Article
Acoustics
Misael Avendano-Camacho, Alejandra Torres-Manotas, Jose Antonio Vallejo
Summary: By combining techniques of singular geometric reduction with the more classical averaging method of Moser, we have proven the existence of closed stable orbits in a strongly coupled Wilberforce pendulum at a 1:2 resonance.
JOURNAL OF VIBRATION AND CONTROL
(2022)
Article
Mathematics
Qi Qiao
Summary: This paper investigates a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By using the geometric singular perturbation theory, the existence of a positive traveling wave connecting two constant steady states is confirmed. The monotonicity of the wave is analyzed for different parameter ranges, and spectral instability is observed in some exponentially weighted spaces.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Engineering, Mechanical
Zhouqian Miao, Nikola Popovic, Thomas Zacharis
Summary: The study focuses on a two-body problem with rapid mass loss formulated by Verhulst, where the dynamical system is singularly perturbed due to the small parameter in the governing equations. By using a combination of geometric singular perturbation theory and the blow-up technique, asymptotic expansions for the solutions of this problem are derived. The unexpected dependence of the expansions on fractional powers of the singular perturbation parameter and the occurrence of logarithmic terms are attributed to a resonance phenomenon in one of the coordinate charts after blow-up.
NONLINEAR DYNAMICS
(2021)
Article
Mathematics, Applied
Pedro Toniol Cardin
Summary: This paper provides a geometric analysis of relaxation oscillations in the context of planar fast-slow systems with a discontinuous right-hand side. The conditions for the existence of a stable crossing limit cycle and the convergence of the cycle to a crossing closed singular trajectory are given. The regularization of the crossing relaxation oscillator and the existence of a relaxation oscillation in the regularized vector field are studied. The results are demonstrated with examples including a model of an arch bridge with nonlinear viscous damping.
Article
Mathematics, Applied
Adam Bauer, Paul Carter
Summary: This study focuses on the one-fluid stellar wind problem for steady, radial outflow, taking into account the effects of heat conduction and viscosity. Using geometric singular perturbation techniques, stellar wind profiles are rigorously constructed in the large Reynolds number limit, identifying transonic solutions that accelerate from subsonic to supersonic speeds. These solutions are identified as folded saddle canard trajectories lying in the intersection of a subsonic saddle slow manifold and a supersonic repelling slow manifold.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2021)
Article
Engineering, Mechanical
Minzhi Wei, Liping He
Summary: This paper investigates the existence and uniqueness of periodic waves for a BBM equation with local strong generic delay convection and weak diffusion. The singular perturbed system is reduced to a regular perturbed system by constructing a locally invariant manifold based on geometric singular perturbation theory. The existence and uniqueness of periodic waves are proved with sufficiently small perturbation parameter. Chebyshev criteria are used to study the ratio of Abelian integrals. Furthermore, the upper and lower bounds of the limiting wave speed are obtained.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics, Applied
Rongsheng Cai, Yuhua Cai, Jianhe Shen
Summary: In this article, we investigate the coexistence of a predator and two prey species under the influence of evolution. We propose a three-time-scale model that incorporates rapid adaptive behavior in the predator's feeding choice, slow growth in the prey species, and extremely slow growth in the predator. By utilizing geometric singular perturbation theory and computing the entry-exit function for multidimensional fast-slow systems, we discover that the predator and the two prey species can coexist through relaxation oscillations in their feeding choice strategies, which is attributed to the delayed loss of stability. Additionally, we demonstrate that the predator can coexist with the two prey species through an interior equilibrium state with locally optimal feeding choice.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Engineering, Mechanical
Jundong Wang, Lijun Zhang, Jibin Li
Summary: This paper investigates solitary wave solutions for a generalized Benjamin-Bona-Mahony equation with distributed delay and dissipative perturbation. The existence of solitary wave solutions with a single crest or trough is established, and a new type of solitary wave solution with coexisting crest and trough is observed theoretically. The selection principle for wave speed of the solitary wave is presented, and numerical simulations confirm the theoretical predictions.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics
Ke Wang, Zengji Du, Jiang Liu
Summary: This paper investigates the traveling pulses in coupled FitzHugh-Nagumo (FHN) equations combined with the mechanics equation, considering the presence of doubly-diffusive effect and local time delay. Singular orbits are constructed by analyzing the limit dynamics of the equations in the traveling wave framework. The main analysis relies on the geometric singular perturbation theory and Exchange Lemma to establish the traveling pulses for the full system.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Liang Zhao, Jianhe Shen
Summary: This paper investigates the number and stability of relaxation oscillations in a slow-fast predator-prey model with weak Allee effect and Holling-IV functional response, using geometric singular perturbation theory and a new entry-exit function. Theoretical predictions are verified by numerical simulations.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
Article
Mathematics, Interdisciplinary Applications
Jian Song, Shenquan Liu, Qixiang Wen
Summary: This paper investigates the electrical excitability of pituitary cells and the triggering relationship between their secretory function and electrical activity using mathematical modeling and analysis. The simplified model and the multi-geometric perspective reveal the intrinsic transients and long-term evolution of the system, as well as the dynamic mechanism and mixed-mode oscillations of cell firing. This study provides a new perspective on the origin of cellular spontaneous firing activity.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Applied
Paul Carter, Arjen Doelman, Kaitlynn Lilly, Erin Obermayer, Shreyas Rao
Summary: This paper studies a class of singularly perturbed 2-component reaction-diffusion equations that have bistable traveling front solutions, which exhibit sharp, slow-fast-slow interfaces between stable homogeneous rest states. In many example systems, such as desertification fronts in dryland ecosystems, these fronts can exhibit an instability that leads to fingering patterns. The paper proposes two versions of a 2D stability criterion for long wavelength perturbations along the interface of these slow-fast-slow fronts, and provides examples illustrating the existence and (in)stability of traveling fronts in different systems/models.
PHYSICA D-NONLINEAR PHENOMENA
(2023)
Article
Mathematics, Applied
Kun Zhu, Jianhe Shen
Summary: In this paper, all traveling wave solutions in the generalized Degasperis-Procesi (gDP) equation are classified using the singular traveling wave method. The existence parameter conditions for these waves are obtained, and the persistence of solitary wave solutions under singular perturbation is analyzed by combining geometric singular perturbation theory with an explicit Melnikov method. Through explicit calculation of homoclinic orbits and associated Melnikov integral, the persistence of solitary wave solutions and determination of wave speed to leading order are demonstrated.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Engineering, Mechanical
Lijun Zhang, Jundong Wang, Elena Shchepakina, Vladimir Sobolev
Summary: In this paper, the perturbed mK(3,1) equation was restudied using the geometric singular perturbation theorem and bifurcation analysis to explore the dynamics of solitary wave solutions. It was found that the equation has a new family of solitary waves that decay to constants determined by wave speeds and a parameter, as well as a new type of solitary waves with coexisting crest and trough. The theoretical results were confirmed through numerical simulations.
NONLINEAR DYNAMICS
(2021)
Article
Mathematics, Applied
Shimin Li, Xiaoling Wang, Xiaoli Li, Kuilin Wu
Summary: This paper explores a Leslie-type model with two characteristic time scales, showing the stability of two relaxation oscillations in a predator-prey system with Holling type I functional response function. The numerical simulations further support the analytic results on the coexistence of these oscillations.
APPLIED MATHEMATICS LETTERS
(2021)
