Homoclinic solutions in mechanical systems with small dissipation. Application to DNA dynamics
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.
perturbed second-order ordinary differential equation
scientific article; zbMATH DE number 1773189