Calculating the Basin of Attraction in Asymptotically Autonomous Dynamical Systems
Lead Research Organisation:
University of Oxford
Department Name: Mathematical Institute
Abstract
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Organisations
People |
ORCID iD |
| Holger Wendland (Principal Investigator) |
Publications
Giesl P
(2011)
Numerical determination of the basin of attraction for exponentially asymptotically autonomous dynamical systems
in Nonlinear Analysis: Theory, Methods & Applications
Giesl P
(2012)
Numerical determination of the basin of attraction for asymptotically autonomous dynamical systems
in Nonlinear Analysis: Theory, Methods & Applications
| Description | Ordinary differential equations (ODEs) are a major modeling 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 was to develop new tools to overcome this problem and to derive numerical methods for the analysis of asymptotically autonomous systems. The first main result deals with exponentially asymptotically autonomous system, where the right-hand side converges exponentially fast to the limiting autonomous system. The second main result develops a similar method for general asymptotically autonomous systems. In both cases, the infinite time interval of the nonautonomous system was transformed into a finite one. This transformation reflects the speed of convergence of the asymptotically autonomous system to the autonomous one as time tends to infinity. These new methods have been studied theoretically, including error estimates, and, moreover, they have been implemented into a computer program, available at the website http://www.maths.sussex.ac.uk/~giesl/asymptaut.html together with a summary of the results. |
| Exploitation Route | Extensions to more general ODEs. |
| Sectors | Agriculture, Food and Drink,Chemicals,Pharmaceuticals and Medical Biotechnology |
| URL | http://www.maths.sussex.ac.uk/~giesl/asymptaut.html |