# Analysing Sensitivity in Basins of Attraction using a Local Contraction Criterion

Lead Research Organisation:
University of Sussex

Department Name: Sch of Mathematical & Physical Sciences

### Abstract

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.

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.

### Planned Impact

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.

- 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.

### Organisations

- University of Sussex, United Kingdom (Lead Research Organisation)
- Reykjavik University, Iceland (Collaboration, Project Partner)
- Humboldt University Berlin, Germany (Collaboration, Project Partner)
- University of Muenster (Munster), Germany (Collaboration, Project Partner)
- University of Birmingham, United Kingdom (Collaboration)

## People |
## ORCID iD |

Peter Giesl (Principal Investigator) |

### Publications

Giesl P
(2013)

*Construction of a CPA contraction metric for periodic orbits using semidefinite optimization*in Nonlinear Analysis: Theory, Methods & Applications
Giesl P
(2015)

*Review on computational methods for Lyapunov functions*in Discrete and Continuous Dynamical Systems - Series B
Giesl P
(2015)

*Converse theorems on contraction metrics for an equilibrium*in Journal of Mathematical Analysis and ApplicationsDescription | In this project we have considered 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 wanted 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. We have used 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 was to construct a suitable Riemannian metric for a particular equation. The main purpose of this project was to derive an algorithm that can always construct such a Riemannian metric. The main idea was to formulate the problem of constructing a Riemannian metric as a semidefinite optimisation problem. The new method has been studied theoretically and, moreover, has been implemented into a computer program. In particular we have - transformed the problem of constructing a Riemannian metric satisfying a local contraction criterion into a semidefinite optimisation problem - proved the equivalence of the two problems, i.e. a feasible point of the optimisation problem defines a contraction metric and if the dynamical system has an exponentially stable periodic orbit, then the optimisation problem has a feasible point - derived an objective function for the optimisation problem which provides a bound on the rate of exponential attraction to the periodic orbit - developed an algorithm and written a computer program to solve the above optimisation problem, using the SDP solver PENSDP - the SDP solvers were not able to solve very large problems in high dimensions |

Exploitation Route | The method can 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. Another potential application area of the research is human walking. One could use the information obtained by the Riemannian metric to identify sensitive points of the motion which need particular attention or control. This could lead to a method which can guide athletes, or patients who relearn walking, at which point in the motion cycle they need to pay particular attention in order to maintain stability. Moreover, this could also lead to the development of training programs for paralysed patients to regain movement patterns; for patients with missing limbs, prostheses could be developed to take over the function of the muscles, and they could be fitted individually to the patient. We have derived the first constructive method for a contraction metric. This approach can be generalised to many other situations, where a local contraction metric needs to be computed. These situations include the basin of attraction of an equilibrium in autonomous systems, the basin of attraction of a periodic orbit in autonomous systems, and many more. The application of semidefinite optimisation to dynamical system is a growing area and further collaboration between these fields, beyond this project, will have a strong influence on both dynamical systems and optimisation, in particular on the algorithmic side. After the findings of the limits of SDP solvers for large problems, new directions of research became obvious, like using other numerical methods to solve the construction problem. |

Sectors | Digital/Communication/Information Technologies (including Software),Financial Services, and Management Consultancy,Healthcare |

Description | The problem of finding a Riemannian metric have successfully be transformed into the framework of CPA (continuous piecewise affine) metrics, and been formulated as a semidefinite programming problem. The performance limits of SDP (semidefinite programming) solvers to solve this optimization problem have influenced new approaches to solve the construction problem and has led to a successful application for a joint EPSRC-University of Sussex funded PhD studentship Algorithmic Construction of Finsler-Lyapunov Functions. In this project we express the Riemannian metric as a Finsler-Lyapunov function, again in the CPA framework, which will lead to a linear programming problem, for which much more powerful solvers are available. Concrete economic and societal impact has not yet materialised from this project. |

Description | DTG studentship, jointly funded by EPSRC and the School of Mathematical and Physical Science, University of Sussex |

Amount | £68,300 (GBP) |

Organisation | University of Leeds |

Department | Faculty of Engineering |

Sector | Academic/University |

Country | United Kingdom |

Start | 10/2014 |

End | 03/2018 |

Description | Hafstein |

Organisation | Reykjavík University |

Country | Iceland |

Sector | Academic/University |

PI Contribution | Collaboration on research papers and projects. Hosting research visits. |

Collaborator Contribution | Collaboration on research papers and projects. Hosting research visits. |

