.0000353147t^{36}-0.0000314301t^{37}+0.0000185176t^{38}-8.6531\times {10}^{-6}{t}^{39}+\hfill \\ 3.3462\times {10}^{-6}{t}^{40}-1.0730\times {10}^{-6}{t}^{41}+2.7797\times {10}^{-7}{t}^{42}-5.4037\times {10}^{-8}{t}^{43}+6.1022\times {10}^{-9}{t}^{44}+\hfill \\ 3.4991\times {10}^{-10}{t}^{45}-3.2282\times {10}^{-10}{t}^{46}+6.4532\times {10}^{-11}{t}^{47}-3.4977\times {10}^{-12}{t}^{48}-1.0500\times {10}^{-12}{t}^{49}+\hfill \\ 2.1211\times {10}^{-13}{t}^{50}-2.2512\times {10}^{-15}{t}^{51}-2.7725\times {10}^{-15}{t}^{52}+1.1619\times {10}^{-16}{t}^{53}+\mathcal{O}\left({10}^{-17}\right),\hfill \\ \\ {\displaystyle y_{iter}\left(t\right)=\sum _{m=0}^8{y}_m\left(t\right)=0.35-1.439t+2.6541924999t^2-3.8238547333t^3+4.4121451122t^4-}\hfill \\ 4.1383848736t^5+2.9764430751t^6-1.1237772134t^7-1.1013690249t^8+3.3584567407t^9-\hfill \\ 5.2921867893t^{10}+6.5920058315t^{11}-7.0496161367t^{12}+6.6123683352t^{13}-5.3917763308t^{14}+\hfill \\ 3.6299361859t^{15}-1.6385096471t^{16}-0.2685389994t^{17}+1.8322814144t^{18}-2.8867970280t^{19}+\hfill \\ 3.3737758593t^{20}-3.3348676893t^{21}+2.8867556994t^{22}-2.1866186092t^{23}+1.3965198622t^{24}-\hfill \\ 0.6541129402t^{25}+0.0543407239t^{26}+0.3565299043t^{27}-0.5761656649t^{28}+0.6342872781t^{29}-\hfill \\ 0.5779082070t^{30}+0.4575016887t^{31}-0.3166976472t^{32}+0.1866084036t^{33}-0.0845273037t^{34}+\hfill \\ 0.0158393053t^{35}+0.0223688034t^{36}-0.0373849249t^{37}+0.0376021329t^{38}-0.0303634153t^{39}+\hfill \\ 0.0209467568t^{40}-0.0124401993t^{41}+0.0061499234t^{42}-0.0022094814t^{43}+0.0001558797t^{44}+\hfill \\ 0.0006472994t^{45}-0.0007656868t^{46}+0.0005975154t^{47}-0.0003714574t^{48}+0.0001886539t^{49}-\hfill \\ 0.0000735870t^{50}+0.0000152342t^{51}+7.0725\times {10}^{-6}{t}^{52}-0.0000112027t^{53}+8.5800\times {10}^{-6}{t}^{54}-\hfill \\ 4.9023\times {10}^{-6}{t}^{55}+2.2043\times {10}^{-6}{t}^{56}-7.2576\times {10}^{-7}{t}^{57}+9.8880\times {10}^{-8}{t}^{58}+8.3700\times {10}^{-8}{t}^{59}-\hfill \\ 9.1256\times {10}^{-8}{t}^{60}+5.6222\times {10}^{-8}{t}^{61}-2.6066\times {10}^{-8}{t}^{62}+9.3324\times {10}^{-9}{t}^{63}-2.2804\times {10}^{-9}{t}^{64}+\hfill \\ 6.7332\times {10}^{-11}{t}^{65}+3.2069\times {10}^{-10}{t}^{66}-2.3231\times {10}^{-10}{t}^{67}+1.1093\times {10}^{-10}{t}^{68}-4.1942\times {10}^{-11}{t}^{69}+\hfill \\ 1.3145\times {10}^{-11}{t}^{70}-3.4372\times {10}^{-12}{t}^{71}+7.3525\times {10}^{-13}{t}^{72}-1.2103\times {10}^{-13}{t}^{73}+1.2440\times {10}^{-14}{t}^{74}+\hfill \\ 2.7705\times {10}^{-16}{t}^{75}-4.3783\times {10}^{-16}{t}^{76}+\mathcal{O}\left({10}^{-17}\right),\hfill \\ {\displaystyle z_{iter}\left(t\right)=\sum _{m=0}^8{z}_m\left(t\right)=1.75-3.3075t+2.693025t^2-0.4683647083t^3-2.0785847237t^4+}\hfill \\ 4.2524821616t^5-5.8568576690t^6+6.8020834634t^7-6.9909543887t^8+6.3497986268t^9-\hfill \\ 4.9091578782t^{10}+2.8497970959t^{11}-0.4648007037t^{12}-1.8875748313t^{13}+3.8606053610t^{14}-\hfill \\ 5.1903753956t^{15}+5.7433093428t^{16}-5.5299170802t^{17}+4.6865410308t^{18}-3.4334808676t^{19}+\hfill \\ 2.0223521182t^{20}-0.6854656961t^{21}-0.4023637415t^{22}+1.1437257688t^{23}-1.5194600368t^{24}+\hfill \\ 1.5741528765t^{25}-1.3926540150t^{26}+1.0745752001t^{27}-0.7125356573t^{28}+0.3776656732t^{29}-\hfill \\ 0.1133764187t^{30}-0.0636709100t^{31}+0.1577267766t^{32}-0.1859477270t^{33}+0.1705715118t^{34}-\hfill \\ 0.1328382255t^{35}+0.0893272168t^{36}-0.0506194738t^{37}+0.0217096906t^{38}-0.0034084197t^{39}-\hfill \\ 0.0059645660t^{40}+0.0090896822t^{41}-0.0086078135t^{42}+0.0065701683t^{43}-0.0042829616t^{44}+\hfill \\ 0.0023988128t^{45}-0.0011141966t^{46}+0.0003724605t^{47}-0.0000195098t^{48}-0.0001020489t^{49}+\hfill \\ 0.0001111473t^{50}-0.0000807002t^{51}+0.0000467754t^{52}-0.0000222035t^{53}+8.1667\times {10}^{-6}{t}^{54}-\hfill \\ 1.6963\times {10}^{-6}{t}^{55}-5.3820\times {10}^{-7}{t}^{56}+8.9183\times {10}^{-7}{t}^{57}-6.4790\times {10}^{-7}{t}^{58}+3.4968\times {10}^{-7}{t}^{59}-\hfill \\ 1.5104\times {10}^{-7}{t}^{60}+5.0847\times {10}^{-8}{t}^{61}-1.0841\times {10}^{-8}{t}^{62}-9.9680\times {10}^{-10}{t}^{63}+2.6212\times {10}^{-9}{t}^{64}-\hfill \\ 1.7431\times {10}^{-9}{t}^{65}+8.3175\times {10}^{-10}{t}^{66}-3.2285\times {10}^{-10}{t}^{67}+1.0545\times {10}^{-10}{t}^{68}-2.9049\times {10}^{-11}{t}^{69}+\hfill \\ 6.6069\times {10}^{-12}{t}^{70}-1.1662\times {10}^{-12}{t}^{71}+1.2962\times {10}^{-13}{t}^{72}+3.1150\times {10}^{-15}{t}^{73}-5.4519\times {10}^{-15}{t}^{74}+\hfill \\ 1.3631\times {10}^{-15}{t}^{75}-1.7328\times {10}^{-16}{t}^{76}+\mathcal{O}\left({10}^{-18}\right).\hfill \end{array}$$

