Peter Giesl - Research

I am working in the following areas: Differential equations model many important processes in biology, physics and other sciences. The theory of Dynamical Systems studies the qualitative properties and long-time behaviour of solutions. Starting with special solutions such as equilibria (constant solutions) or periodic solutions, one investigates whether the solution is stable and attractive, i.e. nearby solutions tend to the special one, and tries to determine its basin of attraction, which consists of all solutions which tend to the special one as time goes to infinity. I am particularly interested in numerical methods for the construction of Lyapunov functions and have recently published a Review on Computational Methods for Lyapunov Functions in Discrete Contin. Dyn. Syst. Ser. B 20 No. 8 (2015), 2291-2331
with Sigurdur Hafstein.

Construction of Lyapunov Functions using Meshless Collocation

In order to determine the basin of attraction of an equilibrium for a concrete equation, the method of a Lyapunov function can be used. A Lyapunov function is characterised by its negative orbital derivative, i.e. the derivative along solutions of the differential equation. In general, the explicit construction of a Lyapunov function is very difficult. I have developed a general method to construct a Lyapunov function. This method uses meshless collocation, in particular Radial Basis Functions to approximate Lyapunov functions with a certain orbital derivative. In particular, one approximately solves a linear partial differential equation. The research includes the description and analysis of the method as well as error bounds. The method has been developed for equilibria of autonomous and time-periodic ordinary differential equations, as well as for discrete dynamical systems, non-smooth systems and asymptotically autonomous systems. This is partly joint work with Holger Wendland, then Oxford, supported by an EPSRC Small Grant. Together with my PhD student Najla Mohammed, we have studied grid refinement and the verification of the constructed Lyapunov function. In a current research project with Sigurdur Hafstein, Holger Wendland and Chris Kellett, we are developing a numerical method to construct complete Lyapunov function, describing the complete dynamics of a system, not only the basin of attraction of one particular attractor.

Construction of Lyapunov Functions using Linear Programming (Optimisation)

Another method for the construction of Lyapunov functions uses linear programming. The Lyapunov function in this case is a piecewise linear function and the optimisation conditions guarantee that its orbital derivative is negative. In a series of paper, we have revised the CPA method to construct Lyapunov functions with negative orbital derivative also near the equilibrium. Recently, we have combined the advantages of the construction methods using meshless collocation and linear programming. This is joint work with Sigurdur Hafstein, Reykjavik, Iceland. Together with my PhD student Julia Jackiewicz we are currently studying Lyapunov functions for time-periodic systems.

Contraction metric

One of my main interests is the determination of periodic orbits in dynamical systems. I further developed a contraction criterion for autonomous differential equations which measures whether adjacent solutions will approach each other with respect to a given Riemannian metric. With the criterion, one can prove existence, uniqueness and stability of a periodic orbit, and determine subsets of its basin of attraction. I have generalized this criterion for unbounded sets of the phase space and proved its necessity. Moreover, I have applied it to time-periodic, almost periodic, as well as to nonsmooth differential equations. This is partly joint work with Martin Rasmussen, Imperial College. Supported by an EPSRC Small Grant, I have developed the first constructive method to find a contraction metric using semi-definite optimisation; this is joint work with Sigurdur Hafstein, Reykjavik, Iceland. Recently, I have worked on converse theorems, proving the existence of contraction metrics with various properties for an equilibrium. Following on from these results, I have used meshless collocation to solve a matrix-valued PDE and thus to construct a contraction metric for an equilibrium point (joint work with Holger Wendland).

Stability in Stochastic Differential Equations and systems with noise

We are studying Lyapunov functions for Stochastic Differential Equations and their implications on stochastic stability. This work involves both new definitions on a stochastic equivalent of a basin of attraction and various numerical methods to compute (stochastic) Lyapunov functions; this is joint work with Sigurdur Hafstein and Enrico Scalas. In applications, the right-hand side of a differential equation is not always known, but could be derived from measured data, which is contaminated with noise. Together with Boumediene Hamzi, Martin Rasmussen and Kevin Webster we study the construction of Lyapunov functions for such systems using meshless collocation.

Attractivity over Finite-time intervals

While dynamical systems are interested in the long-time behaviour of systems, real-world applications are more concerned about the development over a finite time. Together with Martin Rasmussen, Imperial, I have achieved an important characterisation of basins and areas of attractions; the latter one is a set of attractive solutions. Beyond this theoretical result, I have developed a construction method for finite-time Lyapunov functions characterising the basin of attraction. Together with my PhD student James McMichen, we have developed a construction method for a Riemannian metric satisfying the contraction criterion using meshless collocation.

Biomechanics of the human muscle-skeletal system

I have studied the biomechanical properties of the muscle-skeletal system and their stabilizing effect on the human elbow joint. This includes a model of the elbow joint by a system of differential equations, the analysis of the influence of a weight held in the hand as well as the existence and the meaning of real and complex eigenvalues. Another application deals with the optimization of different parameters concerning the stability of juggling. This is partly joint work with Heiko Wagner, Münster, and Katja Mombaur, Heidelberg, Germany. Currently, together with my former PhD student Pascal Stiefenhofer and Heiko Wagner, we are developing a method to prevent falling in elderly persons.

Calculating the Basin of Attraction in Asymptotically Autonomous Dynamical Systems Results and Matlab files of an EPSRC funded project

Curriculum Vitae

Publications of P. Giesl

Back to the Homepage of P. Giesl

Department of Mathematics

Page last updated 29/11/2016