Approximate Closed-Form Solutions for a Class of 3D Dynamical Systems Involving a Hamilton–Poisson Part

Abstract

The goal of this paper is to build some approximate closed-form solutions for a class of dynamical systems involving a Hamilton–Poisson part. The chaotic behaviors are neglected. These solutions are obtained by means of a new version of the optimal parametric iteration method (OPIM), namely, the modified optimal parametric iteration method (mOPIM). The effect of the physical parameters is investigated. The Hamilton–Poisson part of the dynamical systems is reduced to a second-order nonlinear differential equation, which is analytically solved by the mOPIM procedure. A comparison between the approximate analytical solution obtained with mOPIM, the analytical solution obtained with the iterative method, and the corresponding numerical solution is presented. The mOPIM technique has more advantages, such as the convergence control (in the sense that the residual functions are smaller than 1), the efficiency, the writing of the solutions in an effective form, and the nonexistence of small parameters. The accuracy of the analytical and corresponding numerical results is illustrated by graphical and tabular representations. The same procedure could be successfully applied to more dynamical systems.

1. Introduction

Many nonlinear phenomena that appear in engineering, chemistry, physics, economics, and biology can be modeled by the nonlinear dynamical systems of the form $\dot{\mathbf{x}}=\mathbf{f}\left(\mathbf{x}\right)$, $\mathbf{f}=(f_1,f_2,f_3)$, where $\mathbf{f}\left(\mathbf{x}\right)=\mathbf{g}\left(\mathbf{x}\right)+\mathbf{h}\left(\mathbf{x}\right)$ such that the system $\dot{\mathbf{x}}=\mathbf{g}\left(\mathbf{x}\right)$ admits a Hamilton–Poisson structure (e.g., is a Hamilton–Poisson system) and $h=(h_1,h_2,h_3)$ is an additive term. There are two functionally independent constants of motion, $H=H\left(\mathbf{x}\right)$ (the Hamiltonian function) and $C=C\left(\mathbf{x}\right)$ (the Casimir function).

In the last decade, the dynamical properties have been examined by several researchers as bifurcation route, Poincar$\acute{e}$ map, frequency spectrum, amplitude modulation, topological horseshoe, the existence of heteroclinic orbit or homoclinic orbit, equilibria, Lyapunov exponent spectrum, a dissipative system, phase portraits, bifurcation diagrams, and Hopf bifurcation. These properties characterize the chaotic behaviors of the dynamical system. Li et al. [1] studied a three-dimensional autonomous chaotic system that is found to possess two nonhyperbolic equilibria. Pham et al. [2] introduced a new system with an infinite number of equilibrium points. Wang et al. [3] presented a watermark encryption algorithm for a new memristive chaotic system. Zhang et al. [4] proposed a numerical scheme for the study of the dislocated projective synchronization (DPS) between the fractional-order and the integer-order chaotic systems. Tong [5] investigated the chaotic attractor for a three-dimensional (3D) chaotic system that possess invariable Lyapunov exponent spectra and controllable signal amplitude. He et al. [6] introduced a new four-dimensional chaotic system with coexisting attractors having three quadratic nonlinearities and only one unstable fixed point. Singh et al. [7] reported a new 4D dissipative chaotic system studying the coexistence of asymmetric hidden chaotic attractors with a curve of equilibria. Sun et al. [8] proposed a novel kind of compound–combination antisynchronization scheme among five chaotic systems. Cicek et al. [9] implemented in practical applications a new three-dimensional continuous time chaotic system by an electronic circuit design. Lai et al. [10] numerically investigated a new 3D autonomous chaotic system with coexisting attractors. Varan et al. [11] implemented a synchronization circuit model of a third-degree Malasoma system with chaotic flow. Su [12] investigated the horseshoe chaos using the topological horseshoe theory, taking into account a three-dimensional (3D) autonomous chaotic system. Zhou et al. [13] introduced and analyzed theoretically the basic dynamical properties of a three-dimensional chaotic system. The result shows the chaotic attractor by the realization of a circuit experiment. Akgul et al. [14] explored a three-dimensional chaotic system with cubic nonlinearities. They applied the electronic circuit implementation for real environment application. Pham et al. [15] introduced a three-dimensional chaotic system displaying both hidden attractors with infinite equilibria and hidden attractors without equilibrium. Zhang [16] investigated a method for generating complex grid multiwing chaotic attractors. Kacar [17] developed a four-dimensional chaotic system and implemented an analogue circuit and microcontroller. Tuna et al. [18] presented numerical, analog, and digital circuit modelings by using a 3D chaotic system with a single equilibrium point. Naderi et al. [19] explored the exponential synchronization of the chaotic system without a linear term and its application in secure communication by using the exponential stability theorem and showing the ability and effectiveness of the proposed method by numerical simulation. Li et al. [20] studied complicated dynamical behaviors of a three-dimensional chaotic system with quadratic nonlinearities.

Recently, Liu et al. [21] developed a new multiwing chaotic system that has an excellent effect on image encryption. Hu et al. [22] designed a circuit implementation to verify the physical feasibility of an asymmetric memristor-based chaotic system with only one equilibrium point. Sun et al. [23] studied a color image encryption scheme base on a 5D memristive chaotic system. Wang et al. [24] explored the problem in image encryption on the basis of a chaotic system with time delay. Guo et al. [25] proposed a multivortex hyperchaotic system, emphasizing its application to image encryption and outstanding anticropping and antinoise performance. Yildirim et al. [26] used the particle swarm optimization (PSO) and ant colony optimization (ACO) to optimize the initial conditions of a continuous-time chaotic system. Ding et al. [27] proposed a cryptosystem and its application in image encryption. Lai et al. [28] proposed a four-dimensional multiscroll chaotic system with application to image encryption. Lu et al. [29] proposed an encryption algorithm for 3D medical models.

Recently, Karimov et al. [30] implemented an analog circuit and proposed a novel technique for reconstructing ordinary differential equations (ODEs) describing the circuit from data. This technique is shown for a well-studied R$\ddot{o}$ssler chaotic system. Karimov et al. [31] studied the synchronization between a circuit modeling the R$\ddot{o}$ssler chaotic system and a computer model by using adaptive generalized synchronization.

Beyond chaotic behaviors, some systems could have nonlinear singularities. Such systems are investigated using the topological degree theory and the qualitative analysis of a Poincar$\acute{e}$ map with action angle variables [32]. Cheng et al. [33] established the existence of homoclinic solutions for a differential inclusion system involving the $p\left(t\right)$-Laplacian by using a variational principle. Fonda et al. [34] proved the existence and multiplicity results for periodic solutions of Hamiltonian systems using the Poincar$\acute{e}$–Birkhoff fixed point theorem.

Many nonlinear differential problems from applied engineering are analytically solved by some methods, namely, the multiple scales technique [35], the optimal iteration parametrization method (OIPM) [36], the optimal homotopy asymptotic method (OHAM) [37,38,39], and the optimal homotopy perturbation method (OHPM) [40,41,42].

The structure of this paper is as follows: In Section 2, we present in detail some dynamical systems involving a Hamilton–Poisson part. The steps of the mOPIM technique are the subject of Section 3. Section 4 presents the semianalytical solutions obtained by the mOPIM method. Section 5 provides the numerical results and emphasizes the validation of the method. The conclusions and perspectives are highlighted in Section 6.

2. A Class of Dynamical Systems Involving a Hamilton–Poisson Part

The T system analyzed in [43] describes the stability of the chaotic behavior by an integrable deformation. This system has the following form:

$$\left\{\begin{array}{c}\dot{x}=-ax+ay\hfill \\ \dot{y}=(c-a)x-axz\hfill \\ \dot{z}=-bz+xy\hfill \end{array}\right.,a,b,c\in \mathbb{R},a\ne 0,$$

(1)

with a chaotic behavior for some positive values, $a=2.1$, $b=0.6$, $c=30$ [44].

A Hamilton–Poisson part of the system (1) is

$$\left\{\begin{array}{c}\dot{x}=ay\hfill \\ \dot{y}=-axz\hfill \\ \dot{z}=xy\hfill \end{array}\right.,a\in \mathbb{R},a\ne 0.$$

(2)

The functionally independent constants of motion of the system (2) are

$$\left\{\begin{array}{c}H(x,y,z)=\frac{1}{2}{x}^2-az\hfill \\ C(x,y,z)=\frac{1}{2}{y}^2+\frac{a}{2}{z}^2\hfill \end{array}\right.,a\in \mathbb{R},a\ne 0.$$

(3)

Remark 1.

Considering the initial conditions:

$$x\left(0\right)=x_0,y\left(0\right)=y_0,z\left(0\right)=z_0,$$

(4)

the phase curves of dynamics (1) are the intersections of the following surfaces:

$$\left\{\begin{array}{c}\frac{1}{2}{x}^2-az=\frac{1}{2}{x}_0^2-a{z}_0\hfill \\ \frac{1}{2}{y}^2+\frac{a}{2}{z}^2=\frac{1}{2}{y}_0^2+\frac{a}{2}{z}_0^2\hfill \end{array}\right.,\mathit{for}a\ne 0.$$

(5)

2.1. Closed-Form Solutions of the T System Involving a Hamilton–Poisson Part

For the system (2), there are following cases:

(i)

In the case $a>0$, the transformations

$$\left\{\begin{array}{c}y\left(t\right)=R\sqrt{2}\frac{2u\left(t\right)}{1+u^2\left(t\right)}\hfill \\ z\left(t\right)=\frac{R\sqrt{2}}{\sqrt{a}}\frac{1-u^2\left(t\right)}{1+u^2\left(t\right)}\hfill \end{array}\right.,$$

(6)

where $R=\sqrt{\frac{1}{2}{y}_0^2+\frac{a}{2}{z}_0^2}$, $u\left(t\right)$ is an unknown smooth function, provide the closed-form solutions.

The third equation from Equation (2) yields

$$x\left(t\right)=-\frac{2}{\sqrt{a}}\frac{\dot{u}\left(t\right)}{1+u^2\left(t\right)}.$$

(7)

From the first Equation (2), we obtain

$$\ddot{u}\left(t\right)[1+u^2\left(t\right)]-2u\left(t\right){\left[\dot{u}\left(t\right)\right]}^2+aR\sqrt{2a}·u\left(t\right)[1+u^2\left(t\right)]=0.$$

(8)

The initial conditions $u\left(0\right)$ and $\dot{u}\left(0\right)$ obtained from Equations (4), (6), and (7) are

$$u\left(0\right)=\sqrt{\frac{R\sqrt{2}-z_0\sqrt{a}}{R\sqrt{2}+z_0\sqrt{a}}},\dot{u}\left(0\right)=-\frac{\sqrt{a}}{2}{x}_0[1+u^2\left(0\right)],$$

(9)

with $R\sqrt{2}+z_0\sqrt{a}>0$.

