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Approximating the basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem

Pages: 259 - 268, Issue Special, September 2009

 Abstract        Full Text (220.5K)              

Peter Giesl - Department of Mathematics, University of Sussex, Brighton, BN1 9RF, United Kingdom (email)
Holger Wendland - Department of Mathematics, University of Sussex, Brighton, BN1 9RF, United Kingdom (email)

Abstract: The basin of attraction of an equilibrium or periodic orbit can be determined by sublevel sets of a Lyapunov function. A Lyapunov function is a function with negative orbital derivative, which is defined by $LV(t,x)= {\nabla_x V(t,x),f(t,x)} + \partial_t V(t,x)$. 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 are periodic 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.

Keywords:  periodic differential equation, periodic orbit, basin of attraction, Lyapunov function, partial differential equation, meshless collocation, radial basis function
Mathematics Subject Classification:  Primary: 37C55, 65N35; Secondary: 34C25, 65N15

Received: July 2008;      Revised: April 2009;      Published: September 2009.