Geometric Singular Perturbation Theory Research Articles

Oscillatory and transient dynamics of a slow–fast predator–prey system with fear and its carry-over effect

In almost every ecological system, growth of various interacting species evolve in different time scales and the implementation of this time scale difference in the corresponding mathematical model exhibits some rich and complex oscillatory dynamics. In this article, we consider a predator–prey model with Beddington–DeAngelis functional response in which the prey reproduction is affected by the predation induced fear and its carry-over effect. Considering the growth of prey species occurs on a faster time scale than that of predator, the proposed system reduces to a ‘slow–fast predator–prey’ system. Using the geometric singular perturbation theory and asymptotic expansion technique, we investigate the system both analytically and numerically, and observe a wide range of rich and complex dynamics such as canard cycles (with or without head) near the singular Hopf-bifurcation threshold and relaxation oscillation cycles. The system experiences a canard explosion through which a rapid transition from small amplitude limit cycle to large amplitude limit cycle occurs in a tiny parametric interval. These types of complex oscillatory dynamics are absent in non slow–fast systems. Furthermore, it is shown that the interplay between fear and its carry-over effect, and the variation of time scale parameter may lead to a regime shift of the oscillatory dynamics. We also study the impact of fear and its carry-over effect on the properties of long transient dynamics. Thus our study provides some valuable biological insights of a slow–fast predator–prey system which will aid in understanding the interplay between fear and its carry-over effect.

Persistence of Kink and Periodic Waves to Singularly Perturbed Two-Component Drinfel’d–Sokolov–Wilson System

This paper concerns the persistence of kink and periodic waves to singularly perturbed two-component Drinfel’d-Sokolov-Wilson system. Geometric singular perturbation theory is first employed to reduce the high-dimensional system to the perturbed planar system. By perturbation analysis and Abelian integrals theory, we then are able to find the sufficient conditions about the wave speed to guarantee the existence of heteroclinic orbit and periodic orbits, which indicates the existence of kink and periodic waves. Furthermore, we also show that the limit wave speed c_0(k) is increasing.

The solitary wave, kink and anti-kink solutions coexist at the same speed in a perturbed nonlinear Schrödinger equation

Persistence of the traveling wave solutions of a perturbed higher order nonlinear Schrödinger equation with distributed delay is studied by the geometric singular perturbation theory. The solitary wave, kink and anti-kink solutions are proved to coexist simultaneously at the same speed c by combing the Melnikov method and the bifurcation analysis. Interestingly, a new type of traveling wave solution possessing crest, trough and kink (anti-kink) is discovered. Further, numerical simulations are carried out to confirm the theoretical results.

Wave propagation and stabilization in the Boussinesq–Burgers system

This paper considers the existence and stability of traveling wave solutions of the Boussinesq–Burgers system describing the propagation of bores. Assuming the fluid is weakly dispersive, we establish the existence of three different wave profiles by the geometric singular perturbation theory alongside phase plane analysis. We further employ the method of weighted energy estimates to prove the nonlinear asymptotic stability of the traveling wave solutions against small perturbations. The technique of taking antiderivative is utilized to integrate perturbation functions because of the conservative structure of the Boussinesq–Burgers system. Using a change of variable to deal with the dispersion term, we perform numerical simulations for the Boussinesq–Burgers system to showcase the generation and propagation of various wave profiles in both weak and strong dispersions. The numerical simulations not only confirm our analytical results, but also illustrate that the Boussinesq–Burgers system can generate numerous propagating wave profiles depending on the profiles of initial data and the intensity of fluid dispersion, where in particular the propagation of bores can be generated from the system in the case of strong dispersion.

Persistence of kink and anti-kink wave solutions for the perturbed double sine-Gordon equation

Based on the geometric singular perturbation theory and the Melnikov method, we study the persistence of kink and anti-kink wave solutions for the perturbed double sine-Gordon equation. The explicit expression of the Melnikov function is given. Moreover, the monotonicity of the period function for unperturbed double sine-Gordon equation is investigated.

A Flux Ratio Independent of the Permanent Charge in PNP Models

PNP models with an arbitrary number of positively charged ion species and one negatively charged ion species are studied in this paper under the assumption that positively charged ion species have the same valence and the permanent charge is a piecewise constant function. The permanent charge plays the key role in many functions of an ion channel, such as selectivity and gating. In this paper, using the geometric singular perturbation theory, a flux ratio independent of the permanent charge is proved.

