Peter Giesl

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The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration xn+1 = g(xn) can be determined through sublevel sets of a Lyapunov function. In [3] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov(More)
In this paper, we derive error estimates for generalized interpolation, in particular collocation, in Sobolev spaces. We employ our estimates to collo-cation problems using radial basis functions and extend and improve previously known results for elliptic problems. Finally, we use meshless colloca-tion to approximate Lyapunov functions for dynamical(More)
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This paper provides an explicit Lyapunov function for a general single-joint muscle-skeletal model. Using this Lyapunov function one can determine analytically large subsets of the basin of attraction of an asymptotically stable equilibrium. Besides providing an analytical tool for the analysis of such a system we consider an elbow model and show that the(More)
— An integral part of the CPA method to compute Continuous and Piecewise Affine Lyapunov functions for nonlinear systems is the generation of a suitable triangulation. Recently, the CPA method was revised by using more advanced triangulations and it was proved that it can compute a CPA Lya-punov function for any nonlinear system possessing an exponentially(More)
— The numerical construction of Lyapunov functions provides useful information on system behavior. In the Continuous and Piecewise Affine (CPA) method, linear programming is used to compute a CPA Lyapunov function for continuous nonlinear systems. This method is relatively slow due to the linear program that has to be solved. A recent proposal was to(More)
In [3] Conley showed that the state-space of a dynamical system can be decomposed into a gradient-like part and a chain-recurrent part, and that this decomposition is characterized by a so-called complete Lyapunov function for the system. In [14] Kalies, Mischaikow, and VanderVorst proposed a combinatorial method to compute discrete approximations to such(More)
Consider a dynamical system given by a system of autonomous ordinary differential equations. In this paper we provide a sufficient local condition for an unbounded subset of the phase space to belong to the basin of attraction of a limit cycle. This condition also guarantees the existence and uniqueness of such a limit cycle, if that subset is compact. If(More)
Lyapunov functions are an important tool to determine the basin of attraction of equilibria in Dy-namical Systems through their sublevel sets. Recently, several numerical construction methods for Lyapunov functions have been proposed, among them the RBF (Radial Basis Function) and CPA (Continuous Piecewise Affine) methods. While the first method lacks a(More)