# Study of Nonlinear Models of Oscillatory Systems by Applying an Intelligent Computational Technique

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## Abstract

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## 1. Introduction

- The main purpose of this study is to formulate mathematical models and investigate the influence of variations in certain parameters of nonlinear oscillators such as a rotational pendulum system, mass attached to an elastic wire, a uniform beam carrying an intermediate lumped mass, a two-mass system with three springs, the van der Pol equation, and a two-mass system with small damping.
- An integrated novel design of soft computing based on neural networks and the backpropagated Levenberg–Marquardt algorithm is utilized to study the displacement, velocity, and acceleration of the models.
- The supervised learning of the NNs-BLM algorithm works effectively on the data set generated by a numerical solution using the Runge–Kutta method.
- The performance of the design scheme is validated by conducting convergence analysis based on mean square error, regression analysis, error histogram, and curve fitting with reference data. Results demonstrate that the proposed algorithm is smooth and easy to implement.

## 2. Proposed Methodology

- An initial data set is generated by using an analytical solution or calculating a numerical solution by using the Runge–Kutta method of order 4 (RK-4), with the ND Solve package in Mathematica.
- In the second phase, the BLM algorithm is executed by using “nftool” in the MATLAB package with appropriate settings of hidden neurons and testing data. Further, BLM uses a reference solution and implements the process of testing, training, and validation to obtain approximate solutions for different cases of nonlinear oscillators. Table 1 shows the parameter setting for the execution of the design scheme.