Existence of periodic wave for a perturbed MEW equation

<abstract><p>A perturbed MEW equation including small backward diffusion, dissipation and nonlinear term is considered by the geometric singular perturbation theory. Based on the monotonicity of the ratio of Abelian integrals, we prove the existence of periodic wave on a manifold for perturbed MEW equation. By Chebyshev system criterion, the uniqueness of the periodic wave is obtained. Furthermore, the monotonicity of the wave speed is proved and the range of the wave speed is obtained. Additionally, the monotonicity of period is given by Picard-Fuchs equation.</p></abstract>

Solitary waves of singularly perturbed generalized KdV equation with high order nonlinearity

<p style='text-indent:20px;'>The paper is concerned on solitary waves for singularly perturbed generalized KdV equation with high order nonlinear terms. We firstly give the phase portraits of system related to the unperturbed equation under various cases by theory of planar dynamical system. Then by using geometric singular perturbation theory and Melnikov's method, the existence of solitary wave solutions of generalized KdV equations with high order nonlinear terms is established. It is proven that some solitary wave solutions with particular wave speeds will persist under small perturbations.</p>

Asymptotic behaviors and existence of traveling wave solutions to the Keller-Segel model with logarithmic sensitivity

<p style='text-indent:20px;'>In this paper, we are concerned with the existence of traveling wave solutions for the Keller-Segel model with logarithmic sensitivity. By the Hopf-Cole transformation and traveling wave transformation, the degenerate Keller-Segel system is transformed into a singularly perturbed system. By constructing an invariant region to prove the existence of the traveling wave solutions for the degenerate system, we obtain the traveling wave solutions for Keller-Segel system with small parameter by using geometric singular perturbation theory and Fredholm theory. Finally, we discuss the asymptotic behaviors of the traveling wave solutions.</p>

Existence and local uniqueness of classical Poisson-Nernst-Planck systems with multi-component permanent charges and multiple cations

<p style='text-indent:20px;'>We consider a quasi-one-dimensional Poisson-Nernst-Planck system for ionic flow through a membrane channel. Multi-component nonzero permanent charges, the major structural quantity of an ion channel, are included in the model. The system includes <i>three</i> ion species, two cations with the same valences and one anion, which provides more correlations/interactions between ions compared to the case included only two oppositely charged particles. The cross-section area of the channel is included in the system, which provides certain information of the geometry of the three-dimensional channel. This is crucial for our analysis. Under the framework of geometric singular perturbation theory, more importantly, the specific structure of the model, the existence and local uniqueness of solutions to the system for nonzero permanent charges is established. Numerical simulations are performed to roughly investigate the effects on ionic flows in terms of both I-V relations and individual fluxes from nonzero permanent charges.</p>

Geometric singular perturbation of a nonlocal partially degenerate model for <i>Aedes aegypti</i>

<p style='text-indent:20px;'>This paper is devoted to investigate the existence of traveling wave solutions for a partially degenerate <i>Aedes aegypti</i> model with nonlocal effects. By taking specific kernel forms and time scale transformation, we transform the nonlocal model into a singularly perturbed system with small parameters. A locally invariant manifold for wave profile system is obtained with the aid of the geometric singular perturbation theory, and then the existence of traveling wave solutions is proved provided that the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_0>1 $\end{document}</tex-math></inline-formula> through utilizing the Fredholm orthogonal method. Furthermore, we study the asymptotic behaviors of traveling wave solution with the help of asymptotic theory. The methods used in this work can help us overcome the difficulty that the solution map associated with the system is not compact. Numerically, we perform simulations to demonstrate the theoretical results.</p>

EXISTENCE AND UNIQUENESS OF PERIODIC WAVES FOR A PERTURBED SEXTIC GENERALIZED BBM EQUATION

This paper is devoted to the existence and uniqueness of periodic waves for a perturbed sextic generalized BBM equation with weak backward diffusion and dissipation effects. By applying geometric singular perturbation theory and analyzing the perturbations of a Hamiltonian system with a hyper-elliptic Hamiltonian of degree seven, we prove the existence and uniqueness of periodic wave solutions with each wave speed in an open interval. It is also proved that the periodic wave solution persists for any energy parameter $h$ in an open interval and sufficiently small perturbation parameter. Furthermore, we prove that the wave speed $c_0(h)$ is strictly monotonically increasing with respect to $h$ by analyzing Abelian integral having three generating elements. Moreover, the upper and lower bounds of the limiting wave speed are obtained.

Single- and Double-Peak Solitary Waves of Two-Component Drinfel’d–Sokolov–Wilson System with Kuramoto–Sivashinsky Perturbation

Solitary wave solutions of two-component Drinfel’d–Sokolov–Wilson system with Kuramoto–Sivashinsky perturbation are considered. We first employ geometric singular perturbation theory to reduce the higher-dimensional system of equations to the perturbed planar system. We then further exploit the Melnikov method to explore the persistence of one homoclinic orbit, and the generation of a new homoclinic orbit, indicating the existence of single- and double-peak solitary waves. Of particular interest is the appearance of the double-peak solitary wave solution. Finally, we include the numerical simulations to verify the theoretical results.

