{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T12:46:28Z","timestamp":1775479588547,"version":"3.50.1"},"reference-count":37,"publisher":"Wiley","license":[{"start":{"date-parts":[[2020,11,8]],"date-time":"2020-11-08T00:00:00Z","timestamp":1604793600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2020,11,8]]},"abstract":"<jats:p>In this paper, the Helmholtz equation with quadratic damping themes is used for modeling the dynamics of a simple prey-predator system also called a simple Lotka\u2013Volterra system. From the Helmholtz equation with quadratic damping themes obtained after modeling, the equilibrium points have been found, and their stability has been analyzed. Subsequently, the harmonic oscillations have been studied by the harmonic balance method, and the phenomena of resonance and hysteresis are observed. The primary and secondary resonances have been researched by the multiple-scale method, and the conditions of stability of the amplitudes of oscillations are determined. Chaos is detected analytically by the Melnikov method and numerically using the basin of attraction, the bifurcation diagram, the Lyapunov exponent, the phase portrait, and the Poincar\u00e9 section. The effects of all the parameters of the system are analyzed in detail, and special emphasis is placed on the new parameters. Through this analysis, the complex phenomena such as hysteresis, bistability, amplitude jump, resonances, and chaos have been obtained. The control of the parameters and the necessary conditions to control the aforementioned phenomena have been found.<\/jats:p>","DOI":"10.1155\/2020\/8822534","type":"journal-article","created":{"date-parts":[[2020,11,9]],"date-time":"2020-11-09T21:05:22Z","timestamp":1604955922000},"page":"1-17","source":"Crossref","is-referenced-by-count":17,"title":["Nonlinear Dynamics of the Quadratic-Damping Helmholtz Oscillator"],"prefix":"10.1155","volume":"2020","author":[{"given":"R.","family":"Fangnon","sequence":"first","affiliation":[{"name":"Laboratoire de M\u00e9canique des Fluides de la Dynamique Nonlin\u00e9aire et de la Mod\u00e9lisation des Syst\u00e8mes Biologiques (LMFDNMSB), Institut de Math\u00e9matiques et de Sciences Physiques, Porto-Novo, Benin"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C.","family":"Ainamon","sequence":"additional","affiliation":[{"name":"Laboratoire de M\u00e9canique des Fluides de la Dynamique Nonlin\u00e9aire et de la Mod\u00e9lisation des Syst\u00e8mes Biologiques (LMFDNMSB), Institut de Math\u00e9matiques et de Sciences Physiques, Porto-Novo, Benin"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A. 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