Homoclinic solutions in mechanical systems with small dissipation. Application to DNA dynamics
type: Item
instance of: scholarly article
title: Homoclinic solutions in mechanical systems with small dissipation. Application to DNA dynamics
zbMATH Open document ID: 1011.34033
author: Bogdan Kazmierczak
author: Tomasz Lipniacki
published in: Journal of Mathematical Biology
publication date: 2002-07-29T00:00:00Z
review text: The authors analyze the persistence of homoclinic solutions to perturbed systems of second-order ODEs \(x''=F(x(t))\), with \(x\in \mathbb{R}^n\) and \(t\in \mathbb{R}^1\). Such equations may arise, e.g., from PDEs, when looking for solutions in the form of traveling waves. Solitary travelling waves correspond to homoclinic or heteroclinic solutions to the corresponding ODEs. Assuming that the original system has a homoclinic solution, the interest is in its persistence for perturbed systems of the type \[ x''=F(x(t))-\delta{\mathcal D}x^\prime(t)+ \mu\psi(x(t),x^\prime(t),t+T), \] where \(\delta,\mu\in \mathbb{R}^1, \psi:\mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R}^1\) is a smooth function satisfying some conditions, \(T\in \mathbb{R}^1\), \({\mathcal D}\) is a diagonal \(n\times n\)-matrix with positive entries \(d_1,\dots,d_n\). It is known that in general the perturbed system ceases to have a homoclinic solution. In opposite, for the considered small perturbations, under some additional assumptions, it is shown that the homoclinic solutions are preserved. An application to the DNA dynamics is given. The existence of dissipative travelling waves along the DNA molecule is proved.
reviewed by: Ivan Ginchev
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 92D20
zbMATH DE Number: 1773189
zbMATH Keywords: homoclinic solutions
zbMATH Keywords: small dissipations
zbMATH Keywords: perturbed second-order ordinary differential equation
zbMATH Keywords: DNA dynamics
MaRDI profile type: Publication
Wikidata QID: Q52041346
full work available at URL: s002850100131
OpenAlex ID: W2043809793
DOI: 10.1007/S002850100131
description: scientific article; zbMATH DE number 1773189