Approximating the basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem
Peter Giesl, Holger Wendland · AIMS Press eBooks · 2009
The basin of attraction of an equilibrium or periodic orbit can be determined by sublevel sets of a Lyapunov function. A Lyapunovfunction is a function with negative orbital derivative, which is defined by $LV(t,x)= { abla_x V(t,x),f(t,x)} + \partial_t V(t,x)$.We construct a Lyapunov function by approximately solving a Cauchy problem with a linear PDE for its orbital derivative and boundary conditions on a non-characteristic hypersurface. For the approximation we use meshless collocation. We describe the general approximate reconstruction of multivariate functions, which areperiodic in one variable, from discrete data sets and derive error estimates.This method has already been applied to autonomous dynamical systems. In this paper, however, we consider a time-periodic ODE $\dot{x}=f(t,x)$, $x\in \mathbb R^n$, and study the basin of attraction of an exponentially asymptotically stable periodic orbit.