Pattern Formation in a Model of an Injured Nerve Fiber

Aronson, D. G.; Padrón, Víctor

doi:10.1137/080732341

Cited by 6 publications

(9 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where the parameter λ is strictly positive, is obtained from the Michelson system by performing a linear change of variables followed by the change of function x 2 → |x|. In [5,6] the authors studied some global connections of system (2). Particularly, they give an analytic proof of the existence of a pair of homoclinic connections and a T-point heteroclinic cycle.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…The main aim of the present work is to analytically prove the local appearance of the RP4-orbits that emerge from the RP2-orbit corresponding to the crossing tangency in system (2). The following theorem, which is the core of the paper, establishes their existence.…”

Section: Theorem 1 (Carmona Et Al [4]mentioning

confidence: 95%

“…). There exists a value λ = λ C > 0 such that system (2) has an RP2-orbit with period less than 4π, which crosses the separation plane tangentially through the origin.…”

Section: Theorem 1 (Carmona Et Al [4]mentioning

confidence: 99%

See 2 more Smart Citations

Noose bifurcation and crossing tangency in reversible piecewise linear systems

et al. 2014

View full text Add to dashboard Cite

The structure of the periodic orbits analysed in this paper was first reported in the Michelson system by Kent and Elgin (1991 Nonlinearity 4 1045-61), who gave it the name noose bifurcation. Recently, it has been found in a piecewise linear system with two linearity zones separated by a plane, which is called the separation plane. In this system, the orbits that take part in the noose bifurcation have two and four points of intersection with the separation plane, and they are arranged in two curves that are connected by a point where the periodic orbit has a crossing tangency with the separation plane. In this work, we analytically prove the local existence of the curve of periodic orbits with four intersections that emerges from the point corresponding to the crossing tangency. Moreover, we add a numerical study of the stability and bifurcations of the periodic orbits involved in the noose curve for the piecewise linear system and check that they exhibit the same configuration as that of the Michelson system.

show abstract

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Theorem 1 (Carmona Et Al [4]mentioning

confidence: 95%

See 1 more Smart Citation

Noose bifurcation and crossing tangency in reversible piecewise linear systems

et al. 2014

View full text Add to dashboard Cite

show abstract

“…Finally, we briefly mention another area of research where similar equations arise, that is in the context of wave propagation or stationary solutions for bistable reaction diffusion equations in excitable media (see, for instance, [24][25][26][27] and the references therein). In the above quoted papers, a typical one-dimensional model equation takes the form = + ( , ) ,…”

Section: Introductionmentioning

confidence: 99%

Complex Dynamics in One‐Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

Zanini

Zanolin

2018

Complexity

View full text Add to dashboard Cite

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated.

show abstract

“…Since then, the applicability of these systems has been fully demonstrated. They have been used to model not only dynamical processes coming from engineering, but also, for instance, social behaviors and financial or biological problems [3,12,13,25]. The analysis of PWL systems revealed that they exhibit rich dynamics as smooth systems, in particular limit cycles and periodic orbits [15,17], homoclinic and heteroclinic connections [5], and strange attractors [25].…”

mentioning

confidence: 99%

A Multiple Time Scale Coupling of Piecewise Linear Oscillators. Application to a Neuroendocrine System

Fernández-García¹,

Desroches²,

Krupa³

et al. 2015

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

We analyze a four-dimensional slow-fast piecewise linear system consisting of two coupled McKean caricatures of the FitzHugh-Nagumo system. Each oscillator is a continuous slow-fast piecewise linear system with three zones of linearity. The coupling is one-way, that is, one subsystem evolves independently and is forcing the other subsystem. In contrast to the original FitzHugh-Nagumo system, we consider a negative slope of the linear nullcline in both the forcing and the forced system. In the forcing system, this lets us, by just changing one parameter, pass from a system having one equilibrium and a relaxation cycle to a system with three equilibria keeping the relaxation cycle. Thus, we can easily control the changes in the oscillation frequency of the forced system. The case with three equilibria and a linear slow nullcline is a new configuration of the McKean caricature, where the existence of the relaxation cycle was not studied previously. We also consider a negative slope of the y-nullcline in the forced system that enables us to reproduce a quasi-steady state called the surge. We analyze not only the qualitative behavior of the four-dimensional system, but also quantitative aspects such as the period, frequency, and amplitude of the oscillations. The system is used to reproduce all the features endowed in a former smooth model and reproduce the secretion pattern of the hypothalamic neurohormone GnRH along the ovarian cycle in different species. Introduction.Piecewise linear (PWL) systems are a family of nonsmooth systems well known for reproducing the dynamics of models coming from applications. The first examples of PWL systems were developed for the modeling of engineering problems (such as mechanical, electronic, and control device problems) [1]. Since then, the applicability of these systems has been fully demonstrated. They have been used to model not only dynamical processes coming from engineering, but also, for instance, social behaviors and financial or biological problems [3,12,13,25]. The analysis of PWL systems revealed that they exhibit rich dynamics as smooth systems, in particular limit cycles and periodic orbits [15,17], homoclinic and heteroclinic connections [5], and strange attractors [25].The celebrated FitzHugh-Nagumo system [14,27] models, in first approximation, the behavior of an excitable system, for example, a neuron. Basically, it is assumed that a neuron behaves as an electronic circuit. The van der Pol oscillator [2] can be considered as a special

show abstract

Pattern Formation in a Model of an Injured Nerve Fiber

Cited by 6 publications

References 10 publications

Noose bifurcation and crossing tangency in reversible piecewise linear systems

Noose bifurcation and crossing tangency in reversible piecewise linear systems

Complex Dynamics in One‐Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

A Multiple Time Scale Coupling of Piecewise Linear Oscillators. Application to a Neuroendocrine System

Contact Info

Product

Resources

About