(ii)

For $a<0$, the closed-form solutions can be written as

$$\left\{\begin{array}{c}y\left(t\right)=R\sqrt{2}·\frac{2·u\left(t\right)}{1-u^2\left(t\right)}\hfill \\ z\left(t\right)=R\sqrt{2\left|a\right|}·\frac{1+u^2\left(t\right)}{1-u^2\left(t\right)}\hfill \end{array}\right.,$$

(10)

where $R=\sqrt{\frac{1}{2}{z}_0^2-\frac{1}{2\left|a\right|}{y}_0^2}$.

Equation (2) yields

$$x\left(t\right)=\frac{2}{\sqrt{\left|a\right|}}\frac{\dot{u}\left(t\right)}{1-u^2\left(t\right)}.$$

(11)

The following nonlinear problem gives the unknown function $u\left(t\right)$:

$$\left\{\begin{array}{c}\ddot{u}\left(t\right)·(1-u^2\left(t\right))+2u\left(t\right)·{\left(\dot{u}\left(t\right)\right)}^2-a\left|a\right|R\sqrt{2}·u\left(t\right)·(1-u^2\left(t\right))=0\hfill \\ u\left(0\right)=\sqrt{\frac{z_0-R\sqrt{2}}{z_0+R\sqrt{2}}},\dot{u}\left(0\right)=\frac{\sqrt{\left|a\right|}}{2}{x}_0·\left[1-u^2\left(0\right)\right],\hfill \end{array}\right.$$

(12)

with $z_0+R\sqrt{2}>0$.

2.2. Other 3D Dynamical Systems Involving a Hamilton–Poisson Part

(i)

A three-dimensional autonomous chaotic system with three multipliers presented in [45] is

$$\left\{\begin{array}{c}\dot{x}=-ax+byz\hfill \\ \dot{y}=cy-dxz\hfill \\ \dot{z}=-kz+mxy\hfill \end{array}\right.,a,b,c,d,k,m\in \mathbb{R},$$

(13)

having a Hamilton–Poisson part, namely,

$$\left\{\begin{array}{c}\dot{x}=byz\hfill \\ \dot{y}=-dxz\hfill \\ \dot{z}=mxy\hfill \end{array}\right.,b,d,m\in \mathbb{R},$$

(14)

with $H=\frac{1}{2}(\frac{1}{b}{x}^2+\frac{1}{d}{y}^2)$ and $C=\frac{1}{2}(\frac{1}{b}{x}^2-\frac{1}{m}{z}^2)$.

The closed-form solutions of the system (14) could be written as

$$\left\{\begin{array}{c}x\left(t\right)=R\sqrt{b}\frac{1+u^2\left(t\right)}{1-u^2\left(t\right)}\hfill \\ z\left(t\right)=R\sqrt{m}\frac{2u\left(t\right)}{1-u^2\left(t\right)}\hfill \\ y\left(t\right)=\frac{2}{\sqrt{bm}}\frac{\dot{u}\left(t\right)}{1-u^2\left(t\right)}\hfill \end{array}\right.,$$

(15)

where $R=\sqrt{\frac{1}{b}{x}_0^2-\frac{1}{m}{z}_0^2}$, for $\frac{1}{b}{x}_0^2-\frac{1}{m}{z}_0^2>0$.

$u\left(t\right)$ is an unknown smooth function, a solution of the nonlinear problem:

$$\begin{array}{c}\ddot{u}\left(t\right)[1-u^2\left(t\right)]+2u\left(t\right){\left[\dot{u}\left(t\right)\right]}^2+bmd{R}^2·u\left(t\right)[1+u^2\left(t\right)]=0\hfill \\ u\left(0\right)=\sqrt{\frac{b}{m}}\frac{|z_0|}{|x_0+R\sqrt{b}|},\dot{u}\left(0\right)=\frac{1}{2}{y}_0\sqrt{bm}(1-u^2\left(0\right)).\hfill \end{array}$$

(16)

If $\frac{1}{b}{x}_0^2-\frac{1}{m}{z}_0^2<0$, then the closed-form solutions are

$$\left\{\begin{array}{c}x\left(t\right)=R\sqrt{b}\frac{2u\left(t\right)}{1-u^2\left(t\right)}\hfill \\ z\left(t\right)=R\sqrt{m}\frac{1+u^2\left(t\right)}{1-u^2\left(t\right)}\hfill \\ y\left(t\right)=\frac{2}{\sqrt{bm}}\frac{\dot{u}\left(t\right)}{1-u^2\left(t\right)}\hfill \end{array}\right.,$$

(17)

where $R=\sqrt{\frac{1}{m}{z}_0^2-\frac{1}{b}{x}_0^2}$, $u\left(t\right)$ is an unknown smooth function, a solution of the nonlinear problem (16), with the initial conditions

$$u\left(0\right)=\sqrt{\frac{m}{b}}\frac{|x_0|}{|z_0+R\sqrt{m}|},\dot{u}\left(0\right)=\frac{1}{2}{y}_0\sqrt{bm}(1-u^2\left(0\right)).$$

(18)

(ii)

The Qi chaotic system [46] has the form

$$\left\{\begin{array}{c}\dot{x}=a(y-x)+yz\hfill \\ \dot{y}=cx-y-xz\hfill \\ \dot{z}=-bz+xy\hfill \end{array}\right.,a,b,c\in {\mathbb{R}}_{+}.$$

(19)

The analysis of energy exchange was examined by transforming into a Kolmogorov-type system. It is shown that this system possesses four forms of energy, by decomposing the vector field of this chaotic system into four forms of torque: inertial, internal, dissipative, and external.

The chaotic system presented in [47] is

$$\left\{\begin{array}{c}\dot{x}=-ax+yz+dsign\left(y\right)\hfill \\ \dot{y}=by-xz\hfill \\ \dot{z}=-cz+xy\hfill \end{array}\right.,a,b,c\in {\mathbb{R}}_{+}^{*},d\ne 0.$$

(20)

The system can have hyperchaotic behaviors. A physically realizable system is shown by a circuit implementation of the chaotic system.

The hyperchaotic system described in [48] is

$$\left\{\begin{array}{c}\dot{x}=a(y-x)+yz\hfill \\ \dot{y}=cx-xz\hfill \\ \dot{z}=-bz+xy\hfill \end{array}\right.,a,b,c\in {\mathbb{R}}_{+}^{*}.$$

(21)

A circuit experiment was implemented, proving rich dynamics, and can exhibit periodic, quasi-periodic, chaos, and hyperchaos behavior.

The last three chaotic systems have the same Hamilton–Poisson part, namely,

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

(22)

with $H=\frac{1}{2}(x^2+y^2)$ and $C=\frac{1}{2}(x^2-z^2)$.

The proposed closed-form solutions of the system (22) are

$$\left\{\begin{array}{c}x\left(t\right)=R\frac{1+u^2\left(t\right)}{1-u^2\left(t\right)}\hfill \\ z\left(t\right)=R\frac{2u\left(t\right)}{1-u^2\left(t\right)}\hfill \\ y\left(t\right)=\frac{2\dot{u}\left(t\right)}{1-u^2\left(t\right)}\hfill \end{array}\right.,$$

(23)

where $R=\sqrt{x_0^2-z_0^2}$, for $x_0^2-z_0^2>0$, $u\left(t\right)$ is an unknown smooth function, a solution of the nonlinear problem (16), being a particular case of the system (14) with $b=d=m=1$.

(iii)

The chaotic system with hyperbolic sine nonlinearity [49]

$$\left\{\begin{array}{c}\dot{x}=-ax+yz\hfill \\ \dot{y}=-sh\left(y\right)+xz\hfill \\ \dot{z}=z-xy\hfill \end{array}\right.,a>0,$$

(24)

has a Hamilton–Poisson part, namely,

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

(25)

with $H=\frac{1}{2}(x^2+z^2)$ and $C=\frac{1}{2}(x^2-y^2)$.

The closed-form solutions of the system (25) could be

$$\left\{\begin{array}{c}x\left(t\right)=R\frac{1+u^2\left(t\right)}{1-u^2\left(t\right)}\hfill \\ y\left(t\right)=R\frac{2u\left(t\right)}{1-u^2\left(t\right)}\hfill \\ z\left(t\right)=\frac{2\dot{u}\left(t\right)}{1-u^2\left(t\right)}\hfill \end{array}\right.,$$

(26)

where $R=\sqrt{x_0^2-y_0^2}$, for $x_0^2-y_0^2>0$, $u\left(t\right)$ is an unknown smooth function, a solution of the nonlinear problem (16) (taking $b=m=d=1$) with the initial conditions

$$u\left(0\right)=\frac{|y_0|}{|x_0+R|},\dot{u}\left(0\right)=\frac{1}{2}{z}_0(1-u^2\left(0\right)).$$

(27)

If $x_0^2-y_0^2<0$, then the closed-form solutions are

$$\left\{\begin{array}{c}x\left(t\right)=R\frac{2u\left(t\right)}{1-u^2\left(t\right)}\hfill \\ y\left(t\right)=R\frac{1+u^2\left(t\right)}{1-u^2\left(t\right)}\hfill \\ z\left(t\right)=\frac{2\dot{u}\left(t\right)}{1-u^2\left(t\right)}\hfill \end{array}\right.,$$

(28)

where $R=\sqrt{y_0^2-x_0^2}$, $u\left(t\right)$ is an unknown smooth function, a solution of the nonlinear problem (16), with the initial conditions

$$u\left(0\right)=\frac{|x_0|}{|y_0+R|},\dot{u}\left(0\right)=\frac{1}{2}{z}_0(1-u^2\left(0\right)).$$

(29)

(iv)

The chaotic system explored in [50] has the form

$$\left\{\begin{array}{c}\dot{x}=-\frac{ab}{a+b}x-yz+c\hfill \\ \dot{y}=ay+xz\hfill \\ \dot{z}=bz+xy\hfill \end{array}\right.,a,b,c\in \mathbb{R},$$

(30)

with the Hamilton–Poisson part:

$$\left\{\begin{array}{c}\dot{x}=-yz\hfill \\ \dot{y}=xz\hfill \\ \dot{z}=xy\hfill \end{array}\right.,a,b,c\in \mathbb{R},$$

(31)

with $H=\frac{1}{2}(x^2+y^2)$ and $C=\frac{1}{2}(y^2-z^2)$.

