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Student Projects in Dynamical Systems and Applied Bifurcation Theory

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.