Impact | Giesl & Hafstein. Construction of Lyapunov functions for nonlinear planar systems by linear programming. J. Math. Anal. Appl. 388 No. 1 (2012), 463-479. Giesl & Hafstein. Existence of piecewise linear Lyapunov functions in arbitrary dimensions. Discrete Contin. Dyn. Syst. 32 No. 10 (2012), 3539-3565. Construction of a finite-time Lyapunov function by Meshless Collocation. Discrete Contin. Dyn. Syst. Ser. B 17 No. 7 (2012), 2387-2412. Giesl & Hafstein. Construction of a CPA contraction metric for periodic orbits using semidefinite optimization. Nonlinear Anal. 86 (2013), 114-134. (with Sigurdur Hafstein) Local Lyapunov Functions for Periodic and Finite-Time ODEs. In: Recent Trends in Dynamical Systems, A. Johann, H.-P. Kruse, F. Rupp and S. Schmitz (eds.), Springer Proceedings in Mathematics & Statistics, Vol. 35, Springer-Verlag, Berlin, 2013, 125-152. Giesl & Hafstein. Revised CPA method to compute Lyapunov functions for nonlinear systems. J. Math. Anal. Appl. 410 (2014), 292-306. Giesl & Hafstein. Computation of Lyapunov functions for nonlinear discrete systems by linear programming. J. Difference Equ. Appl. 20 (2014), 610-640. Giesl & Hafstein. Implementation of a Simplicial Fan Triangulation for the CPA Method to compute Lyapunov Functions. In: Proceedings of the 2014 American Control Conference, Portland (OR), USA, 2014, no. 0202, pp. 2989-2994. Björnsson, Giesl & Hafstein. Algorithmic verification of approximations to complete Lyapunov functions. In: Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS), Groningen, The Netherlands, 2014, no. 0180, pp. 1181-1188. Björnsson, Giesl, Hafstein, Kellett & Li. Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction. accepted at the 53rd IEEE Conference on Decision and Control, Los Angeles, California, USA, December 15-17, 2014. Giesl and Hafstein: Review on Computational Methods for Lyapunov Functions. Discrete Contin. Dyn. Syst. Ser. B 20 No. 8 (2015), 2291-2331. Björnsson, Giesl, Hafstein and Kellett: Computation of Lyapunov functions for Systems with Multiple Attractors. Discrete Contin. Dyn. Syst. 35 (2015), 4019-4039. Giesl and Hafstein: Computation and Verification of Lyapunov functions. SIAM J. Appl. Dyn. Syst. 14 No. 4 (2015), 1663-1698. Grants: EPSRC Small Grant, Icelandic Research Fund: "Algorithms to compute Lyapunov functions", 2013-2016 and "Lyapunov Methods and Stochastic Stability", 2015-2017, for both Hafstein is PI, Giesl Co-I Workshop: - joint organisation of "Workshop on Algorithms for Dynamical Systems and Lyapunov Functions" 2013 in Reykjavik, Iceland - joint organisation of a Minisymposium "Numerical Methods in Dynamical Systems", at the annual meeting of the Deutsche Mathematiker-Vereinigung (DMV) at the University of Hamburg, Germany, 21.-25. 9. 2015 Editorial work: joint editors of a Special Section Computational methods for Lyapunov functions for the Journal Discrete and Continuous Dynamical Systems Series B, Volume 20, Number 8, October 2015, pp. 2291-2526. |

Start Year | 2009 |

Description | Hintermueller |

Organisation | Humboldt University of Berlin |

Country | Germany |

Sector | Academic/University |

PI Contribution | Collaboration. |

Collaborator Contribution | Collaboration and hositing research visit. |

Impact | Contribution to a journal publication. |

Start Year | 2012 |

Description | Kocvara |

Organisation | University of Birmingham |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Collaboration and seminar. |

Collaborator Contribution | Hosting research visits, free use of computer program. |

Impact | Contribution to journal publication. |

Start Year | 2013 |

Description | Wagner |

Organisation | University of Münster |

Country | Germany |

Sector | Academic/University |

PI Contribution | Collaboration on research papers, hosting research visits. |

Collaborator Contribution | Collaboration on research papers, hosting research visits. |

Impact | Multi-disciplinary: Mathematics and Biomechanics/Sport Science. Organisation of Interdisciplinary Workshop "Mathematical Stability Analysis in Biomechanics and Robotics" 2007, ZiF University of Bielefeld, Germany Giesl, Meisel, Scheurle & Wagner. Stability Analysis of the Elbow with a Load. Journal of Theoretical Biology 228 No. 1 (2004), 115-125. Wagner & Giesl. Self-stability in Biological Systems - Studies based on Biomechanical Models, in: Fast Motions in Biomechanics and Robotics, M. Diehl and K. Mombaur (eds.), Springer-Verlag, LNCIS 340 (2006), 403-410. Giesl & Wagner. On the Determination of the Basin of Attraction for Stationary and Periodic Movements, in: Fast Motions in Biomechanics and Robotics, M. Diehl and K. Mombaur (eds.), Springer-Verlag, LNCIS 340 (2006), 147-166. Giesl & Wagner. Lyapunov functions and the basin of attraction for a single-joint muscle-skeletal model. J. Math. Biol. 54 No. 4 (2007), 453-464. Blickhan, Giesl & Wagner. Musculoskeletal Stabilization of the Elbow - Complex or Real. Journal of Mechanics in Medicine and Biology 7 No. 3 (2007), 275-296. Giesl & Wagner. Mathematical Stability Analysis in Biomechanical Applications, in: Mathematical Biology Research Trends, L. Wilson (ed.), Nova Science Publishers, 2008, 261-273. Giesl, Mombaur & Wagner. Stability Optimization of Juggling, in: Modeling, Simulation and Optimization of Complex Processes, H. G. Bock, E. Kostina, X. P. Hoang, R. Rannacher (eds.), Springer, Berlin, 2008, 419-432. |