The closed-form solutions of the system (31) are

$$\left\{\begin{array}{c}y\left(t\right)=R\frac{1+u^2\left(t\right)}{1-u^2\left(t\right)}\hfill \\ z\left(t\right)=R\frac{2u\left(t\right)}{1-u^2\left(t\right)}\hfill \\ x\left(t\right)=\frac{2\dot{u}\left(t\right)}{1-u^2\left(t\right)}\hfill \end{array}\right.,$$

(32)

where $R=\sqrt{y_0^2-z_0^2}$, for $y_0^2-z_0^2>0$, $u\left(t\right)$ is an unknown smooth function, a solution of the nonlinear problem (16) (taking b = m = d = 1) with the initial conditions

$$u\left(0\right)=\frac{|z_0|}{|y_0+R|},\dot{u}\left(0\right)=\frac{1}{2}{x}_0(1-u^2\left(0\right)).$$

(33)

If $y_0^2-z_0^2<0$, then the closed-form solutions are

$$\left\{\begin{array}{c}y\left(t\right)=R\frac{2u\left(t\right)}{1-u^2\left(t\right)}\hfill \\ z\left(t\right)=R\frac{1+u^2\left(t\right)}{1-u^2\left(t\right)}\hfill \\ x\left(t\right)=\frac{2\dot{u}\left(t\right)}{1-u^2\left(t\right)}\hfill \end{array}\right.,$$

(34)

where $R=\sqrt{z_0^2-y_0^2}$, $u\left(t\right)$ is an unknown smooth function, a solution of the nonlinear problem (16) (taking b = m = d = 1), with the initial conditions

$$u\left(0\right)=\frac{|y_0|}{|z_0+R|},\dot{u}\left(0\right)=\frac{1}{2}{x}_0(1-u^2\left(0\right)).$$

(35)

(v)

A hyperchaotic system [51] explores the phase portraits, Lyapunov exponents, bifurcation diagram, and Poincar$\acute{e}$ map:

$$\left\{\begin{array}{c}\dot{x}=-ay-xz\hfill \\ \dot{y}=-x+xz\hfill \\ \dot{z}=-d-xy\hfill \end{array}\right.,a,d\in \mathbb{R}.$$

(36)

The Hamilton–Poisson part is

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

(37)

with $H=x+y$ and $C=\frac{1}{2}(y^2+z^2)$.

An electronic circuit was designed. This system generates multiwing nonequilibrium attractors.

The closed-form solutions of the system (37) could be

$$\left\{\begin{array}{c}y\left(t\right)=R\frac{2u\left(t\right)}{1+u^2\left(t\right)}\hfill \\ z\left(t\right)=R\frac{1-u^2\left(t\right)}{1+u^2\left(t\right)}\hfill \\ x\left(t\right)=\frac{2\dot{u}\left(t\right)}{1+u^2\left(t\right)}\hfill \end{array}\right.,$$

(38)

where $R=\sqrt{y_0^2+z_0^2}$, $u\left(t\right)$ is an unknown smooth function, a solution to the nonlinear problem

$$\begin{array}{c}\ddot{u}\left(t\right)[1+u^2\left(t\right)]-2u\left(t\right){\left[\dot{u}\left(t\right)\right]}^2+R·\dot{u}\left(t\right)[1-u^2\left(t\right)]=0\hfill \\ u\left(0\right)=\frac{|y_0|}{|z_0+R|},\dot{u}\left(0\right)=\frac{1}{2}{x}_0(1+u^2\left(0\right)).\hfill \end{array}$$

(39)

(vi)

A three-dimensional autonomous chaotic system with only one positive term was explored in [52]:

$$\left\{\begin{array}{c}\dot{x}=-x-2y\hfill \\ \dot{y}=-xz-by-ax\hfill \\ \dot{z}=xy-cz\hfill \end{array}\right.,a,b,c\in {\mathbb{R}}_{+}^{*},$$

(40)

with a Hamilton–Poisson part, namely,

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

(41)

with $H=\frac{1}{2}(y^2+z^2)$ and $C=\frac{1}{2}{x}^2+2z$.

The closed-form solutions of the system (41) could be written as

$$\left\{\begin{array}{c}y\left(t\right)=R\frac{2u\left(t\right)}{1+u^2\left(t\right)}\hfill \\ z\left(t\right)=R\frac{1-u^2\left(t\right)}{1+u^2\left(t\right)}\hfill \\ x\left(t\right)=-\frac{2\dot{u}\left(t\right)}{1+u^2\left(t\right)}\hfill \end{array}\right.,$$

(42)

where $R=\sqrt{y_0^2+z_0^2}$, $u\left(t\right)$ is an unknown smooth function, a solution of the nonlinear problem

$$\begin{array}{c}\ddot{u}\left(t\right)[1+u^2\left(t\right)]-2u\left(t\right){\left[\dot{u}\left(t\right)\right]}^2-2R·u\left(t\right)[1+u^2\left(t\right)]=0\hfill \\ u\left(0\right)=\frac{|y_0|}{|z_0+R|},\dot{u}\left(0\right)=-\frac{1}{2}{x}_0(1+u^2\left(0\right)).\hfill \end{array}$$

(43)

(vii)

An autonomous chaotic system with cubic nonlinearity was presented in [53]:

$$\left\{\begin{array}{c}\dot{x}=-ax+byz\hfill \\ \dot{y}=-c{y}^3+dxz\hfill \\ \dot{z}=ez-fxy\hfill \end{array}\right.,a,b,c,d,e,f\in {\mathbb{R}}_{+}^{*}.$$

(44)

with the Hamilton–Poisson part, namely,

$$\left\{\begin{array}{c}\dot{x}=byz\hfill \\ \dot{y}=dxz\hfill \\ \dot{z}=-fxy\hfill \end{array}\right.,b,d,f\in {\mathbb{R}}_{+}^{*},$$

(45)

with $H=\frac{1}{2}(\frac{1}{b}{x}^2-\frac{1}{d}{y}^2)$ and $C=\frac{1}{2}(\frac{1}{d}{y}^2+\frac{1}{f}{z}^2)$.

The closed-form solutions of the system (45) are

$$\left\{\begin{array}{c}x\left(t\right)=R\sqrt{b}\frac{1+u^2\left(t\right)}{1-u^2\left(t\right)}\hfill \\ y\left(t\right)=R\sqrt{d}\frac{2u\left(t\right)}{1-u^2\left(t\right)}\hfill \\ z\left(t\right)=\frac{2}{\sqrt{bd}}\frac{\dot{u}\left(t\right)}{1-u^2\left(t\right)}\hfill \end{array}\right.,$$

(46)

where $R=\sqrt{\frac{1}{b}{x}_0^2-\frac{1}{d}{y}_0^2}$, for $\frac{1}{b}{x}_0^2-\frac{1}{d}{y}_0^2>0$, $u\left(t\right)$ is an unknown smooth function, a solution of the nonlinear problem:

$$\begin{array}{c}\ddot{u}\left(t\right)[1-u^2\left(t\right)]+2u\left(t\right){\left[\dot{u}\left(t\right)\right]}^2+bdf{R}^2·u\left(t\right)[1+u^2\left(t\right)]=0\hfill \\ u\left(0\right)=\sqrt{\frac{b}{d}}\frac{|y_0|}{|x_0+R\sqrt{b}|},\dot{u}\left(0\right)=\frac{1}{2}{z}_0\sqrt{bd}(1-u^2\left(0\right)).\hfill \end{array}$$

(47)

If $\frac{1}{b}{x}_0^2-\frac{1}{m}{y}_0^2<0$, then the closed-form solutions are

$$\left\{\begin{array}{c}x\left(t\right)=R\sqrt{b}\frac{2u\left(t\right)}{1-u^2\left(t\right)}\hfill \\ y\left(t\right)=R\sqrt{d}\frac{1+u^2\left(t\right)}{1-u^2\left(t\right)}\hfill \\ z\left(t\right)=\frac{2}{\sqrt{bd}}\frac{\dot{u}\left(t\right)}{1-u^2\left(t\right)}\hfill \end{array}\right.,$$

(48)

where $R=\sqrt{\frac{1}{d}{y}_0^2-\frac{1}{b}{x}_0^2}$, $u\left(t\right)$ is an unknown smooth function, a solution of the nonlinear problem (47), with the initial conditions

$$u\left(0\right)=\sqrt{\frac{d}{b}}\frac{|x_0|}{|y_0+R\sqrt{d}|},\dot{u}\left(0\right)=\frac{1}{2}{z}_0\sqrt{bd}(1-u^2\left(0\right)).$$

(49)

(viii)

A three-dimensional chaotic system with a large scope was illustrated in [54]:

$$\left\{\begin{array}{c}\dot{x}=ax+dxz+g{y}^2\hfill \\ \dot{y}=by+exz+hz\hfill \\ \dot{z}=cz+fxy\hfill \end{array}\right.,a,b,c,d,e,f,g,h\in {\mathbb{R}}^{*},$$

(50)

with the Hamilton–Poisson part

$$\left\{\begin{array}{c}\dot{x}=dyz\hfill \\ \dot{y}=exz\hfill \\ \dot{z}=fxy\hfill \end{array}\right.,d,e,f\in {\mathbb{R}}^{*},$$

(51)

with $H=\frac{1}{2}(\frac{1}{d}{x}^2-\frac{1}{e}{y}^2)$ and $C=\frac{1}{2}(\frac{1}{e}{y}^2-\frac{1}{f}{z}^2)$.

