Nonlinear dynamics

Chaos in differential equations

I first got interested in this because of the famous one cusp difference map in Lorenz's 1963 paper. It's 1-d rather than 2-d because of the strong contraction; this is the result of an asymptotic limit, which turns out to be that of large sigma, and it turns out you can derive a map using singular perturbation methods, which predicts chaos (and in that sense solves the Lorenz equations). Jim Keener thought you could use the method to prove rigorously the existence of a chaotic attractor in the Lorenz equations, and while I suppose this to be true, I've never bothered to try. The nuts and bolts of chaos in differential equations are homoclinic bifurcations, and they come in three basic varieties: Lorenz, Shil'nikov, and bifocal. You can extend the analysis formally to n dimensions, to countably infinite dimensions (convection in a box) and even to partial differential equations in unbounded domains. The issue is then (i) to find an example, which requires orbit following for pdes, and (ii) to prove the existence of a strange invariant set, which needs some functional analysis. This is the way in to the study of...

Turbulence

Most particularly in shear flows, the onset of turbulence is very similar to the onset of chaos in the Lorenz equations. One would like to hypothesise that indeed, it is caused by the occurrence of a homoclinic bifurcation. Orszag and colleagues' work in the 1980s suggested that this might be associated with the violent inviscid 3-d instability of a slowly decaying 2-d Tollmien-Schlichting wave, and a fable for this might be the interaction of an evolving line vortex in a shear flow.

Convection offers a different paradigm, as here the transitions to oscillations and chaos occur at lower Rayleigh number, before the transition to full convective turbulence. I've made efforts to analyse the chaotic trajectories at high Rayleigh and Prandtl number (cf. Lorenz above), and Simon Acomb's thesis dealt with this. It's a difficult problem though, and to be realistic needs three dimensions, which makes life horrendous.

Time series analysis

The methods of time series analysis based on dynamical systems concepts, and particularly the idea of an embedding phase space, do two things, more or less: prognosis (prediction) and diagnosis. The latter includes such procedures as signal processing, noise filtering, and I am particularly interested in this in the context of medical data, for example in EEG, ECG records, and also in fetal heart rate monitoring, baroreceptor feedback, pulse oximeter measurements, etc. It seems that there is a great deal of medical technology which routinely produces masses of data, most of it looks chaotic, and there is not much in the way of sophisticated diagnostic procedures. I'm not sure if `nonlinear' methods will really help, but they shouldn't do worse than statistical, Fourier based methods. What interests me is whether one can ask coherent questions in this way. For example, two uncoupled periodic oscillators can be separately recognised from a combined power spectrum. What if two oscillators are nonlinearly coupled, for example heart rate and respiration; can one determine from one signal which bit is due to which oscillator? Does this question even make sense? There seems no doubt that there are major questions of significance which could be addressed in this context, but whether nonlinear data analysis can provide coherent answers (as opposed to just producing methodologies) is less clear.

Differential-delay equations

The question of how chaos is produced in delay-differential equations is the other big question which interests me. The canonical problem is the delay-recruitment equation, sometimes known as the Mackey-Glass equation. When the delay is large, the equation looks like a singularly perturbed problem, with the `outer' limit being a difference equation. The basic question is then how the chaos in the map relates to the chaos in the dde. Apparently this is an impossible problem. Jonathan Wattis's dissertation and Giri Kalamangalam's thesis were concerned with this question. In the singular limit, one can derive from the equation an approximating functional map. But the epsilons disappear and there seems little that one can easily do. Future progress would seem to need a combination of much numerical experimentation and some functional analytic insight. There are other interesting models in Giri's thesis, with two or more delays, or variable delays.