Article
Mathematics, Applied
Wang Xiaoling, Li Shimin
Summary: In this paper, we studied a susceptible-infectious-recovered model with a nonlinear incidence rate and found that it exhibits rich dynamics.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2022)
Article
Mathematics
K. Uldall Kristiansen
Summary: In this paper, the birth of canard limit cycles in slow-fast systems in R-3 is rigorously described through the folded saddle-node of type II and the singular Hopf bifurcation. The paper proves the existence of a family of periodic orbits born in the (singular) Hopf bifurcation and extending to (1) cycles that follow the strong canard of the folded saddle-node. The results show that unlike the explosive family of periodic orbits in R-2, the family in R-3 is not explosive and is called the dud canard.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Engineering, Multidisciplinary
A. M. A. El-Sayed, S. M. Salman, A. M. A. Abo-Bakr
Summary: The Henon map, introduced by Henon, is a rich dynamical model that has been compared with other dynamical systems. In this study, we investigate the dynamics of the difference equation with continuous arguments corresponding to the Henon map and its singularly perturbed counterpart. We analyze the local stability of fixed points and observe various types of bifurcation. Theoretical analysis is confirmed through numerical simulations, showcasing the complex dynamics of the system.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Mathematics, Interdisciplinary Applications
Munehisa Sekikawa, Naohiko Inaba
Summary: The study explores the bifurcation structures of nested MMOIB-generated MMOs, revealing complex phenomena with strong order characteristics. Through numerical methods, two- and one-parameter bifurcation diagrams were prepared, showcasing the nested MMOs and their orderly nature. The research defines the first return maps for nested MMOs, shedding light on the appearance of successively nested MMOIBs.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2021)
Article
Engineering, Multidisciplinary
A. M. A. EL-Sayed, S. M. Salman, A. M. A. Abo-Bakr
Summary: The dynamical properties of a class of difference equations with continuous arguments of the form x(t) = g(x(t -r1), x(t -r2)) and its singularly perturbed counterpart epsilon dx dt = -x(t) + g(x(t -r1), x(t -r2)) are investigated in this study. The effect of time delays r1 and r2 on the behavior of the dynamical systems is discussed. Hopf bifurcation is proved to occur in the systems, indicating the creation of periodic orbits by varying the delays. Comparison is made between the singularly perturbed equation and the associated difference equation with continuous arguments as well as the corresponding delay differential equation. It is found that the singularly perturbed equation behaves qualitatively the same as the difference equation and the delay differential equation under certain conditions.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Mathematics, Applied
Yiorgos Patsios, Renato Huzak, Peter De Maesschalck, Nikola Popovic
Summary: MMOs are complex oscillatory patterns found in singularly perturbed systems, typically related to folded singularities and canard trajectories. We introduced a new type of MMOs based on a jump mechanism in a canonical family of slow-fast systems.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Physics, Multidisciplinary
Yaru Liu, Shenquan Liu, Bo Lu, Juergen Kurths
Summary: This article explores the dynamics of mixed-mode oscillations (MMOs) in the auditory cortex based on the calcium-based inner hair cells (IHCs) model, revealing the mechanism of MMOs generation using the geometric singular perturbation theory (GSPT). The analysis shows that system parameters control the oscillation patterns in the IHCs model, with many new oscillations occurring. The study also conducts dynamic analysis using slow-fast analysis and bifurcation analysis, uncovering the underlying dynamic properties of perturbed systems under singular perturbation theory.
Article
Mathematics, Applied
Kaito Kato, Naohiko Inaba, Kuniyasu Shimizu, Takuji Kousaka, Hideaki Okazaki
Summary: The existence of nested mixed-mode oscillation (MMO) generated by a driven slow-fast Bonhoeffer-van der Pol (BVP) oscillator has been confirmed in previous studies. It is asserted that nested MMOs can occur regardless of the type of Hopf bifurcation when no perturbation is applied, suggesting that this phenomenon could be widespread. The study demonstrates that weak periodic perturbations in a classical BVP oscillator can result in at least doubly nested MMOs, which is supported by first return plots.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Engineering, Electrical & Electronic
Yue Yu, Cong Zhang, Zhenyu Chen, Zhengdi Zhang
Summary: This paper investigates the singular Hopf bifurcation and mixed mode oscillations (MMOs) in the perturbed Bonhoeffer-van der Pol (BVP) circuit. The authors use a generalized fast/slow analysis to show the generation mechanism of two distinct kinds of MMOs. By using parametric modulation and bifurcation theory, the authors analyze the transition mechanism and characteristic features in the BVP circuit.
Article
Engineering, Mechanical
Chunyan Gao, Fangqi Chen
Summary: The study demonstrates that transcription and translation delays act as bifurcation parameters driving oscillation behavior in a gene expression model, with their length determining the amplitude and period of the oscillations. Optimal parameter rates are also crucial for inducing limit-cycle oscillations. Additionally, transcription factor concentration serves as a signal inducing bifurcations and affecting delay effects on the system, with subcritical Hopf bifurcation occurring under small signal strength.
NONLINEAR DYNAMICS
(2021)
Article
Mathematics, Interdisciplinary Applications
Chun Zhang, Xindong Ma, Qinsheng Bi
Summary: This paper proposes and analyzes complex mixed-mode oscillation patterns in a modified Rayleigh-Duffing oscillator based on bifurcation theory. Four different types of mixed-mode oscillations are discussed and the paper highlights the sensitivity of the system parameters in determining the oscillation patterns.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Computer Science, Interdisciplinary Applications
Fei Yu, Yuanshi Wang
Summary: This paper investigates an extended predator-prey model with the consideration that predators' fear reduces prey reproduction and the search speed of predators is influenced by prey density. The results show that high levels of fear can stabilize the coexistence steady state, while low levels lead to periodic oscillation. The analysis also reveals that a relatively small search speed of predators promotes the stability of the coexistence steady state, while a large speed results in periodic oscillation. Enhancing prey's sensitivity to predation risk or slowing the predator search speed can stabilize the coexistence steady state.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2022)
Article
Optics
Katarzyna Krupa, Tomasz M. Kardas, Yuriy Stepanenko
Summary: This study demonstrates the real-time observation of double-Hopf-type breathers in an erbium-doped fiber laser and discusses their possible explanation. It provides important insights for understanding laser physics and optimizing fiber cavity design.
LASER & PHOTONICS REVIEWS
(2022)
Article
Mathematics, Applied
P. De Maesschalck, G. Kiss, A. Kovacs
Summary: The study uses geometric singular perturbation theory to investigate a two-gene system with an autoregulatory feedback loop, identifying relaxation oscillations, singular Hopf bifurcations, homoclinic loops, and presenting a new method to compute the criticality of the singular Hopf bifurcations.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Susmita Sadhu
Summary: The ecological model consists of two predator species competing for a common prey, exhibiting rich dynamics such as mixed-mode oscillations and relaxing oscillations. Near the singular Hopf bifurcation, long lasting transient dynamics like chaotic MMOs are observed, persisting for hundreds or thousands of generations before a regime shift occurs. Bifurcation analysis and geometric singular perturbation theory are used to explain the irregular oscillations arising from variations in intraspecific competition among predators.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2021)
