Homoclinic orbits in Hamiltonian systems and heteroclinic orbits in gradient and gradient-like systems
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 relationship between some heteroclinic orbits of gradient-like systems and some homoclinic orbits of Hamiltonian systems. In particular, for \(\theta\neq 0\) there exists a simple Morse decomposition of the set of bounded solutions to (1) and there is a simple symmetry relation between these solutions when \(\theta =\theta_ 0\) and \(\theta =-\theta_ 0\). Although at \(\theta =0\) the Morse decomposition changes dramatically, the author succeeds to exploit the symmetry and to obtain results relating the structure of the set of bounded solutions of (1) for \(\theta =0\) and for \(\theta\neq 0.\) Although the results obtained are independent of the dimension of (1), the indices of the isolating neighborhoods are determined essentially from considering the simplest case, where \(x,y\in {\mathbb{R}}\). This shows the usefulness of isolating neighborhoods: in a system that is genuinely 2n-dimensional, by using an isolating neighborhood one is able to concentrate only on a small subset of the system and show that the flow on this subset is similar to that in an essentially 2-dimensional system.
homoclinic orbits in Hamiltonian systems
scientific article; zbMATH DE number 4132599
scientific article