Sigurdur F. Hafstein

Efficient Sampling of Saddle Points with the Minimum-Mode Following Method

The problem of sampling low lying, first-order saddle points on a high dimensional surface is discussed and a method presented for improving the sampling efficiency. The discussion is in the context of an energy surface for a system of atoms and thermally activated transitions in solids treated within the harmonic approximation to transition state theory. Given a local minimum as an initial state and a small, initial displacement, the minimum-mode following method is used to climb up to a saddle point. The goal is to sample as many of the low lying saddle points as possible when such climbs are repeated from different initial displacements. Various choices for the distribution of initial displacements are discussed and a comparison made between (1)displacements along eigenmodes at the minimum, (2)purely random displacements with a maximum cutoff, and (3) Gaussian distribution of displacements. The last choice is found to give best overall results in two test problems studied, a heptamer island on a surface and a grain boundary in a metal. A method referred to as “skipping-path method” is presented to reduce redundant calculations when a climb heads towards a saddle point that has already been identified. The method is found to reduce the computational effort of finding new saddle points to as little as a third, especially when a thorough sampling is performed.

Computation of Lyapunov functions for nonlinear discrete time systems by linear programming

Given an autonomous discrete time system with an equilibrium at the origin and a hypercube containing the origin, we state a linear programming problem, of which any feasible solution parameterizes a continuous and piecewise affine (CPA) Lyapunov function for the system. The linear programming problem depends on a triangulation of the hypercube. We prove that if the equilibrium at the origin is exponentially stable, the hypercube is a subset of its basin of attraction, and the triangulation fulfils certain properties, then such a linear programming problem possesses a feasible solution. We present an algorithm that generates such linear programming problems for a system, using more and more refined triangulations of the hypercube. In each step the algorithm checks the feasibility of the linear programming problem. This results in an algorithm that is always able to compute a Lyapunov function for a discrete time system with an exponentially stable equilibrium. The domain of the Lyapunov function is only limited by the size of the equilibriums basin of attraction. The system is assumed to have a right-hand side, but is otherwise arbitrary. Especially, it is not assumed to be of any specific algebraic type such as linear, piecewise affine and so on. Our approach is a non-trivial adaptation of the CPA method to compute Lyapunov functions for continuous time systems to discrete time systems.

Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction

We present a novel numerical technique for the computation of a Lyapunov function for nonlinear systems with an asymptotically stable equilibrium point. Our proposed approach constructs a continuous piecewise affine (CPA) function given a suitable partition of the state space, called a triangulation, and values at the vertices of the triangulation. The vertex values are obtained from a Lyapunov function in a classical converse Lyapunov theorem and verification that the obtained CPA function is a Lyapunov function is shown to be equivalent to verification of several simple inequalities. Furthermore, by refining the triangulation, we show that it is always possible to construct a CPA Lyapunov function. Numerical examples are presented demonstrating the effectiveness of the proposed method.

Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction

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 the CPA Lyapunov function based on a Lyapunov function in a converse Lyapunov theorem by Yoshizawa. In this paper we propose computing CPA Lyapunov functions using a Lyapunov function construction in a classic converse Lyapunov theorem by Massera. We provide the theory for such a computation and present several examples to illustrate the utility of this approach.

Computation and Verification of Lyapunov Functions

Lyapunov functions are an important tool to determine the basin of attraction of equilibria in Dynamical 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 verification that the constructed function is a valid Lyapunov function, the second method is rigorous, but computationally much more demanding. In this paper, we propose a combination of these two methods, using their respective strengths: we use the RBF method to compute a potential Lyapunov function. Then we interpolate this function by a CPA function. Checking a finite number of inequalities, we are able to verify that this interpolation is a Lyapunov function. Moreover, sublevel sets are arbitrarily close to the basin of attraction. We show that this combined method always succeeds in computing and verifying a Lyapunov function, as well as in determini...

Computation of Lyapunov functions for discrete-time systems using the Yoshizawa construction

In this paper, we present a new approach for computing Lyapunov functions for nonlinear discrete-time systems with an asymptotically stable equilibrium at the origin. The proposed method constructs a continuous piecewise affine (CPA) function on a compact subset of the state space containing the origin, given a suitable triangulation or partition of the compact set and values at the vertices of the triangulation. Here, the vertex values are fixed using a function from a classical converse Lyapunov theorem originally due to Yoshizawa. Several numerical examples are presented to illustrate the proposed approach.

TRAFFIC FORECAST IN LARGE SCALE FREEWAY NETWORKS

Traffic flow in large and complex freeway networks is a highly nonlinear phenomenon, which makes traffic forecast a difficult task. In this article an approach to traffic forecast is presented, whi...

Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems

In this paper, we present a new approach for computing Lyapunov functions for nonlinear discrete-time systems with an asymptotically stable equilibrium at the origin. Given a suitable triangulation of a compact neighbourhood of the origin, a continuous and piecewise affine function can be parameterized by the values at the vertices of the triangulation. If these vertex values satisfy system-dependent linear inequalities, the parameterized function is a Lyapunov function for the system. We propose calculating these vertex values using constructions from two classical converse Lyapunov theorems originally due to Yoshizawa and Massera. Numerical examples are presented to illustrate the proposed approach.

Implementation of a fan-like triangulation for the CPA method to compute Lyapunov functions

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 stable equilibrium. This more advanced triangulation scheme includes a simplicial fan close to the equilibrium of the system, for which the Lyapunov function is computed. In this paper we prove a result that allows for a simpler, more general and more efficient generation of this simplicial fan and thus improves the implementation of the CPA method. Moreover, the simplicial fan subdivision at the origin is also of great importance in methods relying on a conic decomposition of the state-space.

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions.

Ordinary differential equations arise in a variety of applications, including e.g. climate systems, and can exhibit complicated dynamical behaviour. Complete Lyapunov functions can capture this behaviour by dividing the phase space into the chain-recurrent set, determining the long-time behaviour, and the transient part, where solutions pass through. In this paper, we present an algorithm to construct complete Lyapunov functions. It is based on mesh-free numerical approximation and uses the failure of convergence in certain areas to determine the chain-recurrent set. The algorithm is applied to three examples and is able to determine attractors and repellers, including periodic orbits and homoclinic orbits.