## 3. Numerical Experimentation and Discussion

#### 3.1. Rotational Pendulum

#### 3.2. Oscillations of a Mass Attached to a Stretched Elastic Wire

#### 3.3. Large-Amplitude Free Vibration of a Restrained Uniform Beam Carrying an Intermediate Lumped Mass

#### 3.4. Van der Pol Equations

#### 3.5. Two-Mass System with Three Springs

#### 3.6. Two-Mass System with Small Damping

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Hosen, M.A.; Chowdhury, M.; Ali, M.Y.; Ismail, A.F. An analytical approximation technique for the duffing oscillator based on the energy balance method. Ital. J. Pur. Appl. Math.
**2017**, 37, 455–466. [Google Scholar] - Gottlieb, H. Harmonic balance approach to periodic solutions of non-linear jerk equations. J. Sound Vib.
**2004**, 271, 671–683. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y.; Lin, J.; Hu, Z.; Khan, N.A.; Sulaiman, M. Analysis of Third-Order Nonlinear Multi-Singular Emden–Fowler Equation by Using the LeNN-WOA-NM Algorithm. IEEE Access
**2021**, 9, 72111–72138. [Google Scholar] [CrossRef] - Hu, H.; Zheng, M.; Guo, Y. Iteration calculations of periodic solutions to nonlinear jerk equations. Acta Mech.
**2010**, 209, 269–274. [Google Scholar] [CrossRef] - He, J.H. Iteration perturbation method for strongly nonlinear oscillations. J. Vib. Control
**2001**, 7, 631–642. [Google Scholar] [CrossRef] - Oliveira, A.R. History of Krylov-Bogoliubov-Mitropolsky Methods of Nonlinear Oscillations. Adv. Hist. Stud.
**2017**, 6, 40. [Google Scholar] [CrossRef] [Green Version] - Marathe, A.; Chatterjee, A. Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales. J. Sound Vib.
**2006**, 289, 871–888. [Google Scholar] [CrossRef] - He, J.H. Preliminary report on the energy balance for nonlinear oscillations. Mech. Res. Commun.
**2002**, 29, 107–111. [Google Scholar] [CrossRef] [Green Version] - Heydari, M.; Loghmani, G.; Hosseini, S. An improved piecewise variational iteration method for solving strongly nonlinear oscillators. Comput. Appl. Math.
**2015**, 34, 215–249. [Google Scholar] [CrossRef] - Adomian, G. Solving Frontier Problems of Physics: The Decomposition Method, with a Preface by Yves Cherruault; Fundamental Theories of Physics; Kluwer Academic: Dordrecht, The Netherlands, 1994. [Google Scholar]
- Big-Alabo, A.; Ekpruke, E.O.; Ossia, C.V. Equivalent oscillator model for the nonlinear vibration of a porter governor. J. King Saud-Univ.-Eng. Sci.
**2021**. [Google Scholar] [CrossRef] - Big-Alabo, A.; Ogbodo, C.O.; Ossia, C.V. Semi-analytical treatment of complex nonlinear oscillations arising in the slider-crank mechanism. World Sci. News
**2020**, 142, 1–24. [Google Scholar] - Big-Alabo, A.; Ossia, C.V. Periodic oscillation and bifurcation analysis of pendulum with spinning support using a modified continuous piecewise linearization method. Int. J. Appl. Comput. Math.
**2019**, 5, 1–16. [Google Scholar] [CrossRef] - Kargar, A.; Akbarzade, M. Frequency Analysis of Large-Amplitude Oscillation of a Rotational Pendulum System Using He’s Amplitude-Frequency Formulation (HFAF) and He’s Energy Balance Method (HEBM). Int. J. Math. Anal.
**2012**, 6, 1147–1152. [Google Scholar] - Bataineh, A.S. Application of adaptation HAM for nonlinear oscillator typified as a mass attached to a stretched elastic wire. Commun. Math. Appl.
**2017**, 8, 157–165. [Google Scholar] - Li, S.; Niu, J.; Li, X. Primary resonance of fractional-order Duffing–van der Pol oscillator by harmonic balance method. Chin. Phys. B
**2018**, 27, 120502. [Google Scholar] [CrossRef] - Lai, S.; Yang, X.; Gao, F. Analysis of large-amplitude oscillations in triple-well non-natural systems. J. Comput. Nonlinear Dyn.
**2019**, 14, 091002. [Google Scholar] [CrossRef] - Razzak, M.A. A simple new iterative method for solving strongly nonlinear oscillator systems having a rational and an irrational force. Alex. Eng. J.
**2018**, 57, 1099–1107. [Google Scholar] [CrossRef] - Koochi, A.; Goharimanesh, M. Nonlinear Oscillations of CNT Nano-resonator Based on Nonlocal Elasticity: The Energy Balance Method. Rep. Mech. Eng.
**2021**, 2, 41–50. [Google Scholar] - Qian, Y.; Pan, J.; Chen, S.; Yao, M. The spreading residue harmonic balance method for strongly nonlinear vibrations of a restrained cantilever beam. Adv. Math. Phys.
