Dr. Neil Balmforth

Large Dynamical Systems:

The atmosphere or the ocean is an extremely large dynamical system with many degrees of freedom. This size makes it impractical to model these systems on a first principles basis without some severe approximations. Morever, state-of-the-art models are highly complicated dynamical systems themselves, which makes it challenging to assess the robustness of the models to variations in all the input parameters and physics, and to test methods of reconstruction and prediction. Instead, one can work with smaller, more accessible systems to examine the viability of a physical process or the utility of a technical method. One option is to gradually build up large complex systems by coupling together a number of much simpler subsystems. These coupled ensembles act as metaphors for the complex systems that we encounter in nature.  For example, ensembles of coupled oscillators can be use to explore the phenomenon of synchronization that is observed in biological populations such as flashing fireflies and cooperative cells. And with lattices of coupled maps and simple chaotic systems,  one can test the feasibility of unravelling and reconstructing  the underlying governing dynamics from an observed signal.

My collaborators in these efforts include Antonello Provenzale, Roberto Sassi and Ed Spiegel

Click to enlarge


Static, steady and chaotic fronts in a network of Lorenz system
(colormaps of the x-coordinate of all the subsystems on the (n,t)-plane for three networks with different coupling strengths)
Moore-Spiegel attractor
(phase portrait and time series of the Moore-Spiegel strange attractor)
Patterns of synchrony in a population of coupled oscillators
(population density on the phase-time plane for 100 oscillators coupled through a mean field)                     
Bifurcation to moving kinks on a lattice
Being stable and discrete
Pulse dynamics review
Noisy maps
Synchronizing Moore and Spiegel Checkerboard maps
Timing maps

Master-slave synchronization
A shocking display of synchrony

Coupled maps
Lorenz-Fermi-Pasta-Ulam experiment
Nonadiabatic stochastic resonance

Department of Mathematics / Fluid Labs / Neil Balmforth / Research Interests