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Enhancing predictability in chaotic systems

Project Description

The goal of this research program – which involves theoretical analysis and numerical simulations -- is to apply fundamental results and recent developments (to which the supervisors significantly contributed) of dynamical system theory to enhance our ability to predict the future evolution of complex and chaotic dynamical systems -- such as weather and climate systems.

Predictability is the degree to which we are able to correctly forecast the future state of a system given the (imperfect) knowledge of its present state. While our ability to accurately describe many physical systems via a deterministic set of equations may naively lead one to the impression that the future of such systems can be determined with a high predictability, this is dramatically untrue for a large class of deterministic system displaying chaos.

Chaos theory is amongst the most fundamental conceptual advances in our
understanding of the physical world, and the fact that general nonlinear dynamical systems are exponentially sensitive to initial conditions strongly constraints our ability to predict the future evolution of chaotic physical systems. Any (unavoidable) imprecision in our knowledge of the present state gets exponentially amplified in time according to the well-known largest Lyapunov exponent. It is clear that after a sufficiently long time any original uncertainty on the initial state, for how small it could be, will have been amplified out of proportion, making the predicted state completely different from the “real” one. This leaves us with a relatively short time window in which our predictions are reliable, a typical predictability timescale roughly given by the inverse of the Lyapunov exponent.

Weather forecasting and climate prediction are amongst the most typical examples of this predictability shortfall. In weather forecasting, for instance, one is faced with an incomplete and noisy knowledge of the present atmospheric state and with an approximate knowledge of the fluid dynamics equations governing its evolution. These equations, as the one governing other typical complex systems, are chaotic and extremely high-dimensional. These systems are characterized by different instabilities, each one corresponding to a different uncertainty direction in configuration space and
each one characterized by a different exponential growth rate (the so-called spectrum of Lyapunov exponents).
Intuitively, one can hope to improve the predictability timescale by identifying these directions and controlling the evolution along specific ones.

We will consider the so-called covariant Lyapunov vectors (CLVs), that
precisely span these unstable directions (as much as the stable ones, also present in such systems). Technically speaking, CLVs are associated to LEs (in a relation that resembles the eigenvectors-eigenvalues pairing), and provide an intrinsic decomposition of the tangent space that links growth (or decay) rates of (small) perturbations to physically based directions in configuration space. They can be used to associate instabilities timescales (the inverse of LEs) to the spatial scales and spatial structures of state perturbations or uncertainties.

CLVs have been recently used to study a simple multiscale systems (Lorenz 96), revealing the presence of a slow bundle in tangent space, composed by a set of vectors with a significant projection on the slow degrees of freedom; they correspond to the smallest (in absolute sense) Lyapunov exponents and thereby to the longer time scales. In this research project, we will explore how these results may find practical applications to ensemble forecasting and data assimilations in weather and climate models.

Candidates should have (or expect to achieve) a UK honours degree at 2.1 or above (or equivalent) in Physics, Applied Mathematics or related discipline along with an MSc in the same subjects.

It is essential that candidates have a background in Nonlinear Dynamics along with knowledge of Numerical simulation.


• Apply for Degree of Doctor of Philosophy in Physics
• State name of the lead supervisor as the Name of Proposed Supervisor
• State ‘Self-funded’ as Intended Source of Funding
• State the exact project title on the application form

When applying please ensure all required documents are attached:

• All degree certificates and transcripts (Undergraduate AND Postgraduate MSc-officially translated into English where necessary)
• Detailed CV
• Details of 2 academic referees

Informal inquiries can be made to Dr F Ginelli () with a copy of your curriculum vitae and cover letter. All general enquiries should be directed to the Postgraduate Research School ()

Funding Notes

This project is advertised in relation to the research areas of the discipline of physics. The successful applicant will be expected to provide the funding for Tuition fees, living expenses and maintenance. Details of the cost of study can be found by visiting View Website. THERE IS NO FUNDING ATTACHED TO THESE PROJECTS. Applicants should also be aware that Additional Research Costs of £1,000 per annum for travel to conferences and £2,000 (total) for specialist computer software, are required (above Tuition Fees and Living Expenses).


• F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi, and A. Politi, Characterizing dynamics with covariants Lyapunov vectors, Phys Rev Lett 99, 130601 (2007).

• F. Ginelli, H. Chaté, R. Livi, A. Politi, Covariant Lyapunov vectors, J. Phys. A 46, 254005 (2013).

• M. Carlu, F. Ginelli, V. Lucarini, A. Politi, Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz ’96 model, submitted to Nonlin. Proc. in Geophys., arXiv:1809.05065 (2018).

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