Article
Mathematics, Applied
Kristian Uldall Kristiansen, Morten Gram Pedersen
Summary: In this paper, we use geometric singular perturbation theory and blowup to study the mixed-mode oscillations (MMOs) occurring in two coupled FitzHugh-Nagumo units with symmetric and repulsive coupling. We demonstrate that the MMOs in this model are not due to folded singularities, but rather due to singularities at a cusp of the critical manifold. Using blowup, we determine the number of small-amplitude oscillations (SAOs) analytically, showing that they are determined by the Weber equation and the ratio of eigenvalues. We also show that the model undergoes a saddle-node bifurcation in the desingularized reduced problem, which occurs on a cusp, and not a fold.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Naziru M. Awal, Irving R. Epstein, Tasso J. Kaper, Theodore Vo
Summary: Symmetry-breaking in coupled, identical, fast-slow systems leads to a diverse range of dynamic behaviors, including significant differences in amplitude and frequency as well as qualitatively distinct rhythms between oscillators associated with different functional states. A novel method is presented to analyze these systems, identifying key geometric structures responsible for the symmetry-breaking and demonstrating the robust emergence of various types of symmetry-breaking rhythms. The method is illustrated with two prototypical fast-slow systems: the van der Pol equation describing electrical circuits and the Lengyel-Epstein model of chemical oscillators.
Article
Engineering, Mechanical
Qinsheng Bi, Shaomin Chen
Summary: The paper investigates the slow-fast behaviors of a higher-dimensional non-smooth system and presents novel bursting attractors and interesting phenomena. The study demonstrates the transition from periodic to quasi-periodic attractor in bursting oscillations by utilizing the overlap of the transformed phase portrait and coexisted attractors. The observation of sliding along the trajectory on the bursting attractor is explained by the non-smooth theory or the in-turn influence of two pseudo-attractors in different regions.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics, Interdisciplinary Applications
Evdokiia Slepukhina, Irina Bashkirtseva, Lev Ryashko, Philipp Kuegler
Summary: This paper studies stochastic phenomena in a model of cardiac activity and reveals the importance of mixed-mode oscillations and canards in the dynamics. It shows that weak additive noise can induce MMOs and investigates the probabilistic mechanism using stochastic analysis techniques. The results also demonstrate noise-driven transitions to chaos in the model.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Physics, Multidisciplinary
Xindong Ma, Heqi Zhao, Qinsheng Bi
Summary: This paper investigates the novel bursting patterns induced by different types of hysteresis loops in a parametrically and externally driven one-degree-of-freedom nonlinear oscillator. The study reveals detailed bursting behaviors and dynamic transitions, as well as the disappearance of hypocritical cycles through two approaches, resulting in the appearance of distinctive hysteresis loops.
EUROPEAN PHYSICAL JOURNAL-SPECIAL TOPICS
(2022)
Article
Mathematics, Applied
Tasso J. J. Kaper, Theodore Vo
Summary: Chimeras are states where incoherent and coherent oscillations coexist. The discovery of mixed-amplitude chimeras, with substantially different oscillation structures, amplitudes, and frequencies in the incoherent and coherent regions, is an important advancement for understanding processes in neuroscience, pattern formation, and physics. These states may also provide insights into interpreting electroencephalogram recordings in animals experiencing unihemispheric slow-wave sleep.
Article
Physics, Multidisciplinary
M. Paul Asir, D. Premraj, K. Sathiyadevi
Summary: In a system of three coupled non-autonomous oscillators, we observed complex forms of mixed-mode oscillations (MMOs) and identified different sequences of oscillations and related dynamics. We also found that, in a specific parameter space, the period adding sequence of these oscillations follows a devil's staircase structure.
EUROPEAN PHYSICAL JOURNAL PLUS
(2022)
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
Physics, Multidisciplinary
B. Kaviya, R. Suresh, V. K. Chandrasekar
Summary: We study the dynamics of a hybrid model and report the emergence of extreme bursting events due to a novel pulse-shaped explosion mechanism. The system exhibits complex periodic and chaotic bursting patterns as a function of excitation frequencies, along with small oscillations. We identify the appearance of a sharp pulse-like transition in the equilibrium points of the system as the underlying mechanism for the development of bursting events. The controlling aspect of extreme events is attempted by incorporating a linear damping term.
EUROPEAN PHYSICAL JOURNAL PLUS
(2022)
Article
Mathematics, Applied
C. Abdulwahed, F. Verhulst
Summary: This study investigates three three-dimensional chaotic systems with linear and quadratic terms and one parameter, analyzing second order asymptotic approximations near the origin of phase-space. The results show the existence of periodic solutions with neutral and asymptotic stability, as well as a new type of relaxation oscillations with pulse behavior. These dynamics coexist with invariant tori and chaos in the systems.
Article
Mathematics, Interdisciplinary Applications
Youhua Qian, Danjin Zhang, Bingwen Lin
Summary: This study investigates the bursting oscillation mechanisms in systems with periodic excitation, analyzing different types of symmetric bursting oscillations and their bifurcation mechanisms through numerical simulations. The results show that these bursting oscillations exhibit symmetry in their patterns.
Article
Mathematics, Interdisciplinary Applications
Juanjuan Huang, Qinsheng Bi
Summary: This paper investigates the mechanism of bursting oscillations with high co-dimensional bifurcations in high-dimensional vector fields. By establishing an eight-dimensional system with the slow-fast effect, the paper reveals different types of bursting phenomena, such as Hopf/Hopf bursting oscillations and Hopf/Hopf/fold bursting oscillations, and analyzes their mechanisms. The paper focuses on the analysis of the fast subsystem and introduces different movement modes to aid in understanding its behavior.
CHAOS SOLITONS & FRACTALS
(2023)
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
J. Penalva, M. Desroches, A. E. Teruel, C. Vich
Summary: The phenomenon of slow passage through a Hopf bifurcation is commonly observed in multiple-timescale dynamical systems. In this study, we investigate this phenomenon in piecewise linear (PWL) slow-fast systems for the first time. We provide conditions for a PWL slow-fast system to exhibit a slow passage through a Hopf-like bifurcation and fully describe the way-in/way-out function. We also examine this effect in the Doi-Kumagai model.
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
Mathematics, Applied
Albert Granados, Maciej Krupa
Article
Engineering, Mechanical
Viktor Avrutin, Enric Fossas, Albert Granados, Michael Schanz
NONLINEAR DYNAMICS
(2011)
Article
Mathematics, Applied
V. Avrutin, A. Granados, M. Schanz
Article
Mathematics, Applied
A. Granados, S. J. Hogan, T. M. Seara
PHYSICA D-NONLINEAR PHENOMENA
(2014)
Article
Mathematics, Applied
A. Granados, S. J. Hogan, T. M. Seara
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2012)
Article
Mathematics, Applied
A. Granados, M. Krupa, F. Clement
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2014)
Article
Mathematics, Applied
Enric Fossas, Albert Granados
DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS
(2013)