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A comparison between mOPIM results obtained using the iterative method (using five and eight iterations). The results are represented by Table 4 and Figure 15 and Figure 16. By analyzing the performance, it is clearly shown that increasing the number of iterations makes the performance of the iterative method competitive with the mOPIM method that used just one iteration.

6. Conclusions

The closed-form solutions for a three-parameter dynamical system are obtained using a new analytical approach, mOPIM, for solving second-order nonlinear differential equations.

This system is explicitly integrated via a smooth function solution of a third–order nonlinear differential equation. It is pointed out that the exact parametric solutions describe the heteroclinical orbit of the analyzed system. The obtained results are validated by graphically comparing them with the corresponding numerical solutions.

The accuracy of the mOPIM results is highlighted by graphically and tabularly comparative representations with the corresponding numerical solutions. A comparison with the iterative method using five and eight iterations is included to emphasize the effectiveness of the mOPIM for large intervals using only one iteration. Moreover, the magnitude of the residual values for mOPIM solution is of the ${10}^{-7}$ order by comparison with the corresponding OPIM solution, which is just of the ${10}^{-4}$ order. The performance indicator of the mOPIM results is between ${10}^{-16}$–${10}^{-11}$ orders of magnitude.

A rigorous qualitative analysis of errors is provided by studying heteroscedasticity, autocorrelation and normality.

The achieved results have high potential and they encourage the study of dynamical systems with similar properties, which described technological applications and natural physical phenomena.