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Mathematics
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Bifurcation Theory and Qualitative Analysis of Dynamical Systems
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Bifurcation Theory and Qualitative Analysis of Dynamical Systems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C2: Dynamical Systems".

Deadline for manuscript submissions: 31 August 2026 | Viewed by 716

Special Issue Editor

Prof. Dr. Cuiping Li

E-Mail Website
Guest Editor
School of Mathematical Sciences, Beihang University, Beijing 100191, China
Interests: qualitative theory of differential equations; dynamical systems and bifurcation problems

Special Issue Information

Dear Colleagues,

Bifurcation theory and qualitative analysis of dynamical systems together constitute a robust methodological framework for addressing nonlinear problems, especially when integrated with emerging interdisciplinary fields such as physics, control theory, biology, biomedical science, engineering, economics, information science, neural networks, and artificial intelligence.

Many natural laws and phenomena can be described as continuous or discrete dynamical systems, which typically lack explicit solutions and often exhibit intricate dynamical behavior. Through qualitative analysis methods—such as linearization, Poincaré mapping, and averaging—we can systematically characterize the long-term behavior of their solutions, examine recurrence properties, and identify abrupt changes caused by parameter variations within the system.

This Special Issue seeks to compile the latest research advances in the fields of dynamical systems and bifurcation theory. We welcome submissions of original research articles and comprehensive review papers covering topics including, but not limited to, the following areas: local and global bifurcation, Hilbert’s 16th problem, qualitative analysis, applications to physics (theorical mechanics, celestial mechanics, and fluid mechanics) and artificial intelligence, as well as ecosystems and biomathematics.

Prof. Dr. Cuiping Li
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • local and global bifurcation
  • Hilbert’s 16th problem
  • qualitative analysis
  • applications to physics (theorical mechanics, celestial mechanics, and fluid mechanics) and artificial intelligence
  • ecosystems and biomathematics

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Published Papers (2 papers)

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Research

42 pages, 10310 KB  
Article
Canards and Homoclinic Bifurcations for a Singularly Perturbed Rosenzweig–MacArthur Model with the Generalist Predator
by Xiao Wu, Shuaiwen Dan and Feng Xie
Mathematics 2026, 14(8), 1329; https://doi.org/10.3390/math14081329 - 15 Apr 2026
Viewed by 160
Abstract
In this paper, we investigate the multi-scale dynamics of a singularly perturbed Rosenzweig–MacArthur model with a generalist predator and identify dynamical phenomena, including equilibrium bifurcations, supercritical or subcritical singular Hopf bifurcations, canard explosion bifurcations and homoclinic bifurcations. Specifically, the system exhibits a globally [...] Read more.
In this paper, we investigate the multi-scale dynamics of a singularly perturbed Rosenzweig–MacArthur model with a generalist predator and identify dynamical phenomena, including equilibrium bifurcations, supercritical or subcritical singular Hopf bifurcations, canard explosion bifurcations and homoclinic bifurcations. Specifically, the system exhibits a globally stable node, a headless canard cycle evolving into a homoclinic cycle, a headed canard cycle encompassing either a headless canard cycle or a homoclinic cycle, and so on. Notably, near the boundary equilibrium, these cycles exhibit a diminutive beard-shaped structure whenever it aligns with the transcritical non-normally hyperbolic point. The numerical simulations confirm the occurrence of a canard explosion, relaxation oscillation, and an inverse canard explosion phenomena not previously reported in singularly perturbed systems with both a transcritical point and a canard point. In brief, our results demonstrate that the generalist predation can cause richer bifurcations and dynamics. Full article
(This article belongs to the Special Issue Bifurcation Theory and Qualitative Analysis of Dynamical Systems)
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22 pages, 17862 KB  
Article
On Duopolistic Competition with the Gradient Adjustment Mechanism Under Constant and Decreasing Returns to Scale
by Ruirui Hou, Xiaoliang Li and Wenshuang Wan
Mathematics 2026, 14(8), 1305; https://doi.org/10.3390/math14081305 - 14 Apr 2026
Viewed by 175
Abstract
This paper investigates duopoly competition under both constant and decreasing returns to scale in a market characterized by an isoelastic demand function, where firms adjust their strategies using a gradient adjustment mechanism. To establish the stability conditions of the model, we adopt different [...] Read more.
This paper investigates duopoly competition under both constant and decreasing returns to scale in a market characterized by an isoelastic demand function, where firms adjust their strategies using a gradient adjustment mechanism. To establish the stability conditions of the model, we adopt different analytical approaches depending on the type of returns to scale. Under constant returns to scale, we employ a traditional approach by deriving the closed-form solution of the Nash equilibrium and analyzing the Jacobian matrix to verify whether the moduli of all eigenvalues are less than one. In contrast, under decreasing returns to scale, we analyze the local stability of the Nash equilibrium using symbolic computation methods without deriving a closed-form solution. The results show that when firms have heterogeneous costs, the model can exhibit both period-doubling and Neimark–Sacker bifurcations under both types of returns to scale. However, when costs are homogeneous, only period-doubling bifurcations occur. Numerical simulations support these analytical results and further demonstrate the emergence of complex dynamics, including chaotic behavior. Full article
(This article belongs to the Special Issue Bifurcation Theory and Qualitative Analysis of Dynamical Systems)
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