{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T22:17:00Z","timestamp":1774477020988,"version":"3.50.1"},"reference-count":63,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,4,28]],"date-time":"2025-04-28T00:00:00Z","timestamp":1745798400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>The study offers a comprehensive investigation of periodic solutions in highly nonlinear oscillator systems, employing advanced analytical and numerical techniques. The motivation stems from the urgent need to understand complex dynamical behaviors in physics and engineering, where traditional linear approximations fall short. This work precisely applies He\u2019s Frequency Formula (HFF) to provide theoretical insights into certain classes of strongly nonlinear oscillators, as illustrated through five broad examples drawn from various scientific and engineering disciplines. Additionally, the novelty of the present work lies in reducing the required time compared to the classical perturbation techniques that are widely employed in this field. The proposed non-perturbative approach (NPA) effectively converts nonlinear ordinary differential equations (ODEs) into linear ones, equivalent to simple harmonic motion. This method yields a new frequency approximation that aligns closely with the numerical results, often outperforming existing approximation techniques in terms of accuracy. To aid readers, the NPA is thoroughly explained, and its theoretical predictions are validated through numerical simulations using Mathematica Software (MS). An excellent agreement between the theoretical and numerical responses highlights the robustness of this method. Furthermore, the NPA enables a detailed stability analysis, an area where traditional methods frequently underperform. Due to its flexibility and effectiveness, the NPA presents a powerful and efficient tool for analyzing highly nonlinear oscillators across various fields of engineering and applied science.<\/jats:p>","DOI":"10.3390\/computation13050105","type":"journal-article","created":{"date-parts":[[2025,4,28]],"date-time":"2025-04-28T11:48:33Z","timestamp":1745840913000},"page":"105","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["A Novel Methodology for Scrutinizing Periodic Solutions of Some Physical Highly Nonlinear Oscillators"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9060-4371","authenticated-orcid":false,"given":"Gamal M.","family":"Ismail","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia"}]},{"given":"Galal M.","family":"Moatimid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11566, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4843-1208","authenticated-orcid":false,"given":"Stylianos V.","family":"Kontomaris","sequence":"additional","affiliation":[{"name":"School of Sciences, European University Cyprus, 2404 Nicosia, Cyprus"},{"name":"Department of Engineering and Construction, Metropolitan College, 15125 Athens, Greece"},{"name":"BioNanoTec Ltd., 2043 Nicosia, Cyprus"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1061-4685","authenticated-orcid":false,"given":"Livija","family":"Cveticanin","sequence":"additional","affiliation":[{"name":"Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1016\/j.ymssp.2005.04.008","article-title":"Present and future of nonlinear system identification in structural dynamics","volume":"20","author":"Kerschen","year":"2006","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"043001","DOI":"10.1088\/1361-665X\/aa550a","article-title":"Structural damage identification using damping: A compendium of uses and features","volume":"26","author":"Cao","year":"2017","journal-title":"Smart Mater. Struct."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1155\/2007\/260183","article-title":"The evaluation of the damping characteristics of a hard coating on titanium","volume":"14","author":"Blackwell","year":"2007","journal-title":"Shock. Vib."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1115\/1.1640356","article-title":"Identifying Coulomb and viscous friction in forced dual-damped oscillators","volume":"126","author":"Liang","year":"2004","journal-title":"J. Vib. Acoust."},{"key":"ref_5","first-page":"110","article-title":"Experimental determination of structural damping of different materials","volume":"144","author":"Mevada","year":"2016","journal-title":"Process Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.jweia.2011.12.005","article-title":"Frequency modulated empirical mode decomposition method for the identification of instantaneous modal parameters of aero elastic systems","volume":"101","author":"Zhang","year":"2012","journal-title":"J. Wind Eng. Ind. Aerodyn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"7061","DOI":"10.1016\/j.matpr.2017.11.370","article-title":"Study of damping in composite beams","volume":"5","author":"Kulkarni","year":"2018","journal-title":"Mater. Today Proc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"103518","DOI":"10.1016\/j.ijnonlinmec.2020.103518","article-title":"Identification and parameter estimation of cubic nonlinear damping using harmonic probing and Volterra series","volume":"125","author":"Chatterjee","year":"2020","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"103750","DOI":"10.1016\/j.mechmachtheory.2019.103750","article-title":"On discontinuous dynamics of a class of friction-influenced oscillators with nonlinear damping under bilateral rigid constraints","volume":"147","author":"Dou","year":"2020","journal-title":"Mech. Mach. