scholarly journals Homoclinic Orbits of the FitzHugh–Nagumo Equation: Bifurcations in the Full System

2010 ◽  
Vol 9 (1) ◽  
pp. 138-153 ◽  
Author(s):  
John Guckenheimer ◽  
Christian Kuehn
Keyword(s):  
Homoclinic Orbits ◽  
Full System ◽  
Nagumo Equation

scholarly journals Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit

2009 ◽  
Vol 2 (4) ◽  
pp. 851-872 ◽  
Author(s):  
John Guckenheimer ◽  
◽  
Christian Kuehn ◽  
Keyword(s):  
Homoclinic Orbits ◽  
Singular Limit ◽  
Nagumo Equation

scholarly journals SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1074
Author(s):  
Vincent Wagner ◽  
Nicole Erika Radde
Keyword(s):  
Reaction System ◽  
Stochastic Simulation Algorithm ◽  
Test Bed ◽  
Biochemical Reaction Networks ◽  
Full System ◽  
Catalytic Systems ◽  
Reaction Systems ◽  
Sample Paths ◽  
Alternative Description ◽  
Linear Differential

The Chemical Master Equation is a standard approach to model biochemical reaction networks. It consists of a system of linear differential equations, in which each state corresponds to a possible configuration of the reaction system, and the solution describes a time-dependent probability distribution over all configurations. The Stochastic Simulation Algorithm (SSA) is a method to simulate sample paths from this stochastic process. Both approaches are only applicable for small systems, characterized by few reactions and small numbers of molecules. For larger systems, the CME is computationally intractable due to a large number of possible configurations, and the SSA suffers from large reaction propensities. In our study, we focus on catalytic reaction systems, in which substrates are converted by catalytic molecules. We present an alternative description of these systems, called SiCaSMA, in which the full system is subdivided into smaller subsystems with one catalyst molecule each. These single catalyst subsystems can be analyzed individually, and their solutions are concatenated to give the solution of the full system. We show the validity of our approach by applying it to two test-bed reaction systems, a reversible switch of a molecule and methyltransferase-mediated DNA methylation.

Higher order accurate numerical solution of Nagumo equation

2021 ◽  
Author(s):  
Sobin Thomas ◽  
Gopika P. B. ◽  
Suresh Kumar Nadupuri
Keyword(s):  
Numerical Solution ◽  
Higher Order ◽  
Nagumo Equation ◽  
Accurate Numerical Solution

A Full-System Approach of the Elastohydrodynamic Line/Point Contact Problem

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
W. Habchi ◽  
D. Eyheramendy ◽  
P. Vergne ◽  
G. Morales-Espejel
Keyword(s):  
Finite Element ◽  
Nonlinear System ◽  
System Approach ◽  
Jacobian Matrix ◽  
Point Contact ◽  
System Of Equations ◽  
Nonlinear System Of Equations ◽  
Full System ◽  
Element Discretization ◽  
Fully Coupled

The solution of the elastohydrodynamic lubrication (EHL) problem involves the simultaneous resolution of the hydrodynamic (Reynolds equation) and elastic problems (elastic deformation of the contacting surfaces). Up to now, most of the numerical works dealing with the modeling of the isothermal EHL problem were based on a weak coupling resolution of the Reynolds and elasticity equations (semi-system approach). The latter were solved separately using iterative schemes and a finite difference discretization. Very few authors attempted to solve the problem in a fully coupled way, thus solving both equations simultaneously (full-system approach). These attempts suffered from a major drawback which is the almost full Jacobian matrix of the nonlinear system of equations. This work presents a new approach for solving the fully coupled isothermal elastohydrodynamic problem using a finite element discretization of the corresponding equations. The use of the finite element method allows the use of variable unstructured meshing and different types of elements within the same model which leads to a reduced size of the problem. The nonlinear system of equations is solved using a Newton procedure which provides faster convergence rates. Suitable stabilization techniques are used to extend the solution to the case of highly loaded contacts. The complexity is the same as for classical algorithms, but an improved convergence rate, a reduced size of the problem and a sparse Jacobian matrix are obtained. Thus, the computational effort, time and memory usage are considerably reduced.

scholarly journals A Melnikov Method for Homoclinic Orbits with Many Pulses

1998 ◽  
Vol 143 (2) ◽  
pp. 105-193 ◽  
Author(s):  
Roberto Camassa ◽  
Gregor Kovačič ◽  
Siu-Kei Tin
Keyword(s):  
Melnikov Method ◽  
Homoclinic Orbits

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

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