Determination of the Basin of Attraction in Dynamical Systems using Meshless Collocation
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Thu, 04/02/2010 14:00 |
Dr Peter Giesl (University of Sussex) |
Computational Mathematics and Applications  |
3WS SR |
| In dynamical systems given by an ODE, one is interested in the basin of attraction of invariant sets, such as equilibria or periodic orbits. The basin of attraction consists of solutions which converge towards the invariant set. To determine the basin of attraction, one can use a solution of a certain linear PDE which can be approximated by meshless collocation. The basin of attraction of an equilibrium can be determined through sublevel sets of a Lyapunov function, i.e. a scalar-valued function which is decreasing along solutions of the dynamical system. One method to construct such a Lyapunov function is to solve a certain linear PDE approximately using Meshless Collocation. Error estimates ensure that the approximation is a Lyapunov function. The basin of attraction of a periodic orbit can be analysed by Borg’s criterion measuring the time evolution of the distance between adjacent trajectories with respect to a certain Riemannian metric. The sufficiency and necessity of this criterion will be discussed, and methods how to compute a suitable Riemannian metric using Meshless Collocation will be presented in this talk. | |||