The closed-form solutions of the system (51) could be

$$\left\{\begin{array}{c}x\left(t\right)=R\sqrt{d}\frac{1+u^2\left(t\right)}{1-u^2\left(t\right)}\hfill \\ y\left(t\right)=R\sqrt{e}\frac{2u\left(t\right)}{1-u^2\left(t\right)}\hfill \\ z\left(t\right)=\frac{2}{\sqrt{de}}\frac{\dot{u}\left(t\right)}{1-u^2\left(t\right)}\hfill \end{array}\right.,$$

(52)

where $R=\sqrt{\frac{1}{d}{x}_0^2-\frac{1}{e}{y}_0^2}$, for $\frac{1}{d}{x}_0^2-\frac{1}{e}{y}_0^2>0$, $u\left(t\right)$ is an unknown smooth function, a solution of the nonlinear problem:

$$\begin{array}{c}\ddot{u}\left(t\right)[1-u^2\left(t\right)]+2u\left(t\right){\left[\dot{u}\left(t\right)\right]}^2-def{R}^2·u\left(t\right)[1+u^2\left(t\right)]=0\hfill \\ u\left(0\right)=\sqrt{\frac{d}{e}}\frac{|y_0|}{|x_0+R\sqrt{d}|},\dot{u}\left(0\right)=\frac{1}{2}{z}_0\sqrt{de}(1-u^2\left(0\right)).\hfill \end{array}$$

(53)

If $\frac{1}{d}{x}_0^2-\frac{1}{e}{y}_0^2<0$, then the closed-form solutions are

$$\left\{\begin{array}{c}x\left(t\right)=R\sqrt{d}\frac{1+u^2\left(t\right)}{1-u^2\left(t\right)}\hfill \\ y\left(t\right)=R\sqrt{e}\frac{2u\left(t\right)}{1-u^2\left(t\right)}\hfill \\ z\left(t\right)=\frac{2}{\sqrt{de}}\frac{\dot{u}\left(t\right)}{1-u^2\left(t\right)}\hfill \end{array}\right.,$$

(54)

where $R=\sqrt{\frac{1}{b}{x}_0^2-\frac{1}{m}{y}_0^2}$, $u\left(t\right)$ is an unknown smooth function, a solution of the nonlinear problem (53), with the initial conditions

$$u\left(0\right)=\sqrt{\frac{e}{d}}\frac{|x_0|}{|y_0+R\sqrt{e}|},\dot{u}\left(0\right)=\frac{1}{2}{z}_0\sqrt{de}(1-u^2\left(0\right)).$$

(55)

In the present paper, a modified version of the OPIM technique, namely, the modified optimal parametric iteration method (mOPIM), is proposed to obtain the approximate closed-form solutions of the system (2) subject to the initial conditions given by Equation (4).

3. The Basic Idea of the mOPIM Technique

Let the second-order nonlinear differential equation be

$$\begin{array}{c}{\displaystyle \mathcal{L}\left[u\left(t\right)\right]+\mathcal{N}[t,u\left(t\right),\dot{u}\left(t\right),\ddot{u}\left(t\right)]-g\left(t\right)=0,t\in \mathcal{I}\subset \mathbb{R},}\hfill \end{array}$$

(56)

subject to the initial conditions

$$\begin{array}{c}{\displaystyle \mathcal{B}[u\left(t\right),\dot{u}\left(t\right)]=0,}\hfill \end{array}$$

(57)

where $\mathcal{L}$ is a linear operator, $\mathcal{N}$ a nonlinear operator, $\mathcal{B}$ a boundary operator, g a known function, u an unknown smooth function depending on the independent variable t, and $\dot{u}\left(t\right)=\frac{du}{dt}$.

Marinca et al. [36] proposed the following iterative scheme, namely, optimal parametric iteration method (OPIM), defined by

$$\begin{array}{c}{\displaystyle \mathcal{L}\left[u_{n+1}\left(t\right)\right]+\mathcal{N}[t,u_n,{\dot{u}}_n,{\ddot{u}}_n]+{\alpha }_n(t,C_i){\mathcal{N}}_u[t,u_n,{\dot{u}}_n,{\ddot{u}}_n]+}\hfill \\ +{\beta }_n(t,C_j){\mathcal{N}}_{\dot{u}}[t,u_n,{\dot{u}}_n,{\ddot{u}}_n]+{\gamma }_n(t,C_k){\mathcal{N}}_{\ddot{u}}[t,u_n,{\dot{u}}_n,{\ddot{u}}_n]+\cdots -g\left(t\right)=0\hfill \\ {\displaystyle \mathcal{B}[u_{n+1}\left(t\right),{\dot{u}}_{n+1}\left(t\right)]=0}\hfill \end{array},n\ge 0,$$

(58)

where ${\alpha }_n(t,C_i)$, ${\beta }_n(t,C_j)$, and ${\gamma }_n(t,C_k)$ are auxiliary continuous functions; ${\mathcal{N}}_F=\frac{\partial \mathcal{N}}{\partial F}$ (obtained from Taylor series expansion of the nonlinear operator $\mathcal{N}[t,u\left(t\right),\dot{u}\left(t\right),\ddot{u}\left(t\right)]$); $u_{n+1}\left(t\right)$ is the (n + 1)-th-order approximate solution of Equations (56) and (57), denoted by $\overline{u}\left(t\right)$; and $u_0\left(t\right)$ is the initial approximation, a solution of the linear differential problem:

$$\begin{array}{c}{\displaystyle \mathcal{L}\left[u_0\left(t\right)\right]-g\left(t\right)=0}\hfill \\ {\displaystyle \mathcal{B}[u_0\left(t\right),{\dot{u}}_0\left(t\right)]=0.}\hfill \end{array}$$

(59)

The real constants $C_i$, $C_j$, are $C_k$ are unknown convergence-control parameters and can be optimally computed.

Remark 2.

(1) 

In the case of nonlinear oscillators, the integration of Equation (58) produces secular terms of the form $t\cos \left({\omega }_{0}t\right)$, $t\sin \left({\omega }_{0}t\right)$, $t^2\cos \left({\omega }_{0}t\right)$, $t^2\sin \left({\omega }_{0}t\right)$, $t\cos \left(2{\omega }_{0}t\right)$, $t\sin \left(2{\omega }_{0}t\right)$, and so on. The presence of ${\lambda }_0(t,C_s)$ has the advantage of avoiding the secular terms that appear through integration with the OPIM method, and that makes the oscillation amplitude tend toward infinity (physically, the resonance phenomenon occurs).

(2) 

The OPIM method was successfully applied in the case of ODEs with boundary conditions (see Ref) [55], such as

(a) 

Thin film flow of a fourth-grade fluid down a vertical cylinder

$$\begin{array}{c}{\displaystyle \eta f^{\prime \prime }\left(\eta \right)+f^{\prime }\left(\eta \right)+k\eta +2b\left[{\left(f^{\prime }\left(\eta \right)\right)}^2+3\eta \left(f^{\prime }{\left(\eta \right)}^2{f}^{\prime \prime }\left(\eta \right)\right)\right]=0}\hfill \\ {\displaystyle f\left(1\right)=0,f^{\prime }\left(d\right)=0,}\hfill \end{array}$$

(60)

where  $f^{\prime }\left(\eta \right)=\frac{df}{d\eta }$. The linear operator is chosen as $\mathcal{L}\left[f\left(\eta \right)\right]=\eta f^{\prime \prime }\left(\eta \right)+f^{\prime }\left(\eta \right)+k\eta$.

(b) 

Thermal radiation on MHD flow over a stretching porous sheet

$$\begin{array}{c}{\displaystyle f^{\prime \prime \prime }\left(\eta \right)+f\left(\eta \right)f^{\prime \prime }\left(\eta \right)-f^{\prime }{\left(\eta \right)}^2-M{f}^{\prime }\left(\eta \right)=0}\hfill \\ {\displaystyle {\theta }^{\prime \prime }\left(\eta \right)+\left(a-b{e}^{-\gamma \eta }\right){\theta }^{\prime }\left(\eta \right)-c{e}^{-\gamma \eta }\theta \left(\eta \right)=0}\hfill \\ {\displaystyle f\left(1\right)=\lambda ,f^{\prime }\left(0\right)=1,\theta \left(0\right)=1}\hfill \\ f^{\prime }\left(\eta \right)\to 0,\theta \left(\eta \right)\to 0as\eta \to \infty .\hfill \end{array}$$

(61)

The initial guess is chosen as  ${\theta }_0\left(\eta \right)=0$  and  $f\left(\eta \right)=\lambda +\frac{1}{\lambda }\left(1-e^{-\gamma \eta }\right)$, with $\gamma =\frac{1}{2}\left(\lambda +\sqrt{{\lambda }^2+4M+4}\right)$.

(c) 

The oscillator with cubic and harmonic restoring force

$$\begin{array}{c}{\displaystyle u^{\prime \prime }\left(t\right)+u\left(t\right)+a{u}^3\left(t\right)+b\sin u\left(t\right)=0}\hfill \\ {\displaystyle u\left(0\right)=A,u^{\prime }\left(0\right)=0.}\hfill \end{array}$$

(62)

The linear operator is chosen as $\mathcal{L}\left[u\left(t\right)\right]=u^{\prime \prime }\left(t\right)+u\left(t\right)$.

(d) 

The Thomas–Fermi equation

$$\begin{array}{c}{\displaystyle y^{\prime \prime }\left(x\right)=\sqrt{\frac{y^3\left(x\right)}{x}}\iff x{\left[y^{\prime \prime }\left(x\right)\right]}^2-y^3\left(x\right)=0}\hfill \\ {\displaystyle y\left(0\right)=1,y\left(x\right)\to 0\mathrm{as}x\to \infty .}\hfill \end{array}$$

(63)

The linear operator is chosen as  $\mathcal{L}\left[y\left(x\right)\right]=y^{\prime \prime }\left(x\right)-{\lambda }^{2}y\left(x\right)$, and the nonlinear operator yields  $\mathcal{N}\left[y\left(x\right)\right]=x{\left[y^{\prime \prime }\left(x\right)\right]}^2-y^3\left(x\right)+y^{\prime \prime }\left(x\right)-{\lambda }^{2}y\left(x\right)$.

(e) 

Lotka–Volterra model with three species

$$\begin{array}{c}{\displaystyle x^{\prime }\left(t\right)=x(1-x-\alpha y-\beta z)}\hfill \\ {\displaystyle y^{\prime }\left(t\right)=y(1-\beta x-y-\alpha z)}\hfill \\ {\displaystyle z^{\prime }\left(t\right)=z(1-\alpha x-\beta y-z)}\hfill \\ {\displaystyle x\left(0\right)=a,y\left(0\right)=b,z\left(0\right)=c.}\hfill \end{array}$$

(64)

The initial approximations are chosen as  $x_0\left(t\right)=a{e}^{-t}$, $y_0\left(t\right)=b{e}^{-t}$, and  $z_0\left(t\right)=c{e}^{-t}$  or  $x_0\left(t\right)=a{e}^{-2t}$, $y_0\left(t\right)=b$, and  $z_0\left(t\right)=c{e}^{-t}$, and so on.

