**Up:**Home Page of Sebastian Wieczorek

**Student Projects in Dynamical Systems and Applied Bifurcation Theory**

Supervisor: Prof. Sebastian Wieczorek

Supervisor: Prof. Sebastian Wieczorek

**Tipping Points in The Natural World: Classical Bifurcations and Beyond**(PhD, Year 4)

The time evolution of real-world systems takes place on multiple time scales and is often subject to external

disturbances such as changing external conditions (external forcing). As the external conditions vary

in time, the position of the stable state changes and the system tries to track the moving stable state.

However, some systems fail to adapt to the changing conditions and undergo a sudden and unexpected

transition to a different, often undesired state. Such

**critical transitions**or

**tipping points**are of great

importance in natural science and technology, but cannot be always captured by the traditional stability

analysis and thus may require an alternative approach. Different streams within this project include:

**(a)**Rate-induced tipping using the example of the ”compost-bomb instability”: a sudden release of soil

carbon from peat (bogs) into the atmosphere above some critical rate of atmospheric warming.

This work involves concepts from geometric singular perturbation theory.

Compost Bomb CBS News

Proceedings of The Royal Society A 467 (May 8 2011) 1243-1269

**(b)**Critical-rate hypothesis in the context of species failing to adapt to contemporary climate

that is characterised by weather anomalies including hot and dry Summers, periods of intense

precipitation, etc. This work involves classical bifurcation analysis of ecosystem models.

M. Scheffer et al., Ecosystems 11(2) (2008) 226-237

**(c)**Critical levels of fresh-water forcing leading to the shutdown of the thermohaline ocean circulation.

This work involves a review of the existing ocean-circulation box models and their stability analysis.

Wikipedia: Shutdown of Thermohaline Circulation

**Mathematical Modelling of Neural Excitability**(Year 4)

The human nervous system is a complex network of various cell types which uses electrical signals to

coordinate our voluntary and involuntary behaviour. Within this system, the single active signalling unit

is the neuron. Neurons are typically quiescent, meaning that they are in a stable equilibrium, but can

generate action potentials in the form of voltage spikes in response to certain stimuli. This project

investigates mathematical models of neuronal excitability - the molecular mechanisms responsible

for strongly nonlinear response to small or slow stimuli. The focus of the project will be on the

Fitz-Hugh-Nagumo and Morris-Lecar models, that give rise to three different types (classes) of

excitability. This work involves a combination of bifurcation analysis and geometric singular perturbations.

Wikipedia: Biological Neuron Models

**Spontaneous Desynchronisation of Sleep and Wake Cycle**(Year 4)

Sleep is essential for our health and the maintenance of the brain and the body, yet many features of sleep are

still poorly understood. This project looks at the existing mathematical models based on the neuronal population

model of seep-wake switching subject to circadian oscillator forcing. The aim of the project is to analyse the

nonlinear phenomenon of synchronization between the homeostatic-driven sleep-wake-process and the circadian

pacemaker. The focus will be on spontaneous synchronization-desynchronisation transitions, which correspond to

a sudden loss of resilience, and are often precursors of various health complications.

A.J.K. Phillips et. al, Journal of Biological Rhythms 26 Oct (2011) 441--453

**Nonlinear Dynamics of New Generation Light Emitters**(PhD, Year 4)

Semiconductor lasers are tiny light emitters used in modern applications ranging from communication

schemes, to random number generation, and medicine. They are cheap and easy to produce but very

sensitive to external disturbances and prone to instabilities. Recent technological advances, such as

those leading to quantum-dot lasers and quantum-cascade lasers, open new directions in mathematical

modelling and nonlinear analysis of these devices. From a nonlinear dynamics point of view there are

two interesting questions:

1) Fundamental changes in the laser active medium or laser transition

make the active-medium polarisation evolve on the same time scale as

population inversion and electric field intensity. Polarisation dynamics

often gives rise to largely unexplored strong optical nonlinearities that

are not present in conventional semiconductor lasers. The project will

focus on laser nonlinearities due to quantum coherence with applications

to photonic integrated circuits.

S. Wieczorek and W. W. Chow, Physical Review Letters 97 113903 (2006)

J. P. Reithmaier et al., Nature 432 (2004) 197-200

2) The future generation of semiconductor lasers is being developed to

emit single photons and produce non-classical forms of light for applications

in quantum information. However, very little is understood about the notion

of stability and instability in such devices. This project will explore new

mathematical approaches to capture nonlinear dynamics in the limits of

strong light-matter coupling and cavity Quantum Electrodynamics (cQED),

where the concept of laser instabilities has not been addressed.

M. Nomura et al., Nature Physics 6 (2010) 279-283

Tutorial on semiconductor laser rate equations

**Exploring Complex Dynamical Behaviour Using Lyapunov Exponents**(MSc, year 4)

Nnonlinear oscillators in science and engineering (such as electronic circuits, lasers, neurons or even a

simple spring or mechanical pendulum) can exhibit a wealth of nonlinear behaviour when subject to

external disturbances. The system response to external forcing will typically depend on the control

parameters and will include periodic and quasiperiodic oscillations, chaos as well as interesting bifurcation

transitions between these different types of behaviour. The aim of this computationally oriented project

is to understand and implement numerically the concept of Lyapunov Exponents to explore dynamical

complexity of selected nonlinear oscillators and unveil the graphical beauty of chaos. The project

will be organised about the following topics:

1. Dynamical systems and phase space. ODEs as simple examples of dynamical systems.

2. Definition of attractor and examples of attractors in the phase space. Bifurcations.

3. One-parameter bifurcation diagrams.

4. Linearisation and theory of Lyapunov Exponents.

5. Numerical computation of Lyapunov Exponents.

6. One-parameter Lyapunov diagrams and comparison with one-parameter bifurcation diagrams.

Identifying different attractor types and their bifurcations from Lyapunov diagrams.

7. Creating two-parameter Lyapunov diagrams; see here and here for examples of visually stunning

Lyapunov diagrams.

Tutorial on theory and numerical methods for Lyapunov Exponents

**Stability, Synchronisation and Cluster Dynamics in Complex Networks with Symmetries**(PhD, year 4)

Synchronisation in networks of nonlinear oscillators is of great importance in electrical power

distribution, telecommunication and biology. Many networks produce patterns of synchronised clusters

which are related to hidden or non-obvious symmetries of the network. This project will investigate the

connection between network symmetries and cluster synchronisation using the framework of equivariant

bifurcation theory that combines group theory and traditional stability theory. Particular focus will be

on stability and cluster synchronisation in future power networks which will have to integrate the number

of small intermittent power sources from wind and solar farms, whose fluctuating outputs will create far

more complex stress on power grid operations.