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"22119","DOI":"10.1007\/s11071-023-08659-5","article-title":"Non-trivial solutions and their stability in a two-degree-of-freedom Mathieu\u2013Duffing system","volume":"111","author":"Barakat","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_11","first-page":"275","article-title":"Oscillations of dynamic equations on time scales with advanced arguments","volume":"6","year":"2016","journal-title":"Int. J. Dyn. Syst. Differ. Equ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"239","DOI":"10.5890\/JAND.2019.06.007","article-title":"On the asymptotic stability behaviours of solutions of non-linear differential equations with multiple variable advanced arguments","volume":"8","year":"2019","journal-title":"J. Appl. Nonlinear Dyn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5051","DOI":"10.1016\/j.aej.2021.09.021","article-title":"Analytical accurate solutions of nonlinear oscillator systems via coupled homotopy-variational approach","volume":"61","author":"Ismail","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1016\/j.jsv.2005.02.018","article-title":"Free vibrations of a mass grounded by linear and non-linear springs in series","volume":"289","author":"Kopmaz","year":"2006","journal-title":"J. Sound Vib."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1140\/epjp\/i2019-12824-6","article-title":"Study of strongly nonlinear oscillators using the Aboodh transform and the homotopy perturbation method","volume":"134","author":"Manimegalai","year":"2019","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3281","DOI":"10.1080\/15376494.2021.1892888","article-title":"The extended Rayleigh-Ritz method for an analysis of nonlinear vibrations","volume":"29","author":"Wang","year":"2022","journal-title":"Mech. Adv. Mater. Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"794","DOI":"10.1080\/15376494.2021.2023922","article-title":"Forced vibration analysis of multi-degree-of-freedom nonlinear systems with the extended Galerkin method","volume":"30","author":"Shi","year":"2023","journal-title":"Mech. Adv. Mater. Struct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1177\/14613484231188756","article-title":"Power series approach to nonlinear oscillators","volume":"43","author":"Asad","year":"2024","journal-title":"J. Low Freq. Noise Vib. Act. Control"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2456","DOI":"10.1016\/j.camwa.2009.03.034","article-title":"The homotopy perturbation method for nonlinear oscillators","volume":"58","author":"Shou","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s40094-014-0133-9","article-title":"Analytical study of nonlinear oscillatory systems using the Hamiltonian approach technique","volume":"8","author":"Hermann","year":"2014","journal-title":"J. Theor. Appl. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1016\/j.joems.2017.07.006","article-title":"An analytical coupled homotopy-variational approach for solving strongly nonlinear differential equation","volume":"25","author":"Ismail","year":"2017","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"101219","DOI":"10.1016\/j.jksus.2020.10.016","article-title":"Analytic approximations to non-linear third order jerk equations via modified global error minimization method","volume":"33","author":"Ismail","year":"2021","journal-title":"J. King Saud Univ.-Sci."},{"key":"ref_23","first-page":"19","article-title":"Comment on He\u2019s frequency formulation for nonlinear oscillators","volume":"29","author":"He","year":"2008","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_24","first-page":"77","article-title":"An approximate amplitude-frequency relationship for a nonlinear oscillator with discontinuity","volume":"7","author":"He","year":"2016","journal-title":"Nonlinear Sci. Lett. