Calculating the Basin of Attraction in Asymptotically Autonomous Dynamical Systems

Ordinary differential equations (ODEs) are a major modelling tool in all sciences, for example mechanical systems in physics, population dynamics in biology, reactions in chemistry, price developments in economics and many more. Both for deriving the model and for its analysis it is crucial to understand the dynamical properties of an ODE. Nonautonomous ODEs are equations where the right-hand side explicitly depends on the time t, e.g. through a time-dependent force. They include asymptotically autonomous systems, where the right-hand side tends to an autonomous, i.e. time-independent, function as time tends to infinity. Dynamical systems are interested in the long-time behaviour of solutions. Although the equation converges to an autonomous one as time tends to infinity, not all properties of the limiting autonomous equation carry over to the nonautonomous one. In particular, the basin of attraction of a solution consisting of all points approaching this solution, is a nonautonomous object and can only be determined by considering the nonautonomous equation.However, not only the analytical, but also the numerical analysis of nonautonomous ODEs faces the difficulty that any interesting set is unbounded in the time direction. This makes a direct application of numerical tools from autonomous dynamical systems difficult, since one would need infinitely many grid points for the approximation. The main purpose of this project is to develop new tools, both in dynamical systems and in numerical analysis, to overcome this problem and to derive numerical methods for the analysis of nonautonomous systems. Within dynamical systems, the infinite time interval of the nonautonomous system will be transformed into a finite one; in numerical analysis a method will be developed to distinguish between the different dependencies with respect to the time and the space variables. The new method will be studied theoretically, including error estimates, and, moreover, be implemented into a computer program.

The proposed research project will benefit - researchers in the theory and numerical analysis of Asymptotically Autonomous Dynamical Systems by establishing a new method to map the infinite time to a finite time interval - researchers in Meshless Collocation by a new method to distinguish between the different dependence on variables, e.g. for solutions of PDEs, periodic and non-smooth ODEs - researchers in Applications to calculate basins of attractions through a computer program which will be developed in this project - people in the UK and abroad by enhancing the quality of life through advances in the sport and health sector through application of the methods to biomechanics of the human muscle-skeletal system The transformation of asymptotically autonomous ODEs to a finite time-interval will have an impact on the theory of asymptotically autonomous ODEs, but also open new ways for their numerical treatment. This includes not only meshless collocation, but any numerical method. Although the project is mainly concerned with asymptotically autonomous dynamical systems, it might lead to the development of similar methods for the treatment of general nonautonomous systems. A method in meshless collocation which distinguishes between different variables will have an impact on general interpolation and collocation. Applications to dynamical systems are not only restricted to the calculation of basins of attraction in asymptotically autonomous systems, but will have an influence on periodic ODEs and nonsmooth systems. The method to calculate basins of attraction in asymptotically autonomous dynamical systems will be transformed into a computer program and will directly benefit all researchers in applied disciplines such as physics, biology, medicine, chemistry and many more who study dynamical systems. This impact will be on a medium timescale. Through these applications, the method will - on a larger timescale - indirectly benefit the people in the UK and abroad. For example, through the collaboration of the PI with Prof. Wagner (Sport Sciences, University of Muenster, Germany), the method will be used for the analysis of time-dependent human movements. This will enable us to make predictions about optimal muscle activation for stable and energy efficient movements in order to achieve a certain goal in sport or, for patients with missing limbs, to develop prostheses and to fit them individually. The immediate impact on researchers in Dynamical Systems and Numerical Analysis will be achieved by publications and presentations at conferences. The researchers involved in this project as well as the students in their groups will benefit from learning new skills through the frequent meetings of the groups which are facilitated by this project. At the end of the project, a computer program will be a visible and applicable product that summarises the methods and will be of interest for applications. Further development of this computer program could lead to an economic product to increase the competitiveness of the UK. The close collaboration of the PI and the CI with researchers in Sports Sciences and industry, e.g. Airbus, are an excellent route of publication of the results to the non-mathematical community.