**2017**, 2017, 5214616. [Google Scholar] [CrossRef] [Green Version] - Liu, C.X.; Yan, Y.; Wang, W.Q. Primary and secondary resonance analyses of a cantilever beam carrying an intermediate lumped mass with time-delay feedback. Nonlinear Dyn.
**2019**, 97, 1175–1195. [Google Scholar] [CrossRef] - Momani, S.; Ertürk, V.S. Solutions of non-linear oscillators by the modified differential transform method. Comput. Math. Appl.
**2008**, 55, 833–842. [Google Scholar] [CrossRef] [Green Version] - He, J.H. Max-min approach to nonlinear oscillators. Int. J. Nonlinear Sci. Numer. Simul.
**2008**, 9, 207–210. [Google Scholar] [CrossRef] - He, J.H. Modified straightforward expansion. Meccanica
**1999**, 34, 287–289. [Google Scholar] [CrossRef] - Ullah, M.S.; Ali, M.Z.; Noor, N. Novel dynamics of wave solutions for Cahn–Allen and diffusive predator–prey models using MSE scheme. Partial Differ. Equ. Appl. Math.
**2021**, 3, 100017. [Google Scholar] [CrossRef] - He, J.H. Modified Lindstedt–Poincare methods for some strongly non-linear oscillations: Part I: Expansion of a constant. Int. J. Non-Linear Mech.
**2002**, 37, 309–314. [Google Scholar] [CrossRef] - Roshid, H.O.; Khatun, M.S.; Baskonus, H.M.; Belgacem, F.B.M. Breather, multi-shock waves and localized excitation structure solutions to the Extended BKP–Boussinesq equation. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 101, 105867. [Google Scholar] [CrossRef] - Liu, Y.; Liao, S.; Li, Z. Symbolic computation of strongly nonlinear periodic oscillations. J. Symb. Comput.
**2013**, 55, 72–95. [Google Scholar] [CrossRef] - Ali, A.; Hamraz, M.; Kumam, P.; Khan, D.M.; Khalil, U.; Sulaiman, M.; Khan, Z. A k-nearest neighbours based ensemble via optimal model selection for regression. IEEE Access
**2020**, 8, 132095–132105. [Google Scholar] [CrossRef] - Huang, W.; Jiang, T.; Zhang, X.; Khan, N.A.; Sulaiman, M. Analysis of beam-column designs by varying axial load with internal forces and bending rigidity using a new soft computing technique. Complexity
**2021**, 2021, 6639032. [Google Scholar] [CrossRef] - Khan, N.A.; Sulaiman, M.; Aljohani, A.J.; Kumam, P.; Alrabaiah, H. Analysis of multi-phase flow through porous media for imbibition phenomena by using the LeNN-WOA-NM algorithm. IEEE Access
**2020**, 8, 196425–196458. [Google Scholar] [CrossRef] - Khan, N.A.; Sulaiman, M.; Aljohani, A.J.; Bakar, M.A. Mathematical models of CBSC over wireless channels and their analysis by using the LeNN-WOA-NM algorithm. Eng. Appl. Artif. Intell.
**2022**, 107, 104537. [Google Scholar] [CrossRef] - Khan, N.A.; Khalaf, O.I.; Romero, C.A.T.; Sulaiman, M.; Bakar, M.A. Application of Euler Neural Networks with Soft Computing Paradigm to Solve Nonlinear Problems Arising in Heat Transfer. Entropy
**2021**, 23, 1053. [Google Scholar] [CrossRef] - Khan, N.A.; Sulaiman, M.; Kumam, P.; Bakar, M.A. Thermal analysis of conductive-convective-radiative heat exchangers with temperature dependent thermal conductivity. IEEE Access
**2021**, 9, 138876–138902. [Google Scholar] [CrossRef] - Khan, N.A.; Sulaiman, M.; Kumam, P.; Aljohani, A.J. A new soft computing approach for studying the wire coating dynamics with Oldroyd 8-constant fluid. Phys. Fluids
**2021**, 33, 036117. [Google Scholar] [CrossRef] - Reddy, V.R.; Reddy, V.; Mohan, V.C.J. Speed control of induction motor drive using artificial neural networks-Levenberg-Marquardt Backpropogation algorithm. Int. J. Appl. Eng. Res.
**2018**, 13, 80–85. [Google Scholar] - Multazam, T.; Putri, R.I.; Pujiantara, M.; Lystianingrum, V.; Priyadi, A.; Heryp, M. Short-Term Wind Speed Prediction Base on Backpropagation Levenberg-Marquardt Algorithm; Case Study Area Nganjuk. In Proceedings of the 2017 5th International Conference on Instrumentation, Communications, Information Technology, and Biomedical Engineering (ICICI-BME), Bandung, Indonesia, 6–7 November 2017; pp. 163–166. [Google Scholar] [CrossRef]
- Wu, S.T. Active pendulum vibration absorbers with a spinning support. J. Sound Vib.
**2009**, 323, 1–16. [Google Scholar] [CrossRef] - Khan, Y.; Mirzabeigy, A.; Arjmand, H. Nonlinear oscillation of the bifilar pendulum: An analytical approximation. Multidiscip. Model. Mater. Struct.
**2017**. [Google Scholar] [CrossRef]

