{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T18:54:55Z","timestamp":1773082495110,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T00:00:00Z","timestamp":1754352000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU)","award":["IMSIU-DDRSP2503"],"award-info":[{"award-number":["IMSIU-DDRSP2503"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study investigates the dynamics of dust acoustic periodic waves in a three-component, unmagnetized dusty plasma system using generalized (r\u22c6,q\u22c6) distributions. First, boundary conditions are applied to reduce the model to a second-order nonlinear ordinary differential equation. The Galilean transformation is subsequently applied to reformulate the second-order ordinary differential equation into an unperturbed dynamical system. Next, phase portraits of the system are examined under all possible conditions of the discriminant of the associated cubic polynomial, identifying regions of stability and instability. The Runge\u2013Kutta method is employed to construct the phase portraits of the system. The Hamiltonian function of the unperturbed system is subsequently derived and used to analyze energy levels and verify the phase portraits. Under the influence of an external periodic perturbation, the quasi-periodic and chaotic dynamics of dust ion acoustic waves are explored. Chaos detection tools confirm the presence of quasi-periodic and chaotic patterns using Basin of attraction, Lyapunov exponents, Fractal Dimension, Bifurcation diagram, Poincar\u00e9 map, Time analysis, Multi-stability analysis, Chaotic attractor, Return map, Power spectrum, and 3D and 2D phase portraits. In addition, the model\u2019s response to different initial conditions was examined through sensitivity analysis.<\/jats:p>","DOI":"10.3390\/axioms14080610","type":"journal-article","created":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T13:44:32Z","timestamp":1754401472000},"page":"610","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Mathematical Modeling, Bifurcation Theory, and Chaos in a Dusty Plasma System with Generalized (r\u22c6, q\u22c6) Distributions"],"prefix":"10.3390","volume":"14","author":[{"family":"Beenish","sequence":"first","affiliation":[{"name":"Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan"}]},{"given":"Maria","family":"Samreen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3110-414X","authenticated-orcid":false,"given":"Fehaid Salem","family":"Alshammari","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Shukla, P.K., and Mamun, A.A. (2015). Introduction to Dusty Plasma Physics, CRC Press.","DOI":"10.1201\/9781420034103"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"033702","DOI":"10.1063\/1.4977447","article-title":"A generalized AZ-non-Maxwellian velocity distribution function for space plasmas","volume":"24","author":"Abid","year":"2017","journal-title":"Phys. Plasmas"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_4","first-page":"C6","article-title":"Ion sounds like solitary waves with density depressions","volume":"5","author":"Cairns","year":"1995","journal-title":"J. Phys. IV"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2246","DOI":"10.1063\/1.1688330","article-title":"Some electrostatic modes are based on non-Maxwellian distribution functions","volume":"11","author":"Zaheer","year":"2004","journal-title":"Phys. Plasmas"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"122902","DOI":"10.1063\/1.2139504","article-title":"Landau damping in space plasmas with generalized (r, q) distribution function","volume":"12","author":"Qureshi","year":"2005","journal-title":"Phys. Plasmas"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1088\/0031-8949\/73\/2\/009","article-title":"Dust-charge fluctuations with non-Maxwellian distribution functions","volume":"73","author":"Rubab","year":"2006","journal-title":"Phys. Scr."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"112104","DOI":"10.1063\/1.2364155","article-title":"Effect of non-Maxwellian particle trapping and dust grain charging on dust acoustic solitary waves","volume":"13","author":"Rubab","year":"2006","journal-title":"Phys. Plasmas"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"053704","DOI":"10.1063\/1.5028290","article-title":"Propagation of symmetric and anti-symmetric surface waves in a self-gravitating magnetized dusty plasma layer with generalized (r, q) distribution","volume":"25","author":"Lee","year":"2018","journal-title":"Phys. Plasmas"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"125604","DOI":"10.1088\/1402-4896\/ab346e","article-title":"Oblique modulation and envelope excitations of nonlinear ion sound waves with cubic nonlinearity and generalized (r, q) distribution","volume":"94","author":"Ullah","year":"2019","journal-title":"Phys. Scr."