Next, we propose a modified version of the OPIM procedure, namely, the modified optimal parametric iteration method (mOPIM), in the following form:

$$\begin{array}{c}{\displaystyle \mathcal{L}\left[u_{n+1}\left(t\right)\right]+{\lambda }_0(t,C_s)\mathcal{N}[t,u_n,{\dot{u}}_n,{\ddot{u}}_n]+{\alpha }_n(t,C_i){\mathcal{N}}_u[t,u_n,{\dot{u}}_n,{\ddot{u}}_n]+}\hfill \\ +{\beta }_n(t,C_j){\mathcal{N}}_{\dot{u}}[t,u_n,{\dot{u}}_n,{\ddot{u}}_n]+{\gamma }_n(t,C_k){\mathcal{N}}_{\ddot{u}}[t,u_n,{\dot{u}}_n,{\ddot{u}}_n]+\cdots -g\left(t\right)=0\hfill \\ {\displaystyle \mathcal{B}[u_{n+1}\left(t\right),{\dot{u}}_{n+1}\left(t\right)]=0}\hfill \end{array},n\ge 0,$$

(65)

where the new auxiliary continuous function ${\lambda }_0(t,C_s)$ is a nonzero function and ${\alpha }_n(t,C_i)$, ${\beta }_n(t,C_j)$, and ${\gamma }_n(t,C_k)$ have the same signification. The unknown real parameters $C_i$, $C_j$, $C_k$, and $C_s$ are optimally computed at least.

The $(n+1)$-order approximate solution of Equation (65) is well determined if the convergence-control parameters are known.

If $u_0\left(t\right)$ is the initial approximation of Equation (59), the nonlinear operators $\mathcal{N}[t,u_0,{\dot{u}}_0,{\ddot{u}}_0]$, ${\mathcal{N}}_u[t,u_0,{\dot{u}}_0,{\ddot{u}}_0]$, ${\mathcal{N}}_{\dot{u}}[t,u_0,{\dot{u}}_0,{\ddot{u}}_0]$, and ${\mathcal{N}}_{\ddot{u}}[t,u_0,{\dot{u}}_0,{\ddot{u}}_0]$ that appear in Equation (65) have the form

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

(66)

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

Using the linearly independent functions $h_1,h_2,\cdots ,h_m$, we introduce some types of approximate solutions of Equation (56).

Definition 1.

A sequence of functions ${\left\{s_m\left(t\right)\right\}}_{m\ge 1}$ of the form

$${\displaystyle s_m\left(t\right)=\sum _{i=1}^m{\alpha }_m^i·h_i\left(t\right),m\ge 1,{\alpha }_m^i\in \mathbb{R},}$$

(67)

is called an mOPIM sequence of Equation (56).

Functions of the mOPIM sequences are called mOPIM functions of Equation (56).

The mOPIM sequences ${\left\{s_m\left(t\right)\right\}}_{m\ge 1}$ with the property

$${\displaystyle \underset{m\to \infty }{lim}\mathcal{R}(t,s_m\left(t\right))=0}$$

are called convergent to the solution of Equation (56), where $\mathcal{R}(t,u(t\left)\right)=\mathcal{L}\left[u\right(t\left)\right]+$$\mathcal{N}[t,u\left(t\right),\dot{u}\left(t\right),\ddot{u}\left(t\right)]-g\left(t\right)$.

Definition 2.

The mOPIM functions $\tilde{F}$ satisfying the conditions

$$\left|\mathcal{R}(t,\tilde{F}\left(t\right))\right|<\epsilon ,\mathcal{B}\left(\tilde{F}(t,C_i),\frac{d\tilde{F}(t,C_i)}{dt}\right)=0$$

(68)

are called ε-approximate mOPIM solutions of Equation (56).

Definition 3.

The mOPIM functions $\tilde{F}$ satisfying the conditions

$${\displaystyle \underset{0}{\overset{\infty }{\int }}{\mathcal{R}}^2(t,\tilde{F}\left(t\right))dt\le \epsilon ,\mathcal{B}\left(\tilde{F}(t,C_i),\frac{d\tilde{F}(t,C_i)}{dt}\right)=0}$$

(69)

are called weak ε-approximate mOPIM solutions of Equation (56) on the real interval $(0,\infty )$.

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

Theorem 1.

Equation (56) admits a sequence of weak ε-approximate mOPIM solutions.

Proof. 

It is similar to the theorem from [56]. □

For $u_{n+1}$, an $(n+1)$-order approximate solution of Equations (56) and (57), the validation of this procedure is highlighted by computing the residual function given by

$$\begin{array}{c}{\displaystyle \mathcal{R}\left(t\right)=\mathcal{L}\left[u_{n+1}\left(t\right)\right]+\mathcal{N}[t,u_{n+1}\left(t\right),{\dot{u}}_{n+1}\left(t\right),{\ddot{u}}_{n+1}\left(t\right)]-g\left(r\right),t\in \mathcal{I}\subset \mathbb{R},}\hfill \end{array}$$

(70)

such that ${\displaystyle \mathcal{R}\left(t\right)<<1}$, for all $t\in \mathcal{I}$.

4. Approximate Analytic Solutions via mOPIM

This section emphasizes the applicability of the mOPIM procedure for the nonlinear differential problems given by Equations (8) and (9) using only one iteration. This problem could be written in the form of Equation (56), taking the following operators ($g\left(t\right)=0$):

$$\begin{array}{c}{\displaystyle \mathcal{L}\left[u\left(t\right)\right]=\ddot{u}\left(t\right)+{\omega }_0^{2}u\left(t\right)}\hfill \\ {\displaystyle \mathcal{N}[t,u\left(t\right),\dot{u}\left(t\right),\ddot{u}\left(t\right)]=\ddot{u}\left(t\right)u^2\left(t\right)-2u\left(t\right){\left[\dot{u}\left(t\right)\right]}^2+aR\sqrt{2a}·u\left(t\right)[1+u^2\left(t\right)]-{\omega }_0^{2}u\left(t\right),t>0.}\hfill \end{array}$$

(71)

Taking into consideration the linear operator given by Equation (71), the initial approximation $u_0\left(t\right)$, the solution of Equation (59) is

$$\begin{array}{c}{\displaystyle u_0\left(t\right)=A\cos \left({\omega }_{0}t\right)+B\sin \left({\omega }_{0}t\right),}\hfill \end{array}$$

(72)

with $A=u\left(0\right)$, $B=\frac{\dot{u}\left(0\right)}{{\omega }_0}$.

Using Equation (71), a simple computation yields the following expressions:

$$\begin{array}{c}{\displaystyle {\mathcal{N}}_u[t,u,\dot{u},\ddot{u}]=2u\ddot{u}-2{\left(\dot{u}\right)}^2+3aR\sqrt{2a}{u}^2+aR\sqrt{2a}-{\omega }_0^2,}\hfill \\ {\mathcal{N}}_{\dot{u}}[t,u,\dot{u},\ddot{u}]=-4u\dot{u},{\mathcal{N}}_{\ddot{u}}[t,u,\dot{u},\ddot{u}]=u^2.\hfill \end{array}$$

(73)

Returning to Equation (65), there are a lot of possibilities to choose the following auxiliary functions:

$$\begin{array}{c}{\displaystyle {\lambda }_0(t,C_s)=D_1\cos \left({\omega }_{0}t\right),{\alpha }_n(t,C_i)=D_2,{\beta }_n(t,C_j)=D_3,}\hfill \\ {\displaystyle {\gamma }_n(t,C_k)=\sum _{i=1}^{N_{max}}{B}_i\cos \left(2i{\omega }_{0}t\right)+C_i\sin \left(2i{\omega }_{0}t\right),}\hfill \end{array}$$

(74)

or $\begin{array}{c}{\displaystyle {\lambda }_0(t,C_s)=D_1\cos \left({\omega }_{0}t\right)}\hfill \\ {\alpha }_n(t,C_i)=D_2\cos \left(2{\omega }_{0}t\right)\hfill \\ {\beta }_n(t,C_j)=D_3\sin \left(2{\omega }_{0}t\right)\hfill \\ {\gamma }_n(t,C_k)=E_1\cos \left(4{\omega }_{0}t\right)+F_1\sin \left(4{\omega }_{0}t\right),\hfill \end{array}$ and so on.

Taking into account Equation (74), for $N_{max}=1$, a simple computation yields

$$\begin{array}{c}{\displaystyle {\lambda }_0(t,C_s)\mathcal{N}[t,u_0,{\dot{u}}_0,{\ddot{u}}_0]=T_0+P_1\cos \left(2{\omega }_{0}t\right)+Q_1\sin \left(2{\omega }_{0}t\right)+P_2\cos \left(4{\omega }_{0}t\right)+Q_2\cos \left(4{\omega }_{0}t\right)}\hfill \\ {\alpha }_n(t,C_i){\mathcal{N}}_u[t,u_0,{\dot{u}}_0,{\ddot{u}}_0]=T_1+M_1\cos \left(2{\omega }_{0}t\right)+N_1\sin \left(2{\omega }_{0}t\right)\hfill \\ {\beta }_n(t,C_i){\mathcal{N}}_{\dot{u}}[t,u_0,{\dot{u}}_0,{\ddot{u}}_0]=M_2\cos \left(2{\omega }_{0}t\right)+N_2\sin \left(2{\omega }_{0}t\right)\hfill \\ {\gamma }_n(t,C_i){\mathcal{N}}_{\ddot{u}}[t,u_0,{\dot{u}}_0,{\ddot{u}}_0]=T_3+G_3\cos \left(2{\omega }_{0}t\right)+H_3\sin \left(2{\omega }_{0}t\right)+G_4\cos \left(4{\omega }_{0}t\right)+H_4\sin \left(4{\omega }_{0}t\right)\hfill \end{array}$$