**Figure 5.**(

**a**) Approximate solutions by the design scheme for different cases of rotational pendulum system, while (

**b**) illustrates the phase plane between velocity and displacement of the system.

**Figure 6.**Three-dimensional plots to study the influence of time in velocity and acceleration of rotational pendulum system.

**Figure 11.**(

**a**) Approximate solutions obtained by proposed algorithm for the system. (

**b**) shows the phase plane between velocity and displacement of the stretched elastic wire.

**Figure 12.**Three-dimensional plots to study the influence of time in velocity and acceleration of mass attached to a stretched elastic wire.

**Figure 14.**Convergence of performance function in terms of mean square error for each case of problem 2.

**Figure 16.**(

**a**) Approximate solutions obtained by proposed algorithm for the system. (

**b**) Phase plane analysis between velocity and displacement for mathematical model of restrained uniform beam carrying an intermediate lumped mass.

**Figure 17.**Three-dimensional plots to study the influence of time in velocity and acceleration of mathematical model given in Equation (6).

**Figure 18.**Error histogram analysis for each case of restrained uniform beam carrying an intermediate lumped mass.

**Figure 19.**Convergence of performance function in terms of mean square error for each case of problem 3.

**Figure 20.**(

**a**–

**c**) Comparison of approximate solutions obtained by designed algorithm with RK-4. (

**d**–

**f**) show the analysis of phase plane between velocity and acceleration for van der Pol equation.

**Figure 23.**Approximate solution and histograms of $u\left(t\right)$ and $v\left(t\right)$ for Case I and II of problem 5.

**Figure 24.**Analysis of performance function in terms of mean square error for different cases of problem 5.

**Figure 27.**Approximate solution and histograms of $u\left(t\right)$ and $v\left(t\right)$ for problem 6.

**Figure 28.**Performance state of training parameters and convergence of fitness function for example 6.

Testing | Training | Valiation | Hidden Neurons | Max. Ilteration | Max. Validation Fails | Performance Function |
---|---|---|---|---|---|---|

75% | 15% | 15% | 60 | 1000 | 6 | Mean Square Error |

**Table 2.**Comparison of approximate solutions obtained by NN-BLM algorithm with He’s Energy Balance Method, Homotopy Analysis Method, Residue Harmonic Balance Method, and Homotopy Perturbation Method.

Problem 1 | Problem 2 | Problem 3 | Problem 5 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

t | Exact | HEBM | NN-BLMA | Exact | HAM | NN-BLMA | Exact | RHBM | NN-BLMA | Exact | HPM | NN-BLMA |

0 | 2.96706 | 3.01652 | 2.96706 | 1.5 | 1.5 | 1.5 | 1 | 1 | 1 | 8 | 7.999 | 8 |

3 | −2.15119 | −2.25409 | −2.15119 | −0.33557 | −0.33647 | −0.33557 | −0.99448 | −0.99442 | −0.99448 | −5.69251 | −5.69215 | −5.69251 |

6 | −1.13609 | −1.13986 | −1.13609 | −1.27728 | −1.218 | −1.27728 | 0.97792 | 0.97364 | 0.97792 | −5.01246 | −5.01245 | −5.01246 |

9 | 2.89554 | 2.99654 | 2.89554 | 0.97753 | 0.96723 | 0.97753 | −0.95033 | −0.95047 | −0.95033 | 8.10448 | 8.10448 | 8.10448 |

12 | −2.69881 | −2.831 | −2.69881 | 0.72292 | 0.722911 | 0.72292 | 0.91174 | 0.91177 | 0.91174 | −6.08405 | −6.08401 | −6.08405 |

15 | 0.12032 | 0.12087 | 0.12032 | −1.4209 | −1.42119 | −1.4209 | −0.86222 | −0.86245 | −0.86222 | −4.07551 | −4.07555 | −4.07551 |

Problem 1 | Problem 2 | Problem 3 | |||||||
---|---|---|---|---|---|---|---|---|---|

t | Case I | Case II | Case III | Case I | Case II | Case III | Case I | Case II | Case III |

0 | 2.96706 | 2.44346 | 1.91986 | 0.5 | 1 | 1.5 | 1 | 0.5 | 0.2 |

3 | −2.15119 | −2.07124 | −1.36661 | −0.49397 | −0.70011 | −0.33557 | −0.99448 | −0.47592 | −0.19447 |

6 | −1.13609 | 0.8869 | −0.14399 | 0.47604 | 0.00452 | −1.27728 | 0.97792 | 0.40334 | 0.17801 |

9 | 2.89554 | 0.71232 | 1.53846 | −0.44663 | 0.69344 | 0.97753 | −0.95033 | −0.2829 | −0.15114 |

12 | −2.69881 | −1.97792 | −1.90429 | 0.40645 | −0.99995 | 0.72292 | 0.91174 | 0.12156 | 0.11487 |

15 | 0.12032 | 2.4389 | 1.16415 | −0.35648 | 0.70671 | −1.4209 | −0.86222 | 0.06023 | −0.07096 |

Mean Square Error | |||||||||
---|---|---|---|---|---|---|---|---|---|

Case | Neurons | Training | Validation | Testing | Gradient | Mu | Epochs | Regression | Time |

