{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:40:19Z","timestamp":1760146819393,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,17]],"date-time":"2024-12-17T00:00:00Z","timestamp":1734393600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science and Technology Council of Taiwan","award":["NSTC113-2112-M-002-032"],"award-info":[{"award-number":["NSTC113-2112-M-002-032"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this work, we investigate the transition from regular dynamics to chaotic behavior in a one-dimensional quartic anharmonic classical oscillator driven by a time-dependent external square-wave force. Owing to energy conservation, the motion of an undriven quartic anharmonic oscillator is regular, periodic, and stable. For a driven quartic anharmonic oscillator, the equations of motion cannot be solved analytically due to the presence of an anharmonic term in the potential energy function. Using the fourth-order Runge\u2013Kutta method to numerically solve the equations of motion for the driven quartic anharmonic oscillator, we find that the oscillator motion under the influence of a sufficiently small driving force remains regular, while by gradually increasing the driving force, a series of nonlinear resonances can occur, grow, overlap, and ultimately disappear due to the emergence of chaos.<\/jats:p>","DOI":"10.3390\/computation12120246","type":"journal-article","created":{"date-parts":[[2024,12,17]],"date-time":"2024-12-17T03:46:02Z","timestamp":1734407162000},"page":"246","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Classical Chaos in a Driven One-Dimensional Quartic Anharmonic Oscillator"],"prefix":"10.3390","volume":"12","author":[{"given":"Yun-Hsi","family":"Lin","sequence":"first","affiliation":[{"name":"Pacific American School, Zhubei 30272, Taiwan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3994-2279","authenticated-orcid":false,"given":"Jeng-Da","family":"Chai","sequence":"additional","affiliation":[{"name":"Department of Physics, National Taiwan University, Taipei 10617, Taiwan"},{"name":"Center for Theoretical Physics and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan"},{"name":"Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lichtenberg, A.J., and Lieberman, M.A. (1983). Regular and Stochastic Motion, Springer.","DOI":"10.1007\/978-1-4757-4257-2"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Guckenheimer, J., and Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer.","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Lorenz, E.N. (1993). The Essence of Chaos, University of Washington Press.","DOI":"10.4324\/9780203214589"},{"key":"ref_4","unstructured":"Ott, E. (1993). Chaos in Dynamical Systems, Cambridge University Press."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Bolotin, Y., Tur, A., and Yanovsky, V. (2009). Chaos: Concepts, Control and Constructive Use, Springer.","DOI":"10.1007\/978-3-642-00937-2"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1250","DOI":"10.3390\/encyclopedia2030084","article-title":"Three kinds of butterfly effects within Lorenz models","volume":"2","author":"Shen","year":"2022","journal-title":"Encyclopedia"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/0370-1573(79)90023-1","article-title":"A universal instability of many-dimensional oscillator systems","volume":"52","author":"Chirikov","year":"1979","journal-title":"Phys. Rep."},{"key":"ref_8","unstructured":"Tabor, M. (1989). Chaos and Integrability in Nonlinear Dynamics: An Introduction, Wiley."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0960-0779(91)90032-5","article-title":"Survey of regular and chaotic phenomena in the forced Duffing oscillator","volume":"1","author":"Ueda","year":"1991","journal-title":"Chaos Solitons Fractals"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5245","DOI":"10.1103\/PhysRevE.56.5245","article-title":"Chaotic one-dimensional harmonic oscillator","volume":"56","author":"Lee","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Lakshmanan, M., and Rajasekar, S. (2003). Nonlinear Dynamics: Integrability, Chaos and Patterns, Springer.","DOI":"10.1007\/978-3-642-55688-3"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/j.cnsns.2003.06.001","article-title":"The approximate solution of a classical quartic anharmonic oscillator with periodic force: A simple analytical approach","volume":"10","author":"Mandal","year":"2005","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1119\/1.1591765","article-title":"Square-wave excitation of a linear oscillator","volume":"72","author":"Butikov","year":"2004","journal-title":"Am. J. Phys."},{"key":"ref_14","unstructured":"Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. (1992). Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Butcher, J.C. (2008). Numerical Methods for Ordinary Differential Equations, Wiley.","DOI":"10.1002\/9780470753767"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/BF01020332","article-title":"Quantitative universality for a class of nonlinear transformations","volume":"19","author":"Feigenbaum","year":"1978","journal-title":"J. Stat. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Saiki, Y., and Yorke, J.A. (2023). Can the flap of a butterfly\u2019s wings shift a tornado into Texas\u2013without chaos?. Atmosphere, 14.","DOI":"10.3390\/atmos14050821"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1016\/S0026-2692(01)00156-2","article-title":"Nonautonomous pulse-driven chaotic oscillator based on Chua\u2019s circuit","volume":"33","author":"Elwakil","year":"2002","journal-title":"Microelectron. J."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1034","DOI":"10.1016\/j.chaos.2005.09.046","article-title":"Stochastic resonance with different periodic forces in overdamped two coupled anharmonic oscillators","volume":"30","author":"Gandhimathi","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_20","unstructured":"Duffing, G. (1918). Erzwungene Schwingungen bei Ver\u00e4nderlicher Eigenfrequenz und Ihre Technische Bedeutung, Vieweg."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Kovacic, I., and Brennan, M.J. (2011). The Duffing Equation: Nonlinear Oscillators and their Behaviour, John Wiley & Sons, Ltd.","DOI":"10.1002\/9780470977859"},{"key":"ref_22","unstructured":"Bender, C.M., and Orszag, S.A. (1978). Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill."},{"key":"ref_23","unstructured":"Blanchard, P., Devaney, R.L., and Hall, G.R. (2012). Differential Equations, Cengage Learning."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2050257","DOI":"10.1142\/S0218127420502570","article-title":"Homoclinic orbits and solitary waves within the nondissipative Lorenz model and KdV equation","volume":"30","author":"Shen","year":"2020","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_25","unstructured":"Whittaker, E.T., and Watson, G.N. (1947). Modern Analysis, Cambridge University Press."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1090\/qam\/110250","article-title":"On the application of elliptic functions in non-linear forced oscillations","volume":"17","author":"Hsu","year":"1960","journal-title":"Q. Appl. Math."},{"key":"ref_27","unstructured":"Coddington, E.A., and Levinson, N. (1955). Theory of Ordinary Differential Equations, McGraw-Hill."},{"key":"ref_28","unstructured":"Strogatz, S.H. (2014). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/12\/246\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:53:33Z","timestamp":1760115213000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/12\/246"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,17]]},"references-count":28,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["computation12120246"],"URL":"https:\/\/doi.org\/10.3390\/computation12120246","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2024,12,17]]}}}