Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation

We consider a non-autonomous ordinary differential equation over a finite time interval \begin{document}$[T_1,T_2]$\end{document} . The area of exponential attraction consists of solutions such that the distance to adjacent solutions exponentially contracts from \begin{document}$T_1$\end{document} to \begin{document}$T_2$\end{document} . One can use a contraction metric to determine an area of exponential attraction and to provide a bound on the rate of attraction. In this paper, we will give the first method to algorithmically construct a contraction metric for finite-time systems in one spatial dimension. We will show the existence of a contraction metric, given by a function which satisfies a second-order partial differential equation with boundary conditions. We then use meshless collocation to approximately solve this equation, and show that the resulting approximation itself defines a contraction metric, if the collocation points are sufficiently dense. We give error estimates and apply the method to an example.