I | 60 | $2.22\times {10}^{-11}$ | $6.60\times {10}^{-11}$ | $3.49\times {10}^{-11}$ | $9.98\times {10}^{-8}$ | $1.00\times {10}^{-11}$ | 985 | 1 | 2 s |

II | 60 | $2.67\times {10}^{-8}$ | $1.31\times {10}^{-8}$ | $3.72\times {10}^{-8}$ | $6.64\times {10}^{-6}$ | $1.00\times {10}^{-11}$ | 472 | 1 | 0.01 s |

III | 60 | $1.93\times {10}^{-9}$ | $9.78\times {10}^{-9}$ | $2.55\times {10}^{-8}$ | $2.08\times {10}^{-6}$ | $1.93\times {10}^{-9}$ | 1000 | 1 | 2 s |

I | 60 | $1.36\times {10}^{-8}$ | $3.58\times {10}^{-8}$ | $9.19\times {10}^{-9}$ | $9.98\times {10}^{-8}$ | $1.00\times {10}^{-10}$ | 620 | 1 | 0.03 s |

II | 60 | $8.57\times {10}^{-8}$ | $3.16\times {10}^{-8}$ | $1.04\times {10}^{-7}$ | $3.23\times {10}^{-6}$ | $1.00\times {10}^{-9}$ | 162 | 1 | 1s |

III | 60 | $1.68\times {10}^{-10}$ | $3.31\times {10}^{-10}$ | $3.40\times {10}^{-10}$ | $9.92\times {10}^{-8}$ | $1.00\times {10}^{-9}$ | 699 | 1 | 0.02 s |

**Table 5.**Values of parameters involved in mathematical model of restrained uniform beam carrying an intermediate lumped mass.

Cases | Amplitude (A) | ${\mathit{\epsilon}}_{1}$ | ${\mathit{\epsilon}}_{2}$ | ${\mathit{\epsilon}}_{3}$ | ${\mathit{\epsilon}}_{4}$ |
---|---|---|---|---|---|

I | 1 | 0.326845 | 0.129579 | 0.232598 | 0.087584 |

II | 0.5 | 1.642033 | 0.913055 | 0.313561 | 0.204297 |

III | 0.2 | 4.051486 | 1.665232 | 0.281418 | 0.149677 |

Mean Square Error | |||||||||
---|---|---|---|---|---|---|---|---|---|

Case | Neurons | Training | Validation | Testing | Gradient | Mu | Epochs | Regression | Time |

I | 60 | 3.67 × 10${}^{-10}$ | 1.26 × 10${}^{-9}$ | 9.37 × 10${}^{-10}$ | 9.98 × 10${}^{-8}$ | 1.00 × 10${}^{-11}$ | 992 | 1 | 0.02 s |

II | 60 | 2.61 × 10${}^{-10}$ | 6.69 × 10${}^{-10}$ | 1.35 × 10${}^{-10}$ | 9.97 × 10${}^{-8}$ | 1.00 × 10${}^{-11}$ | 283 | 1 | 0.01 s |

III | 60 | 9.69 × 10${}^{-11}$ | 4.27 × 10${}^{-10}$ | 4.63 × 10${}^{-10}$ | 9.97 × 10${}^{-8}$ | 1.00 × 10${}^{-11}$ | 281 | 1 | 0.01 s |

I | 60 | 3.76 × 10${}^{-10}$ | 2.36 × 10${}^{-9}$ | 3.16 × 10${}^{-9}$ | 9.95 × 10${}^{-8}$ | 1.00 × 10${}^{-12}$ | 100 | 1 | 0.005 s |

II | 60 | 5.27 × 10${}^{-8}$ | 2.08 × 10${}^{-7}$ | 2.99 × 10${}^{-7}$ | 1.72 × 10${}^{-5}$ | 1.00 × 10${}^{-9}$ | 1000 | 1 | 2 s |

III | 60 | 1.02 × 10${}^{-4}$ | 9.57 × 10${}^{-4}$ | 6.91 × 10${}^{-4}$ | 6.09 × 10${}^{-3}$ | 1.00 × 10${}^{-8}$ | 340 | 1 | 0.02 s |

