Analysing Sensitivity in Basins of Attraction using a Local Contraction Criterion

In this project we will consider time-periodic ordinary differential equations (ODEs), which are a major modeling tool in all sciences, for example mechanical systems in physics with periodic forces, population dynamics in biology with seasonal influences or periodic movements of the human muscle-skeletal system like walking. A periodic solution is a solution that repeats itself after a certain time. Both for deriving the model and for its analysis, it is crucial to understand the dynamical properties of an ODE.

Dynamical systems are interested in the long-time behaviour of solutions. In particular, we want to determine and analyse the basin of attraction of a certain periodic solution consisting of all trajectories that finally show the periodic behaviour of this periodic solution. In particular, we will use a local contraction criterion which indicates whether trajectories of adjacent solutions approach each other with respect to a given notion of distance, called a Riemannian metric. Moreover, the metric indicates points which are particularly sensitive to perturbations. The challenge is to find a suitable Riemannian metric for a particular equation.

The main purpose of this project is to derive an algorithm that can always construct such a Riemannian metric. The main idea is to formulate the problem of constructing a Riemannian metric as a convex optimisation problem and to use algorithms to solve it. The new method will be studied theoretically and, moreover, be implemented into a computer program.

Furthermore, we will investigate how the information obtained by the Riemannian metric can be used to gain further insight regarding the sensitivity. This will enable us to identify sensitive points of the motion which need particular attention or control. Finally, we will apply the results to human walking and compare them to real-world experiments. The results will lead to a method which can guide researchers to tell athletes, or patients who relearn walking, at which point in the motion cycle they need to pay particular attention in order to maintain stability.

The proposed research project will benefit

- Researchers in Applications, the Public sector and Business/Industry to analyse basins of attractions through a computer program which will be developed in this project. Examples include:
- Decision makers in environmental agencies by developing models and analysing their behaviour using the computer program developed in this project.
- Policy makers in the public sector by developing economic models and analysing their behaviour using the computer program developed in this project.
- The research will benefit people in the UK and abroad by enhancing the quality of life through advances in the sport and health sector through application of the method to biomechanics of the human muscle-skeletal system.

- Researchers in Dynamical Systems by establishing a new method to analyse the basin of attraction for time-periodic systems, that can be generalised to many other situations.

- Researchers in Convex Optimisation by application of their theory and algorithms to Dynamical Systems, thus advancing their field through new challenges and input, as well as opening new applications.


The method developed in this project will make Borg's criterion accessible for calculations for the first time, and will further investigate the use of the Riemannian metric to gain sensitivity information for the analysis of basins of attraction for time-periodic systems. This method can be generalised to many other situations including the basin of attraction of periodic orbits and equilibria in autonomous systems, systems on a finite time interval, which are particularly important for applications, and the basin of attraction of more complicated attractors.

The application of convex optimisation to dynamical system is completely new and will advance both fields through the exchange of ideas. This already starts within this project, but further collaboration between these fields will have a strong influence on both dynamical systems and convex optimisation.

In this project, the method will be transformed into a computer program and will directly benefit a variety of users: researchers in applied disciplines such as physics, biology, medicine, chemistry and many more who study dynamical systems, decision makers in environmental agencies to derive and study biological and chemical models, policy makers in the public sector to study economic models or, for example, disease transmission in animal or human populations.

Through these applications, the method will benefit the people in the UK and abroad. The collaboration of the PI with Prof. Heiko Wagner (Sport Science, University of Muenster, Germany) has already produced a number of results concerning the stability and the basin of attraction of the human muscle-skeletal system, in particular the elbow joint. Whereas in the past mostly time-independent forcing was considered, in this project we will now be able to study time-periodic applications such as walking. We will analyse specific movements, make predictions about optimal muscle activation for stable and energy efficient movements to achieve a certain goal in sport and identify sensitive points in the movement, where particular attention is needed with respect to perturbations. Moreover, for paralysed patients, individual training programs can be developed to regain movement patterns, for patients with missing limbs, prostheses can be developed to take over the function of the muscles, and they can be fitted individually to the patient.