(75)

where

$$\begin{array}{c}{T}_0=\frac{a^{3/2}A{D}_{1}R}{\sqrt{2}}-\frac{3a^{3/2}{A}^3{D}_{1}R}{4\sqrt{2}}-\frac{3a^{3/2}A{B}^2{D}_{1}R}{4\sqrt{2}}-\frac{1}{2}A{D}_1{\omega }_0^2-\frac{5}{8}{A}^3{D}_1{\omega }_0^2-\frac{5}{8}A{B}^2{D}_1{\omega }_0^2\hfill \\ P_1=\frac{a^{3/2}A{D}_{1}R}{\sqrt{2}}-\frac{a^{3/2}{A}^3{D}_{1}R}{\sqrt{2}}-\frac{1}{2}A{D}_1{\omega }_0^2-\frac{1}{2}{A}^3{D}_1{\omega }_0^2-A{B}^2{D}_1{\omega }_0^2\hfill \\ Q_1=\frac{a^{3/2}B{D}_{1}R}{\sqrt{2}}-\frac{3a^{3/2}{A}^{2}B{D}_{1}R}{2\sqrt{2}}-\frac{a^{3/2}{B}^3{D}_{1}R}{2\sqrt{2}}-\frac{1}{2}B{D}_1{\omega }_0^2-\frac{1}{4}{A}^{2}B{D}_1{\omega }_0^2-\frac{3}{4}{B}^3{D}_1{\omega }_0^2\hfill \\ P_2=-\frac{a^{3/2}{A}^3{D}_{1}R}{4\sqrt{2}}+\frac{3a^{3/2}A{B}^2{D}_{1}R}{4\sqrt{2}}+\frac{1}{8}{A}^3{D}_1{\omega }_0^2-\frac{3}{8}A{B}^2{D}_1{\omega }_0^2\hfill \\ Q_2=-\frac{3a^{3/2}{A}^{2}B{D}_{1}R}{4\sqrt{2}}+\frac{a^{3/2}{B}^3{D}_{1}R}{4\sqrt{2}}+\frac{3}{8}{A}^{2}B{D}_1{\omega }_0^2-\frac{1}{8}{B}^3{D}_1{\omega }_0^2\hfill \\ T_1=\frac{1}{2}\left[2\sqrt{2}{a}^{3/2}{D}_{2}R-3\sqrt{2}{a}^{3/2}{A}^2{D}_{2}R-3\sqrt{2}{a}^{3/2}{B}^2{D}_{2}R-2D_2{\omega }_0^2-4A^2{D}_2{\omega }_0^2-4B^2{D}_2{\omega }_0^2\right]\hfill \\ M_1=\frac{3\sqrt{2}{a}^{3/2}{D}_{2}R}{2}\left(-A^2+B^2\right)\hfill \\ N_1=-3\sqrt{2}{a}^{3/2}AB{D}_{2}R\hfill \\ M_2=-4AB{D}_3{\omega }_0\hfill \\ N_2=2D_3{\omega }_0(A^2-B^2)\hfill \\ T_3=\frac{1}{4}\left(A^2{E}_1-B^2{E}_1+2AB{F}_1\right)\hfill \\ G_3=\frac{E_1}{2}\left(A^2+B^2\right)\hfill \\ H_3=\frac{F_1}{2}\left(A^2+B^2\right)\hfill \\ G_4=\frac{1}{4}\left(A^2{E}_1-B^2{E}_1-2AB{F}_1\right)\hfill \\ H_4=\frac{1}{4}\left(2AB{E}_1+A^2{F}_1-B^2{F}_1\right).\hfill \end{array}$$

By the integration of Equation (65) and using the expressions given by Equations (71)–(75), the first-order approximate solution $u_1$ could be obtained:

$$\begin{array}{c}{\displaystyle u_1\left(t\right)={\tilde{B}}_0+A\cos \left({\omega }_{0}t\right)+B\sin \left({\omega }_{0}t\right)+{\tilde{B}}_1\cos \left(2{\omega }_{0}t\right)+{\tilde{C}}_1\sin \left(2{\omega }_{0}t\right)+{\tilde{B}}_2\cos \left(4{\omega }_{0}t\right)+{\tilde{C}}_2\sin \left(4{\omega }_{0}t\right),}\hfill \end{array}$$

(76)

with the unknown real parameters ${\tilde{B}}_0$, ${\tilde{B}}_1$, ${\tilde{B}}_2$$C_0$, ${\tilde{B}}_1$, and ${\tilde{B}}_2$, depending on the parameters $T_0$, $T_1$, $T_3$, $P_1$, $Q_1$, $P_2$, $Q_2$, $M_1$, $N_1$, $M_2$, $N_2$, $G_3$, $H_3$, $G_4$, and $H_4$, and can be optimally identified.

Analogously, for the value $N_{max}=2$, the expression ${\gamma }_n(t,C_i){\mathcal{N}}_{\ddot{u}}[t,u_0,{\dot{u}}_0,{\ddot{u}}_0]$ is a linear combination between the elementary functions 1, $\cos \left({\omega }_{0}t\right)$, $\sin \left({\omega }_{0}t\right)$, $\cos \left(2{\omega }_{0}t\right)$, $\sin \left(2{\omega }_{0}t\right)$, $\cos \left(4{\omega }_{0}t\right)$, $\sin \left(4{\omega }_{0}t\right)$, $\cos \left(6{\omega }_{0}t\right)$, and $\sin \left(6{\omega }_{0}t\right)$.

Then the first-order approximate solution $u_1$ obtained from Equation (65) is a linear combination between the elementary functions 1, $\cos \left({\omega }_{0}t\right)$, $\sin \left({\omega }_{0}t\right)$, $\cos \left(2{\omega }_{0}t\right)$, $\sin \left(2{\omega }_{0}t\right)$, $\cos \left(4{\omega }_{0}t\right)$, $\sin \left(4{\omega }_{0}t\right)$, $\cos \left(6{\omega }_{0}t\right)$, and $\sin \left(6{\omega }_{0}t\right)$.

For an arbitrary integer number $N_{max}$, inductively, the expression ${\gamma }_n(t,C_i){\mathcal{N}}_{\ddot{u}}[t,u_0,{\dot{u}}_0,{\ddot{u}}_0]$ is a linear combination between the elementary functions 1, $\cos \left(2(i+1){\omega }_{0}t\right)$ and $\sin \left(2(i+1){\omega }_{0}t\right)$, for $i=\overline{0,N_{max}}$. Therefore, the the first-order approximate solution $u_1$ will be of the form

$$\begin{array}{c}{\displaystyle u_1\left(t\right)=T_0+P_0\cos \left({\omega }_{0}t\right)+Q_0\sin \left({\omega }_{0}t\right)+\sum _{i=1}^{N_{max}}{B}_i\cos \left(2i{\omega }_{0}t\right)+C_i\sin \left(2i{\omega }_{0}t\right),}\hfill \end{array}$$

(77)

where the unknown convergence-control parameters $T_0$, $P_0$, $Q_0$, $B_i$, and $C_i$ for $i=\overline{1,N_{max}}$ could be optimally computed.

Using the same procedure, the approximate closed-form solutions of the nonlinear problems presented in Section 2.2 could be obtained by means of the mOPIM method.

5. Numerical Results and Discussions

This section illustrates the validation of the applied method by a comparison between the obtained analytic results and the corresponding numerical ones. Additionally, the corresponding absolute errors are graphically and tabularly presented.

The unknown convergence-control parameters ${\omega }_0$, $T_0$, $P_0$, $Q_0$, $B_i$, and $C_i$ for $i=\overline{0,N_{max}}$ from Equation (77) are optimally computed for some values of the index number $N_{max}$ and are exposed in Appendix A.

From

Table 1. Values of the absolute errors: ${ϵ}_u=|u_{numerical}-{\overline{u}}_{mOPIM}|$ for $a=0.25$, the initial conditions $x_0=0.25$, $y_0=0.5$, $z_0=1.5$, and different values of the index $N_{max}\in \{5,10,15\}$; ${\overline{u}}_{mOPIM}$ analytic approximate solution of Equations (8) and (9) obtained from Equations (77) and (A1)–(A5).

t	${\mathit{N}}_{\mathit{max}}=5$	${\mathit{N}}_{\mathit{max}}=10$	${\mathit{N}}_{\mathit{max}}=15$
0	1.665334 $\times {10}^{-16}$	7.986389 $\times {10}^{-13}$	5.800859 $\times {10}^{-12}$
3	2.792333 $\times {10}^{-5}$	1.071748 $\times {10}^{-5}$	3.355308 $\times {10}^{-8}$
6	1.996162 $\times {10}^{-4}$	1.299889 $\times {10}^{-5}$	3.921769 $\times {10}^{-8}$
9	5.557844 $\times {10}^{-4}$	4.809530 $\times {10}^{-6}$	8.765128 $\times {10}^{-8}$
12	1.658452 $\times {10}^{-3}$	1.859888 $\times {10}^{-5}$	1.230130 $\times {10}^{-7}$
15	2.828080 $\times {10}^{-3}$	1.196188 $\times {10}^{-5}$	1.274314 $\times {10}^{-7}$
18	3.438877 $\times {10}^{-3}$	1.547642 $\times {10}^{-5}$	9.088403 $\times {10}^{-8}$
21	2.390172 $\times {10}^{-3}$	1.324309 $\times {10}^{-5}$	7.922086 $\times {10}^{-8}$
24	2.233670 $\times {10}^{-3}$	1.330894 $\times {10}^{-6}$	1.494354 $\times {10}^{-7}$
27	1.605828 $\times {10}^{-3}$	5.178678 $\times {10}^{-6}$	1.768828 $\times {10}^{-7}$
30	7.452653 $\times {10}^{-4}$	8.405924 $\times {10}^{-6}$	4.481940 $\times {10}^{-8}$

and

Table 2. Values of the absolute errors: ${ϵ}_u=|u_{numerical}-{\overline{u}}_{mOPIM}|$ for $a=0.25$, the initial conditions $x_0=0.25$, $y_0=0.5$, $z_0=1.5$, and different values of the index $N_{max}\in \{20,25\}$; ${\overline{u}}_{mOPIM}$ analytic approximate solution of Equations (8) and (9) obtained from Equations (77) and (A1)–(A5).

t	${\mathit{N}}_{\mathit{max}}=20$	${\mathit{N}}_{\mathit{max}}=25$
0	3.382294 $\times {10}^{-13}$	6.677991 $\times {10}^{-14}$
3	7.806301 $\times {10}^{-11}$	3.911580 $\times {10}^{-10}$
6	3.264915 $\times {10}^{-9}$	1.459806 $\times {10}^{-10}$
9	1.069987 $\times {10}^{-8}$	3.221471 $\times {10}^{-10}$
12	6.903293 $\times {10}^{-9}$	3.189073 $\times {10}^{-10}$
15	1.322873 $\times {10}^{-8}$	1.252072 $\times {10}^{-10}$
18	1.967300 $\times {10}^{-9}$	5.024303 $\times {10}^{-11}$
21	1.064146 $\times {10}^{-8}$	8.643594 $\times {10}^{-11}$
24	7.988264 $\times {10}^{-9}$	5.610625 $\times {10}^{-11}$
27	5.097681 $\times {10}^{-9}$	7.915236 $\times {10}^{-10}$
30	8.908084 $\times {10}^{-9}$	3.644332 $\times {10}^{-10}$

, it is easy to see that a good agreement between the obtained analytic results and the corresponding numerical ones is revealed for $N_{max}=25$. For this value of $N_{max}$ in Figure 1, the variation of absolute error is depicted.