Problem 4 | Problem 5 | Problem 6 | |||||||
---|---|---|---|---|---|---|---|---|---|

t | Case I | Case II | Case III | Case I | Case II | Case I | |||

$\mathit{u}\left(\mathit{t}\right)$ | $\mathit{v}\left(\mathit{t}\right)$ | $\mathit{u}\left(\mathit{t}\right)$ | $\mathit{v}\left(\mathit{t}\right)$ | $\mathit{u}\left(\mathit{t}\right)$ | $\mathit{v}\left(\mathit{t}\right)$ | ||||

0 | 0.01 | 0.01 | 0.01 | 8 | 10 | 5 | 1 | 1 | 0.5 |

3 | −0.0099 | −0.02498 | 1.50214 | −5.69251 | −3.7239 | −3.07566 | −4.7436 | −0.84116 | −0.42692 |

6 | 0.00925 | −0.00824 | 0.44609 | −5.01246 | −3.13545 | 2.24812 | 4.25251 | 0.67892 | 0.36974 |

9 | −0.00791 | 0.42076 | −1.83206 | 8.10448 | 9.83682 | −2.90897 | 1.08233 | −0.51973 | −0.327 |

12 | 0.00571 | −1.03501 | −1.56265 | −6.08405 | −4.53945 | 0.00641 | 1.33001 | 0.37165 | 0.29367 |

15 | −0.00244 | 1.39666 | −1.03589 | −4.07551 | −2.75077 | 1.67698 | −0.64469 | −0.24233 | −0.26364 |

Mean Square Error | |||||||||
---|---|---|---|---|---|---|---|---|---|

Case | Neurons | Training | Validation | Testing | Gradient | Mu | Epochs | Regression | Time |

I | 60 | 1.03 × 10${}^{-7}$ | 7.39 × 10${}^{-6}$ | 7.68 × 10${}^{-6}$ | 2.63 × 10${}^{-5}$ | 1.00 × 10${}^{-10}$ | 526 | 1 | 0.06 s |

60 | 5.48 × 10${}^{-8}$ | 7.46 × 10${}^{-6}$ | 4.58 × 10${}^{-6}$ | 2.17 × 10${}^{-5}$ | 1.00 × 10${}^{-9}$ | 1000 | 1 | 2 s | |

II | 60 | 1.43 × 10${}^{-6}$ | 5.09 × 10${}^{-6}$ | 9.57 × 10${}^{-6}$ | 1.54 × 10${}^{-4}$ | 1.00 × 10${}^{-8}$ | 1000 | 1 | 2 s |

60 | 1.46 × 10${}^{-6}$ | 7.02 × 10${}^{-6}$ | 7.38 × 10${}^{-6}$ | 2.55 × 10${}^{-5}$ | 1.00 × 10${}^{-8}$ | 1000 | 1 | 2 s | |

I | 60 | 4.96 × 10${}^{-11}$ | 3.42 × 10${}^{-8}$ | 8.76 × 10${}^{-8}$ | 2.58 × 10${}^{-7}$ | 1.00 × 10${}^{-12}$ | 1000 | 1 | 2 s |

60 | 6.18 × 10${}^{-7}$ | 8.63 × 10${}^{-6}$ | 9.52 × 10${}^{-6}$ | 1.13 × 10${}^{-6}$ | 1.00 × 10${}^{-11}$ | 57 | 1 | 0.001 s |

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**MDPI and ACS Style**

Khan, N.A.; Alshammari, F.S.; Romero, C.A.T.; Sulaiman, M.
Study of Nonlinear Models of Oscillatory Systems by Applying an Intelligent Computational Technique. *Entropy* **2021**, *23*, 1685.
https://doi.org/10.3390/e23121685

**AMA Style**

Khan NA, Alshammari FS, Romero CAT, Sulaiman M.
Study of Nonlinear Models of Oscillatory Systems by Applying an Intelligent Computational Technique. *Entropy*. 2021; 23(12):1685.
https://doi.org/10.3390/e23121685

**Chicago/Turabian Style**

Khan, Naveed Ahmad, Fahad Sameer Alshammari, Carlos Andrés Tavera Romero, and Muhammad Sulaiman.
2021. "Study of Nonlinear Models of Oscillatory Systems by Applying an Intelligent Computational Technique" *Entropy* 23, no. 12: 1685.
https://doi.org/10.3390/e23121685