{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:19:58Z","timestamp":1760059198801,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,27]],"date-time":"2025-05-27T00:00:00Z","timestamp":1748304000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this research, we investigate the phenomenon of multistability and complex dynamic behaviors in plasma waves by utilizing advanced mathematical techniques. We examine how fractional-order derivatives influence plasma wave stability by applying the fractional diffusion\u2013reaction model, the framework of nonlinear dynamical systems, and the (G\u2032G2) method. The principal direction of our work is associated with different forms of oscillations in the plasma wave: non-linear periodic, solitons, and kink waves. This leads to the study of small amplitude pulses and solitary waves, which are significant in plasma activities. Using bifurcation analysis, we discuss how these waves appear and develop under different conditions, as well as determine which conditions generate the chaotic behavior or highly complex patterns of waves. We study the details of transitions between waves and their chaotic behavior to characterize the laws that govern their plasma environment. Moreover, we have used non-linear modeling and numerical simulations to understand in detail the complex patterns and the factors of stability underlying the phenomena of plasma waves. In addition, our study also investigates the correspondence between non-linearity, multi-stability, and the birth of complex structures such as solitons and kink waves. The solutions of the dynamical system produced by the proposed nonlinear model generate different patterns of response based on system parameter variation. These patterns include oscillations and decay behaviors. Research results about system stability and solution convergence under various parameter settings provide an extended performance evaluation of the proposed method through a better understanding of system dynamics. They increase our understanding of chaotic behavior in plasma systems and pave the way for applications in plasma physics and energy systems, as well as advanced technologies.<\/jats:p>","DOI":"10.3390\/axioms14060405","type":"journal-article","created":{"date-parts":[[2025,5,27]],"date-time":"2025-05-27T09:29:36Z","timestamp":1748338176000},"page":"405","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dynamical Systems with Fractional Derivatives: Focus on Phase Portraits and Plasma Wave Propagation Using Lakshmanan\u2013Porsezian\u2013Daniel Model"],"prefix":"10.3390","volume":"14","author":[{"given":"Abdul Ghaffar","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, University of Engineering and Technology, Taxila 47050, Pakistan"}]},{"given":"Muhammad","family":"Muddassar","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Engineering and Technology, Taxila 47050, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3937-5101","authenticated-orcid":false,"given":"Sultan","family":"Shoaib","sequence":"additional","affiliation":[{"name":"Department of Engineering, Gulf University of Science and Technology, Hawally 32093, Kuwait"},{"name":"GUST Engineering & Applied Innovation Research Centre (GEAR), Gulf University for Science and Technology, Hawally 32093, Kuwait"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-0036-887X","authenticated-orcid":false,"given":"Zia Ur","family":"Rehman","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Engineering and Technology, Taxila 47050, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8613-2017","authenticated-orcid":false,"given":"Muhammad","family":"Zahid","sequence":"additional","affiliation":[{"name":"Department of Engineering, Gulf University of Science and Technology, Hawally 32093, Kuwait"},{"name":"Department of Telecommunication Engineering, University of Engineering and Technology, Taxila 47050, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Rehman, Z.U., Zahid, M., Bolla, D.P., Shoaib, S., Amin, Y., Khattak, R., and Muddassar, M. 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