Homoclinic orbits in Hamiltonian systems and heteroclinic orbits in gradient and gradient-like systems
type: Item
instance of: scholarly article
title: Homoclinic orbits in Hamiltonian systems and heteroclinic orbits in gradient and gradient-like systems
zbMATH Open document ID: 0692.34029
DOI: 10.1016/0022-0396(89)90182-4
author: K. Mischaikow
published in: Journal of Differential Equations
publication date: 1989-01-01T00:00:00Z
review text: The author describes a relationship between homoclinic orbits in Hamiltonian systems, heteroclinic orbits in damped systems, and heteroclinic orbits in gradient systems. He considers the family of systems (1) \(\dot x=y\), \(\dot y=\theta y-\nabla V(x)\), where \(x,y\in {\mathbb{R}}^ n\), and \(\theta\in {\mathbb{R}}\) is taken to be a parameter value. For \(\theta =0\), (1) is a Hamiltonian system with a Hamiltonian function \(H(x,y)=(1/2)<y,y>+V(x).\) For \(\theta\neq 0\), (1) is called a damped system, because \(dH/dt=\theta <y,y>\), i.e., energy is no longer conserved. It is shown that, when letting \(\theta\) \(\to \infty\), one studies the heteroclinic solutions of \(\dot x=\nabla V(x).\) The results given in this paper extend those of (*) \textit{K. Mischaikow} [Classification of traveling wave solutions of reaction diffusion systems (preprint, LCDS No.86-5, Brown University) (1985)], where the connection matrix was used to classify the possible traveling wave solutions of a system of reaction diffusion equations of the form \((2)\quad x_{\tau}=x_{\alpha \alpha}+\nabla V(x).\) A traveling wave with wave speed \(\theta\) is a solution of the form \(x(t)=x(\alpha +\theta \tau)=x(\alpha,\tau).\) Finding traveling wave solutions with wave speed \(\theta\) is equivalent with finding bounded solutions of (1). Although (*) gives a classification scheme, it does not guarantee existence. The results given in the paper under review overcome this deficiency, at least for certain types of traveling wave solutions. As the author mentions, \textit{D. Terman} has studied similar problems [see, e.g., Trans. Am. Math. Soc. 308, No.1, 391-412 (1988; Zbl 0681.35046), ibid. 301, 537-556 (1987; Zbl 0657.35070)]. However, while Terman's technique involve examining the behavior of solutions to (1) as \(\theta\) \(\to 0\), in this paper, the author begins with the Hamiltonian system \(\theta =0\) and perturbs away from it to \(\theta >0\). The author determines an explicit relations…
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 4132599
zbMATH Keywords: homoclinic orbits in Hamiltonian systems
zbMATH Keywords: damped systems
zbMATH Keywords: heteroclinic orbits in gradient systems
zbMATH Keywords: traveling wave solutions
zbMATH Keywords: system of reaction diffusion equations
MaRDI profile type: Publication
full work available at URL: 0022-0396(89)90182-4
OpenAlex ID: W2089791893
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description: scientific article; zbMATH DE number 4132599