A comparison between the approximate analytic solution ${\overline{u}}_{mOPIM}$ of Equations (8) and (9) given by Equation (77) and the corresponding numerical solution for $a>0$ is highlighted in

Table 3. The approximate analytic solution ${\overline{u}}_{mOPIM}$ (77) of Equations (8) and (9) and the corresponding numerical solution for $a=0.25$, the initial conditions $x_0=0.25$, $y_0=0.5$, $z_0=1.5$, and $N_{max}=25$ (absolute errors: ${ϵ}_u=|u_{numerical}-{\overline{u}}_{mOPIM}|$).

t	${\mathit{u}}_{\mathit{numerical}}$	${\overline{\mathit{u}}}_{\mathit{mOPIM}}$	${\mathit{ϵ}}_{\mathit{u}}$
0	0.3027756377	0.3027756377	6.677991 $\times {10}^{-14}$
3	0.0032845067	0.0032845071	3.911580 $\times {10}^{-10}$
6	−0.2988484691	−0.2988484693	1.459806 $\times {10}^{-10}$
9	−0.3403690488	−0.3403690485	3.221471 $\times {10}^{-10}$
12	−0.0787141615	−0.0787141618	3.189073 $\times {10}^{-10}$
15	0.2466436711	0.2466436712	1.252072 $\times {10}^{-10}$
18	0.3620311865	0.3620311864	5.024303 $\times {10}^{-11}$
21	0.1511730954	0.1511730953	8.643594 $\times {10}^{-11}$
24	−0.1840136731	−0.1840136731	5.610625 $\times {10}^{-11}$
27	−0.3662950369	−0.3662950376	7.915236 $\times {10}^{-10}$
30	−0.2177176347	−0.2177176343	3.644332 $\times {10}^{-10}$

and

Table 4. The approximate analytic solution ${\overline{x}}_{mOPIM}$ (7) and the corresponding numerical solution for $a=0.25$, the initial conditions $x_0=0.25$, $y_0=0.5$, $z_0=1.5$, and $N_{max}=25$ (absolute errors: ${ϵ}_x=|x_{numerical}-{\overline{x}}_{mOPIM}|$).

t	${\mathit{x}}_{\mathit{numerical}}$	${\overline{\mathit{x}}}_{\mathit{mOPIM}}$	${\mathit{ϵ}}_{\mathit{x}}$
0	0.25	0.2499999985	1.445188 $\times {10}^{-9}$
3	0.4624590237	0.4624590276	3.972243 $\times {10}^{-9}$
6	0.2570637655	0.2570636473	1.181607 $\times {10}^{-7}$
9	−0.1634564945	−0.1634563481	1.463877 $\times {10}^{-7}$
12	−0.4503182221	−0.4503184807	2.585838 $\times {10}^{-7}$
15	−0.3324278454	−0.3324277961	4.932482 $\times {10}^{-8}$
18	0.0705998931	0.0705998473	4.577343 $\times {10}^{-8}$
21	0.4166639663	0.4166640231	5.680127 $\times {10}^{-8}$
24	0.3935012294	0.3935012981	6.871170 $\times {10}^{-8}$
27	0.0249032354	0.0249032010	3.434804 $\times {10}^{-8}$
30	−0.3637331399	−0.3637329877	1.521911 $\times {10}^{-7}$

and qualitatively represented in Figure 2 and Figure 3. Similarly, for $a<0$, the comparative solutions are exposed in

Table 5. The approximate analytic solution ${\overline{u}}_{mOPIM}$ (77) of Equation (12) and the corresponding numerical solution for $a=-0.15$, the initial conditions $x_0=0.25$, $y_0=0.55$, $z_0=1.5$, and $N_{max}=25$ (absolute errors: ${ϵ}_u=|u_{numerical}-{\overline{u}}_{mOPIM}|$).

t	${\mathit{u}}_{\mathit{numerical}}$	${\overline{\mathit{u}}}_{\mathit{mOPIM}}$	${\mathit{ϵ}}_{\mathit{u}}$
0	0.7161175138	0.7161175138	1.417754 $\times {10}^{-13}$
6	0.7032944482	0.7032944694	2.122879 $\times {10}^{-8}$
12	0.3234806148	0.3234806047	1.015607 $\times {10}^{-8}$
18	−0.3427164082	−0.3427164089	7.483426 $\times {10}^{-10}$
24	−0.7086126654	−0.7086126721	6.768228 $\times {10}^{-9}$
30	−0.7113611793	−0.7113611530	2.630958 $\times {10}^{-8}$
36	−0.3530400106	−0.3530400526	4.201062 $\times {10}^{-8}$
42	0.3128079068	0.3128079500	4.321215 $\times {10}^{-8}$
48	0.7002348666	0.7002348370	2.953453 $\times {10}^{-8}$
54	0.7185639058	0.7185638813	2.452961 $\times {10}^{-8}$
60	0.3815739146	0.3815739314	1.683062 $\times {10}^{-8}$

and

Table 6. The approximate analytic solution ${\overline{y}}_{mOPIM}$ (10) and the corresponding numerical solution for $a=-0.15$, the initial conditions $x_0=0.25$, $y_0=0.55$, $z_0=1.5$, and $N_{max}=25$ (absolute errors: ${ϵ}_y=|y_{numerical}-{\overline{y}}_{mOPIM}|$).

t	${\mathit{y}}_{\mathit{numerical}}$	${\overline{\mathit{y}}}_{\mathit{mOPIM}}$	${\mathit{ϵ}}_{\mathit{y}}$
0	0.55	0.550000000000338	3.379518 $\times {10}^{-13}$
6	0.5206977593	0.5206979193	1.599459 $\times {10}^{-7}$
12	0.1351805604	0.1351806190	5.861116 $\times {10}^{-8}$
18	−0.1452987372	−0.1452987338	3.429793 $\times {10}^{-9}$
24	−0.5325474695	−0.5325478658	3.963163 $\times {10}^{-7}$
30	−0.5388373225	−0.5388373757	5.322350 $\times {10}^{-8}$
36	−0.1509038596	−0.1509037212	1.383424 $\times {10}^{-7}$
42	0.1297363561	0.1297365890	2.328451 $\times {10}^{-7}$
48	0.5140640684	0.5140645556	4.872576 $\times {10}^{-7}$
54	0.5558836780	0.5558836348	4.317333 $\times {10}^{-8}$
60	0.1671020584	0.1671016723	3.860081 $\times {10}^{-7}$

, respectively, in Figure 4, Figure 5 and Figure 6.

For the first dynamical system described in Section 2.2, the obtained solutions by the mOPIM technique and the corresponding numerical results are presented in detail by the comparison in

Table 7. The approximate analytic solution ${\overline{u}}_{mOPIM}$ (77) of Equation (16) and the corresponding numerical solution for $b=0.250$, $d=0.45$, and $m=0.75$, the initial conditions $x_0=1.25$, $y_0=0.25$, $z_0=0.35$, and $N_{max}=25$ (absolute errors: ${ϵ}_u=|u_{numerical}-{\overline{u}}_{mOPIM}|$).

t	${\mathit{u}}_{\mathit{numerical}}$	${\overline{\mathit{u}}}_{\mathit{mOPIM}}$	${\mathit{ϵ}}_{\mathit{u}}$
0	0.0813641358	0.0813641358	1.204730 $\times {10}^{-13}$
2	0.0833334819	0.0833334819	3.184762 $\times {10}^{-11}$
4	−0.0614111799	−0.0614111795	4.478518 $\times {10}^{-10}$
6	−0.0980368958	−0.0980368959	3.483953 $\times {10}^{-11}$
8	0.0379370221	0.0379370221	4.286203 $\times {10}^{-11}$
10	0.1071197387	0.1071197380	6.304264 $\times {10}^{-10}$
12	−0.0122874830	−0.0122874833	2.706000 $\times {10}^{-10}$
14	−0.1100615848	−0.1100615850	1.659267 $\times {10}^{-10}$
16	−0.0140666728	−0.0140666729	1.289021 $\times {10}^{-10}$
18	0.1066938318	0.1066938312	5.206623 $\times {10}^{-10}$
20	0.0396142006	0.0396141999	6.878308 $\times {10}^{-10}$

,

Table 8. The approximate analytic solution ${\overline{x}}_{mOPIM}$ (15) and the corresponding numerical solution for $b=0.250$, $d=0.45$, and $m=0.75$, the initial conditions $x_0=1.25$, $y_0=0.25$, $z_0=0.35$, and $N_{max}=25$ (absolute errors: ${ϵ}_x=|x_{numerical}-{\overline{x}}_{mOPIM}|$).

t	${\mathit{x}}_{\mathit{numerical}}$	${\overline{\mathit{x}}}_{\mathit{mOPIM}}$	${\mathit{ϵ}}_{\mathit{x}}$
0	1.25	1.2499999999	4.884981 $\times {10}^{-14}$
2	1.2508111827	1.2508111669	1.576702 $\times {10}^{-8}$
4	1.2428981332	1.2428980804	5.279530 $\times {10}^{-8}$
6	1.2575006684	1.2575006885	2.008940 $\times {10}^{-8}$
8	1.2371144259	1.2371143740	5.186460 $\times {10}^{-8}$
10	1.2621963761	1.2621964234	4.725770 $\times {10}^{-8}$
12	1.2339311236	1.2339310851	3.854395 $\times {10}^{-8}$
14	1.2638104780	1.2638105495	7.145415 $\times {10}^{-8}$
16	1.2340468263	1.2340468062	2.005757 $\times {10}^{-8}$
18	1.2619664732	1.2619665313	5.814743 $\times {10}^{-8}$
20	1.2374362328	1.2374362326	1.857536 $\times {10}^{-10}$

,

Table 9. The approximate analytic solution ${\overline{u}}_{mOPIM}$ (77) of Equations (16) and (27) and the corresponding numerical solution for the initial conditions $x_0=1.55$, $y_0=0.75$, $z_0=0.35$, and $N_{max}=25$ (absolute errors: ${ϵ}_u=|u_{numerical}-{\overline{u}}_{mOPIM}|$).

t	${\mathit{u}}_{\mathit{numerical}}$	${\overline{\mathit{u}}}_{\mathit{mOPIM}}$	${\mathit{ϵ}}_{\mathit{u}}$
0	0.2580453378	0.2580453378	5.551115 $\times {10}^{-17}$
1	0.1368008897	0.1368008898	9.801889 $\times {10}^{-11}$
2	−0.2307222259	−0.2307222257	1.978850 $\times {10}^{-10}$
3	−0.1830075275	−0.1830075275	2.248004 $\times {10}^{-11}$
4	0.1941280553	0.1941280555	1.241964 $\times {10}^{-10}$
5	0.2218381157	0.2218381158	1.300534 $\times {10}^{-10}$
6	−0.1497146227	−0.1497146226	1.003465 $\times {10}^{-10}$
7	−0.2517490655	−0.2517490655	3.825217 $\times {10}^{-11}$
8	0.0992567298	0.0992567296	2.150129 $\times {10}^{-10}$
9	0.2715597650	0.2715597651	5.533706 $\times {10}^{-11}$
10	−0.0447838420	−0.0447838418	1.849511 $\times {10}^{-10}$