Existence of traveling waves in a delayed convecting shallow water fluid model

<abstract><p>This paper investigates a delayed shallow water fluid model that has not been studied in previous literature. Applying geometric singular perturbation theory, we prove the existence of traveling wave solutions for the model with a nonlocal weak delay kernel and local strong delay convolution kernel, respectively. When the convection term contains a nonlocal weak generic delay kernel, the desired heteroclinic orbit is obtained by using Fredholm theory and linear chain trick to prove the existence of two kink wave solutions under certain parametric conditions. When the model contains local strong delay convolution kernel and weak backward diffusion, under the same parametric conditions to the previous case, the corresponding traveling wave system can be reduced to a near-Hamiltonian system. The existence of a unique periodic wave solution is established by proving the uniqueness of zero of the Melnikov function. Uniqueness is proved by utilizing the monotonicity of the ratio of two Abelian integrals.</p></abstract>

Effects of ion sizes on Poisson–Nernst–Planck systems with multiple ions

In this work, we analyze a one‐dimensional version of steady‐state Poisson–Nernst–Planck models for ionic flow through a membrane channel including ionic interactions modeled by the density functional theory. The model includes an arbitrary number of positively changed ions with the same valences and one negatively charged ion and ignores the permanent charge. The model can be viewed as a singularly perturbed differential system; therefore, our analysis is mainly based on the geometric singular perturbation theory. The existence and the uniqueness result for small ion sizes are established, and treating the sizes as small parameters, we also derive an approximation of the individual flux, the – (current–voltage) relation and the individual flux difference. Critical potentials are identified, and their roles in characterizing ionic flow properties are studied.

Slow-fast torus knots

The aim of this paper is to study global dynamics of $C^\infty$-smooth slow-fast systems on the $2$-torus of class $C^\infty$ using geometric singular perturbation theory and the notion of slow divergence integral. Given any $m\in\mathbb{N}$ and two relatively prime integers $k$ and $l$, we show that there exists a slow-fast system $Y_{\epsilon}$ on the $2$-torus that has a $2m$-link of type $(k,l)$, i.e. a (disjoint finite) union of $2m$ slow-fast limit cycles each of $(k,l)$-torus knot type, for all small $\epsilon>0$. The $(k,l)$-torus knot turns around the $2$-torus $k$ times meridionally and $l$ times longitudinally. There are exactly $m$ repelling limit cycles and $m$ attracting limit cycles. Our analysis: a) proves the case of normally hyperbolic singular knots, and b) provides sufficient evidence to conjecture a similar result in some cases where the singular knots have regular nilpotent contact with the fast foliation.

Solitary Waves of the Perturbed KdV Equation with Nonlocal Effects

In this paper, the Korteweg–de Vries (KdV) equation is considered, which is a shallow water wave model in fluid mechanic fields. First the existence of solitary wave solutions for the original KdV equation and geometric singular perturbation theory are recalled. Then the existence of solitary wave solutions is established for the equation with two types of delay convolution kernels by using the method of dynamical system, especially the geometric singular perturbation theory, invariant manifold theory and Melnikov method. Finally, the asymptotic behaviors of solitary wave solution are discussed by applying the asymptotic theory. Moreover, an interesting result is found for the equation without backward diffusion effect, there is no solitary wave solution in the case of local delay, but there is a solitary wave solution in the case of nonlocal delay.

Traveling waves and their spectral stability in Keller–Segel system with large cell diffusion

Keller–Segel system models prototype of the population-based chemotaxis, which describes the oriented or partially oriented movement of cells in response to a chemical signal produced by the cells themselves. This paper focuses on dynamics of a class of Keller–Segel systems with a large diffusion. We obtain several different kinds of traveling waves and characterize their spectral stability. The main tools are geometric singular perturbation theory, qualitative theory and spectral analysis via the method of nonlocal eigenvalue problem.

Dynamics in diffusive Leslie–Gower prey–predator model with weak diffusion

This paper is concerned with the diffusive Leslie–Gower prey–predator model with weak diffusion. Assuming that the diffusion rates of prey and predator are sufficiently small and the natural growth rate of prey is much greater than that of predators, the diffusive Leslie–Gower prey–predator model is a singularly perturbed problem. Using travelling wave transformation, we firstly transform our problem into a multiscale slow-fast system with two small parameters. We prove the existence of heteroclinic orbit, canard explosion phenomenon and relaxation oscillation cycle for the slow-fast system by applying the geometric singular perturbation theory. Thus, we get the existence of travelling waves and periodic solutions of the original reaction–diffusion model. Furthermore, we also give some numerical examples to illustrate our theoretical results.

A traveling wave bifurcation analysis of turbulent pipe flow

Using various techniques from dynamical systems theory, we rigorously study an experimentally validated model by [Barkley et al 2015 Nature 526 550–3], which describes the rise of turbulent pipe flow via a PDE system of reduced complexity. The fast evolution of turbulence is governed by reaction-diffusion dynamics coupled to the centerline velocity, which evolves with advection of Burgers’ type and a slow relaminarization term. Applying to this model a spatial dynamics ansatz and geometric singular perturbation theory, we prove the existence of a heteroclinic loop between a turbulent and a laminar steady state and establish a cascade of bifurcations of various traveling waves mediating the transition to turbulence. The most complicated behaviour can be found in an intermediate Reynolds number regime, where the traveling waves exhibit arbitrarily long periodic-like dynamics indicating the onset of chaos. Our analysis provides a systematic mathematical approach to identifying the transition to spatio–temporal turbulent structures that may also be applicable to other models arising in fluid dynamics.