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1557","DOI":"10.1007\/s40819-016-0160-0","article-title":"Amplitude-frequency relationship for conservative nonlinear oscillators with odd nonlinearities","volume":"3","author":"He","year":"2017","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2379","DOI":"10.1142\/S0217979211100370","article-title":"He\u2019s frequency-amplitude formulation for nonlinear oscillators","volume":"25","author":"Ren","year":"2011","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_27","first-page":"86","article-title":"Theoretical basis of He\u2019s frequency-amplitude formulation for nonlinear oscillators","volume":"9","author":"Ren","year":"2018","journal-title":"Nonlinear Sci. Lett. A"},{"key":"ref_28","first-page":"340","article-title":"An amplitude-period formula for a second order nonlinear oscillator","volume":"8","year":"2017","journal-title":"Nonlinear Sci. Lett. A"},{"key":"ref_29","first-page":"348","article-title":"Remarks on an approximate formula for the period of conservative oscillations in nonlinear second order ordinary differential equations","volume":"8","author":"Antola","year":"2017","journal-title":"Nonlinear Sci. Lett. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3222","DOI":"10.1016\/j.camwa.2010.03.013","article-title":"Frequency analysis of strongly nonlinear generalized Duffing oscillators using He\u2019s frequency-amplitude formulation and He\u2019s energy balance method","volume":"59","author":"Younesian","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1050","DOI":"10.1177\/1461348418812327","article-title":"He\u2019s frequency-amplitude formulation with average residuals for nonlinear oscillators","volume":"38","author":"Ren","year":"2019","journal-title":"J. Low Freq. Noise Vib. Act. Control"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2154","DOI":"10.29020\/nybg.ejpam.v17i3.5339","article-title":"Periodic solutions of strongly nonlinear oscillators using He\u2019s frequency formulation","volume":"17","author":"Ismail","year":"2024","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Moatimid, G.M., and Amer, T.S. (2023). Dynamical system of a time-delayed \u03d56-Van der Pole oscillator: A non-perturbative approach. Sci. Rep., 13.","DOI":"10.1038\/s41598-023-38679-5"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1177\/14613484231195276","article-title":"A novel methodology for a time-delayed controller to prevent nonlinear system oscillations","volume":"43","author":"Moatimid","year":"2024","journal-title":"J. Low Freq. Noise Vib. Act. Control"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Moatimid, G.M., Amer, T.S., and Galal, A.A. (2023). Studying highly nonlinear oscillators using the non-perturbative methodology. Sci. Rep., 13.","DOI":"10.1038\/s41598-023-47519-5"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Moatimid, G.M., El-Sayed, A.T., and Salman, H.F. (2024). Different controllers for suppressing oscillations of a hybrid oscillator via non-perturbative analysis. Sci. Rep., 14.","DOI":"10.1038\/s41598-023-50750-9"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3185","DOI":"10.29020\/nybg.ejpam.v17i4.5479","article-title":"The non-perturbative approach in examining the motion of a simple pendulum associated with a rolling wheel with a time-delay","volume":"17","author":"Alluhydan","year":"2024","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_38","first-page":"2878","article-title":"A novel inspection of a time-delayed rolling of a rigid rod","volume":"17","author":"Alluhydan","year":"2024","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Alluhydan, K., Moatimid, G.M., Amer, T.S., and Galal, A.A. (2024). Inspection of a time-delayed excited damping Duffing oscillator. Axioms, 13.","DOI":"10.3390\/axioms13060416"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2358","DOI":"10.59400\/sv2358","article-title":"Scrutinizing highly nonlinear oscillators using He\u2019s frequency formula","volume":"59","author":"Ismail","year":"2025","journal-title":"Sound Vib."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1211","DOI":"10.1007\/s42417-024-01469-y","article-title":"Inspection of some extremely nonlinear oscillators using an inventive approach","volume":"12","author":"Moatimid","year":"2024","journal-title":"J. Vib. Eng. Technol."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/s42417-024-01688-3","article-title":"Insightful examination of some nonlinear classification linked with Mathieu oscillators","volume":"13","author":"Moatimid","year":"2025","journal-title":"J. Vib. Eng. Technol."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Alanazy, A., Moatimid, G.M., Amer, T.S., Mohamed, M.A.A., and Abohamer, M.K. (2025). A novel procedure in scrutinizing a cantilever beam with tip mass: Analytic and bifurcation. Axioms, 14.","DOI":"10.3390\/axioms14010016"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"S1831","DOI":"10.1007\/s42417-024-01506-w","article-title":"An innovative approach in inspecting a damped Mathieu cubic\u2013quintic Duffing oscillator","volume":"12","author":"Moatimid","year":"2024","journal-title":"J. Vib. Eng. Technol."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Moatimid, G.M., and Mohamed, M.Y. (2024). Nonlinear electro-rheological instability of two moving cylindrical fluids: An innovative approach. Phys. Fluids, 36.","DOI":"10.1063\/5.0188061"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"120924","DOI":"10.1016\/j.oceaneng.2025.120924","article-title":"Innovative methodology in scrutinizing nonlinear rolling ship in longitudinal waves","volume":"327","author":"Alanazy","year":"2025","journal-title":"Ocean Eng."