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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 collocation problems using radial basis functions and extend and improve previously known results for elliptic problems. Finally, we use meshless collocation to approximate Lyapunov functions for dynamical systems.
Recently the authors proved the existence of piecewise affine Lyapunov functions for dynamical systems with an exponentially stable equilibrium in two dimensions [7]. Here, we extend these results by designing an algorithm to explicitly construct such a Lyapunov function. We do this by modifying and extending an algorithm to construct Lyapunov functions(More)
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 Lyapunov function for any nonlinear system possessing an exponentially(More)
Lyapunov functions are a main tool to determine the domain of attraction of equilibria in dynamical systems. Recently, several methods have been presented to construct a Lyapunov function for a given system. In this paper, we improve the construction method for Lyapunov functions using Radial Basis Functions. We combine this method with a new grid(More)
We present a novel method to compute Lyapunov functions for continuous-time systems with multiple local attractors. In the proposed method one first computes an outer approximation of the local attractors using a graphtheoretic approach. Then a candidate Lyapunov function is computed using a Massera-like construction adapted to multiple local attractors. In(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 compute(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)