,

Table 10. The approximate analytic solution ${\overline{z}}_{mOPIM}$ (26) and the corresponding numerical solution for the initial conditions $x_0=1.55$, $y_0=0.75$, $z_0=0.35$, and $N_{max}=25$ (absolute errors: ${ϵ}_z=|z_{numerical}-{\overline{z}}_{mOPIM}|$).

t	${\mathit{z}}_{\mathit{numerical}}$	${\overline{\mathit{z}}}_{\mathit{mOPIM}}$	${\mathit{ϵ}}_{\mathit{z}}$
0	0.35	0.3500000044	4.438894 $\times {10}^{-9}$
1	−0.7361776001	−0.7361776290	2.890399 $\times {10}^{-8}$
2	−0.4979055030	−0.4979056583	1.552408 $\times {10}^{-7}$
3	0.6489382094	0.6489379874	2.220426 $\times {10}^{-7}$
4	0.6208728951	0.6208728159	7.920362 $\times {10}^{-8}$
5	−0.5332290474	−0.5332288509	1.964807 $\times {10}^{-7}$
6	−0.7158051879	−0.7158052698	8.184599 $\times {10}^{-8}$
7	0.3915070650	0.3915068641	2.008447 $\times {10}^{-7}$
8	0.7816900986	0.7816900816	1.693372 $\times {10}^{-8}$
9	−0.2288523021	−0.2288520752	2.269252 $\times {10}^{-7}$
10	−0.8186446175	−0.8186446186	1.138482 $\times {10}^{-9}$

,

Table 11. The approximate analytic solution ${\overline{u}}_{mOPIM}$ (77) of Equations (16) and (33) and the corresponding numerical solution for the initial conditions $x_0=0.75$, $y_0=0.95$, $z_0=0.25$, and $N_{max}=25$ (absolute errors: ${ϵ}_u=|u_{numerical}-{\overline{u}}_{mOPIM}|$).

t	${\mathit{u}}_{\mathit{numerical}}$	${\overline{\mathit{u}}}_{\mathit{mOPIM}}$	${\mathit{ϵ}}_{\mathit{u}}$
0	0.1339394440	0.1339394440	3.202993 $\times {10}^{-13}$
1	0.3678152638	0.3678152642	3.647225 $\times {10}^{-10}$
2	0.2279650712	0.2279650715	2.869146 $\times {10}^{-10}$
3	−0.1454902169	−0.1454902166	2.825168 $\times {10}^{-10}$
4	−0.3693950508	−0.3693950509	7.337930 $\times {10}^{-11}$
5	−0.2180138015	−0.2180138015	8.116590 $\times {10}^{-12}$
6	0.1568748321	0.1568748321	3.115582 $\times {10}^{-11}$
7	0.3705660013	0.3705660013	2.219724 $\times {10}^{-11}$
8	0.2078152196	0.2078152196	3.170524 $\times {10}^{-11}$
9	−0.1680803111	−0.1680803110	8.022660 $\times {10}^{-11}$
10	−0.3713269352	−0.3713269352	3.649314 $\times {10}^{-11}$

and

Table 12. The approximate analytic solution ${\overline{y}}_{mOPIM}$ (32) and the corresponding numerical solution for the initial conditions $x_0=0.75$, $y_0=0.95$, $z_0=0.25$, and $N_{max}=25$ (absolute errors: ${ϵ}_y=|y_{numerical}-{\overline{y}}_{mOPIM}|$).

t	${\mathit{y}}_{\mathit{numerical}}$	${\overline{\mathit{y}}}_{\mathit{mOPIM}}$	${\mathit{ϵ}}_{\mathit{y}}$
0	0.95	0.9499999999	1.630917 $\times {10}^{-13}$
1	1.2033009149	1.2033009655	5.063425 $\times {10}^{-8}$
2	1.0169960446	1.0169959880	5.662388 $\times {10}^{-8}$
3	0.9561549652	0.9561546963	2.689327 $\times {10}^{-7}$
4	1.2061597339	1.2061598849	1.509541 $\times {10}^{-7}$
5	1.0079868679	1.0079867476	1.203212 $\times {10}^{-7}$
6	0.9627637335	0.9627636506	8.295834 $\times {10}^{-8}$
7	1.2082915777	1.2082918531	2.753882 $\times {10}^{-7}$
8	0.9992517322	0.9992516775	5.466931 $\times {10}^{-8}$
9	0.9698056708	0.9698055682	1.025970 $\times {10}^{-7}$
10	1.2096831387	1.2096832205	8.182391 $\times {10}^{-8}$

.

mOPIM Solutions versus Iterative Solutions

To emphasize the advantages of the presented method, the iterative solutions are obtained by the iterative method [57].

If the system (2) is integrated over the interval $[0,t]$, it results in

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

(78)

The iterative procedure leads to

$$\begin{array}{c}{\displaystyle x_0\left(t\right)=x\left(0\right),x_1\left(t\right)=N_1(x_0,y_0,z_0)=\underset{0}{\overset{t}{\int }}a{y}_0\left(s\right)ds,}\hfill \\ {\displaystyle y_0\left(t\right)=y\left(0\right),y_1\left(t\right)=N_2(x_0,y_0,z_0)=-a\underset{0}{\overset{t}{\int }}{x}_0\left(s\right)z_0\left(s\right)ds,}\hfill \\ {\displaystyle z_0\left(t\right)=z\left(0\right),z_1\left(t\right)=N_3(x_0,y_0,z_0)=\underset{0}{\overset{t}{\int }}{x}_0\left(s\right)y_0\left(s\right)ds,}\hfill \\ \cdots \hfill \\ x_m\left(t\right)=N_1\left({\displaystyle \sum _{i=0}^{m-1}{x}_i,\sum _{i=0}^{m-1}{y}_i,\sum _{i=0}^{m-1}{z}_i}\right)-N_1\left({\displaystyle \sum _{i=0}^{m-2}{x}_i,\sum _{i=0}^{m-2}{y}_i,\sum _{i=0}^{m-2}{z}_i}\right),\hfill \\ y_m\left(t\right)=N_2\left({\displaystyle \sum _{i=0}^{m-1}{x}_i,\sum _{i=0}^{m-1}{y}_i,\sum _{i=0}^{m-1}{z}_i}\right)-N_2\left({\displaystyle \sum _{i=0}^{m-2}{x}_i,\sum _{i=0}^{m-2}{y}_i,\sum _{i=0}^{m-2}{z}_i}\right),\hfill \\ z_m\left(t\right)=N_3\left({\displaystyle \sum _{i=0}^{m-1}{x}_i,\sum _{i=0}^{m-1}{y}_i,\sum _{i=0}^{m-1}{z}_i}\right)-N_3\left({\displaystyle \sum _{i=0}^{m-2}{x}_i,\sum _{i=0}^{m-2}{y}_i,\sum _{i=0}^{m-2}{z}_i}\right),\hfill \\ m\ge 2.\hfill \end{array}$$

(79)

The solutions of Equation (2), using the iterative algorithm, can be written 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 six iterations and considering the initial conditions, $x\left(0\right)=0.25$, $y\left(0\right)=0.5$, and $z\left(0\right)=1.5$ (presented in

Table 13. Values of the approximate analytical solution $\overline{x}{\left(t\right)}_{mOPIM}$ (A5), the iterative solution $x_{iter}\left(t\right)$ (80), and the corresponding numerical solution.

t	${\mathit{x}}_{\mathit{numerical}}$	${\overline{\mathit{x}}}_{\mathit{mOPIM}}$	${\mathit{x}}_{\mathit{iter}}$
0	0.25	0.2499999985	0.25
1	0.3610049696	0.3610048962	0.3610050309
2	0.4353207343	0.4353207328	0.4353288189
3	0.4624590237	0.4624590276	0.4626173487
4	0.4382012681	0.4382013071	0.4394406708
5	0.36632873959	0.3663287291	0.3719271924
6	0.2570637655	0.2570636473	0.2745357196
7	0.1235690104	0.1235688481	0.1667502201
8	−0.0208470389	−0.0208471012	0.0778136191
9	−0.1634564945	−0.1634563481	0.0783493199
10	−0.2914024236	−0.2914022766	0.3597669503

), and the physical constant $a=0.250$, taking into account the algorithm (79), become

$$\begin{array}{c}{\displaystyle x_{iter}\left(t\right)=\sum _{m=0}^6{x}_m\left(t\right)=0.25+0.125t-0.01171875t^2-0.0022786458t^3-}\hfill \\ -0.0000152587t^4+0.0000139872t^5\hfill \\ \\ {\displaystyle y_{iter}\left(t\right)=\sum _{m=0}^6{y}_m\left(t\right)=0.5-0.0937499999t-0.0273437499t^2-0.0002441406t^3+}\hfill \\ +0.0002797444t^4+0.0000886917t^5\hfill \\ {\displaystyle z_{iter}\left(t\right)=\sum _{m=0}^6{z}_m\left(t\right)=1.5+0.1249999999t+0.0195312499t^2-0.0081380208t^3-}\hfill \\ -0.0008799235t^4+0.0001131693t^5+0.0000217888t^6.\hfill \end{array}$$

(80)

In Figure 7 and Table 13, respectively, is presented a parallel between the mOPIM solutions ${\overline{x}}_{mOPIM}$ and the corresponding iterative solutions $x_{iter}$ given in Equation (80). This comparative analysis highlights the efficiency and the accuracy of the modified mOPIM method using only one iteration.

The precision and efficiency of the mOPIM method (using just one iteration) against the iterative method are described in [57] (using six iterations), arising from the presented comparison.

6. Conclusions

A new analytical approach, namely, the modified optimal parametric iteration method (mOPIM), for solving second-order nonlinear differential equations is developed using only one iteration.

In this way, the closed-form analytical approximate solutions are built for a class of nonlinear dynamical systems that possess a Hamilton–Poisson structure.

The obtained results are validated by graphically comparing them with the corresponding numerical solutions. The corresponding absolute errors are tabulated.

A comparison between the approximate analytical solution obtained with mOPIM, the analytical solution obtained with the iterative method, and the corresponding numerical solution highlights the advantages of the mOPIM method.

These comparisons prove the precision of the applied method in the sense that the semianalytical solutions are approaching the exact solution; e.g., the residual functions are much smaller than 1.

The achieved results have high potential, especially given the strong alignment demonstrated between the analytical and numerical outcomes, and they encourage the study of other dynamical systems with similar properties.

The possibility of a comparison between our results and some experiments based on the dynamical systems having a Hamilton–Poisson structure could be the subject of a future work.