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"120975","DOI":"10.1016\/j.oceaneng.2025.120975","article-title":"Novel approach in inspecting nonlinear rolling ship in longitudinal waves","volume":"327","author":"Moatimid","year":"2025","journal-title":"Ocean Eng."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1706","DOI":"10.1121\/1.1918794","article-title":"Equivalent linearization techniques","volume":"35","author":"Caughey","year":"1963","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/0020-7462(78)90017-3","article-title":"On the existence and uniqueness of solutions generated by equivalent linearization","volume":"13","author":"Spanos","year":"1979","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1007\/s40819-021-01026-1","article-title":"Special functions for solving nonlinear differential equations","volume":"7","author":"He","year":"2021","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1177\/14613484241278421","article-title":"Analytical solution for strongly nonlinear vibration of a stringer shell","volume":"44","author":"Farea","year":"2025","journal-title":"J. Low Freq. Noise Vib. Act. Control."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.rinp.2016.04.012","article-title":"Harmonic balance method for solving a large-amplitude oscillation of a conservative system with inertia and static non-linearity","volume":"6","author":"Molla","year":"2016","journal-title":"Results Phys."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1006\/jsvi.1996.0672","article-title":"On the large amplitude free vibrations of restrained uniform beam carring an intermediate lumped mass","volume":"199","author":"Hamdan","year":"1997","journal-title":"J. Sound Vib."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1435","DOI":"10.1007\/s00170-012-4113-4","article-title":"Dynamic response of a beam carrying a lumped mass along its span","volume":"64","author":"Sfahani","year":"2013","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1140\/epjp\/i2019-12621-3","article-title":"Higher-order approximate periodic solution for the oscillator with strong nonlinearity of polynomial type","volume":"134","author":"Cveticanin","year":"2019","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Ismail, G.M., Hosen, M.A., Mohammadian, M., El-Moshneb, M.M., and Bayat, M. (2022). Nonlinear Vibration of Electrostatically Actuated Microbeam. Mathematics, 10.","DOI":"10.3390\/math10244762"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1590\/S1679-78252014000300009","article-title":"Nonlinear vibration of an electrostatically actuated microbeam","volume":"11","author":"Bayat","year":"2014","journal-title":"Lat. Am. J. Solids Struct."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1016\/j.cap.2010.08.037","article-title":"Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS)","volume":"11","author":"Fu","year":"2011","journal-title":"Curr. Appl. Phys."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/s13204-015-0445-3","article-title":"Nonlinear oscillation of nanoelectro-mechanical resonators using energy balance method: Considering the size effect and the van der Waals force","volume":"6","author":"Ghalambaz","year":"2016","journal-title":"Appl. Nanosci."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1140\/epjp\/i2019-12399-2","article-title":"Analytical approximations to nonlinear oscillation of nanoelectro-mechanical resonators","volume":"134","author":"Ismail","year":"2019","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"2155","DOI":"10.1007\/s42417-022-00693-8","article-title":"Analytical approximate solutions of a magnetic spherical pendulum: Stability analysis","volume":"11","author":"Moatimid","year":"2023","journal-title":"J. Vib. Eng. Technol."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1091","DOI":"10.1016\/S0022-460X(02)01385-8","article-title":"Comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach","volume":"264","author":"Amabili","year":"2003","journal-title":"J. Sound Vib."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1023\/A:1004119928867","article-title":"A new approach to the relativistic Schr\u00f6dinger equation with central potential: Ansatz method","volume":"40","author":"Dong","year":"2001","journal-title":"Int. J. Theor. Phys."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/5\/105\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:23:36Z","timestamp":1760030616000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/5\/105"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,28]]},"references-count":63,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["computation13050105"],"URL":"https:\/\/doi.org\/10.3390\/computation13050105","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,28]]}}}