{
  "abstract": "In this article, we use truncated M-fractional derivatives to analyze the bifurcation and chaotic behavior of and traveling-wave solutions to the Zhiber–Shabat equation. By introducing truncated M-fractional derivatives, the equation exhibits richer dynamic properties. Based on phase diagram analysis and dynamical system theory, the bifurcation behavior of the equilibrium point of a two-dimensional dynamical system is discussed. At the same time, the dynamical behavior of a two-dimensional dynamical system with periodic disturbances is considered, revealing the complex chaotic phenomena of the system under specific parameters. A planar phase diagram, a three-dimensional phase diagram, a sensitivity analysis, and a maximum Lyapunov exponent diagram of the perturbed two-dimensional dynamical system were employed. Furthermore, various forms of accurate analytical solutions were obtained through traveling-wave transformation and numerical simulation. The three-dimensional, two-dimensional, density, and polar coordinates of the solutions were plotted using mathematical software. The results indicate that the fractional order and system parameters have a significant impact on the morphology and chaotic characteristics of the solution. This study provides new theoretical insights into the nonlinear dynamics of fractional-order Zhiber–Shabat equations.",
  "abstract_shown": true,
  "affiliations": [
    "College of Computer Science, Chengdu University, Chengdu 610106, China",
    "Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Zhao Li",
      "orcid": "0000-0002-2502-5330"
    },
    {
      "affs": [
        1
      ],
      "name": "Ejaz Hussain",
      "orcid": "0009-0002-7410-3267"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2fnn2fmzsj245.bib",
  "cited_by": 16,
  "doi": "10.3390/fractalfract10050335",
  "fulltext_url": null,
  "id": "2fnn2fmzsj245",
  "issns": [
    "2504-3110"
  ],
  "landing_url": "https://doi.org/10.3390/fractalfract10050335",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2611|Fractional Differential Equations Solutions"
    ]
  ],
  "openalex": "W7161293329",
  "pdf_url": "https://www.mdpi.com/2504-3110/10/5/335/pdf?version=1778838775",
  "pmcid": null,
  "pmid": null,
  "published": "2026-05-15",
  "publisher": "Multidisciplinary Digital Publishing Institute",
  "ris": "https://research.draconitic.com/paper/2fnn2fmzsj245.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7161293329",
      "source": "openalex",
      "url": "https://doi.org/10.3390/fractalfract10050335"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2504-3110",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Bifurcation Analysis and Chaotic Behaviors of and a Traveling-Wave Solution to the Zhiber–Shabat Equation with a Truncated M-Fractional Derivative",
  "title_text": "Bifurcation Analysis and Chaotic Behaviors of and a Traveling-Wave Solution to the Zhiber–Shabat Equation with a Truncated M-Fractional Derivative",
  "type": "peer-reviewed",
  "updated_at": "2026-11-08T13:20:35Z",
  "url": "https://research.draconitic.com/paper/2fnn2fmzsj245",
  "venue": "Fractal and Fractional",
  "versions": []
}