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1007\/s11082-023-05958-4","article-title":"Bifurcations, chaotic behavior, sensitivity analysis and soliton solutions of the extended Kadometsev\u2013Petviashvili equation","volume":"56","author":"Xu","year":"2024","journal-title":"Opt. Quantum Electron."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"San, S., and Alshammari, F.S. (2025). Analytical and Dynamical Study of Solitary Waves in a Fractional Magneto-Electro-Elastic System. Fractal Fract., 9.","DOI":"10.3390\/fractalfract9050309"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"015279","DOI":"10.1088\/1402-4896\/ad9e41","article-title":"Analytical solutions and dynamical behaviors of the extended Bogoyavlensky-Konopelchenko equation in deep water dynamics","volume":"100","author":"Jhangeer","year":"2024","journal-title":"Phys. Scr."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"073705","DOI":"10.1063\/1.4991406","article-title":"Super-soliton dust-acoustic waves in four-component dusty plasma using non-extensive electron and ion distributions","volume":"24","author":"Abulwafa","year":"2017","journal-title":"Phys. Plasmas"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"103703","DOI":"10.1063\/1.4990849","article-title":"Effect of dust ion collision on dust ion acoustic waves in the framework of the damped Zakharov-Kuznetsov equation in the presence of external periodic force","volume":"24","author":"Ali","year":"2017","journal-title":"Phys. Plasmas"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"e202000022","DOI":"10.1002\/ctpp.202000022","article-title":"Bifurcation analysis of nonlinear and supernonlinear dust\u2013acoustic waves in a dusty plasma using the generalized (r, q) distribution function for ions and electrons","volume":"60","author":"Taha","year":"2020","journal-title":"Contrib. Plasma Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Kop\u00e7as\u0131z, B., and Ya\u015far, E. (J. Ocean. Eng. Sci., 2022). Novel exact solutions and bifurcation analysis to the dual-mode nonlinear Schr\u00f6dinger equation, J. Ocean. Eng. Sci., in press.","DOI":"10.1016\/j.joes.2022.06.007"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"103178","DOI":"10.1016\/j.asej.2024.103178","article-title":"Symmetry analysis, dynamical behavior, and conservation laws of the dual-mode nonlinear fluid model","volume":"16","author":"Jhangeer","year":"2025","journal-title":"Ain Shams Eng. J."},{"key":"ref_19","unstructured":"Sidorov, N., Loginov, B., Sinitsyn, A.V., and Falaleev, M.V. (2013). Lyapunov\u2013Schmidt Methods in Nonlinear Analysis and Applications, Springer Science Business Media."},{"key":"ref_20","first-page":"653","article-title":"Ensemble Classifier Design Based on Perturbation Binary Salp Swarm Algorithm for Classification","volume":"135","author":"Zhu","year":"2023","journal-title":"CMES-Comput. Model. Eng. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1007\/s12591-019-00511-w","article-title":"Basins of attraction and stability of nonlinear systems\u2019 equilibrium points","volume":"31","author":"Sidorov","year":"2023","journal-title":"Differ. Equ. Dyn. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"110856","DOI":"10.1016\/j.chaos.2021.110856","article-title":"Multiple and generic bifurcation analysis of a discrete Hindmarsh-Rose model","volume":"146","author":"Li","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"101766","DOI":"10.1016\/j.najef.2022.101766","article-title":"Multi-scale systemic risk and spillover networks of commodity markets in the bullish and bearish regimes","volume":"62","author":"Zhang","year":"2022","journal-title":"N. Am. J. Econ. Financ."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"115089","DOI":"10.1016\/j.cam.2023.115089","article-title":"Bifurcation analysis and complex dynamics of a Kopel triopoly model","volume":"426","author":"Li","year":"2023","journal-title":"J. Comput. Appl. Math."},{"key":"ref_25","unstructured":"and Samreen, M. (2025). Qualitative Behavior and Travelling Wave Solutions of the (n+1)-Dimensional Camassa-Holm Kadomtsev-Petviashvili Equation. Int. J. Geom. Methods Mod. Phys., 2550275."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Li, Z., Lyu, J., and Hussain, E. (2024). Bifurcation, chaotic behaviors, and solitary wave solutions for the fractional Twin-Core couplers with Kerr law non-linearity. Sci. Rep., 14.","DOI":"10.1038\/s41598-024-74044-w"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/610\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:23:46Z","timestamp":1760034226000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/610"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,5]]},"references-count":26,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["axioms14080610"],"URL":"https:\/\/doi.org\/10.3390\/axioms14080610","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,5]]}}}