| EPSRC Reference: |
EP/J014532/1 |
| Title: |
Analysing Sensitivity in Basins of Attraction using a Local Contraction Criterion |
| Principal Investigator: |
Giesl, Professor P |
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| Department: |
Sch of Mathematical & Physical Sciences |
| Organisation: |
University of Sussex |
| Scheme: |
Standard Research |
| Starts: |
01 August 2012 |
Ends: |
31 July 2013 |
Value (£): |
23,588
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Summary on Grant Application Form |
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.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.sussex.ac.uk |