We work in the broad area of nonlinear science, the Wordle image gives a sample of the words in titles of our recent papers.
Over the past few years, our interests have been in applying concepts of nonlinear dynamics to various systems of current interest such as models of coupled oscillators, synchronization, time-delay dynamics, and so on.
Research Summary
Systems under the effect of parametric modulation – periodic, quasiperiodic, chaotic, or even stochastic forcing – exhibit many interesting phenomena and different dynamical properties that are obtained in the unforced systems are modified, leading to the bifurcations and attractors with novel dynamical properties.
Over the years we have examined a number of such dynamical systems, particularly a variety of quasiperiodically driven oscillators. One interest was the phenomenon of strange nonchaotic motion on attractors (SNAs) that were seen in nonlinear systems driven by aperiodic forcing. Such motion is strange, in the sense of being on fractal attractors, but nonchaotic, in the sense of having non-positive Lyapunov exponents.
We have studied a number of issues relating to SNAs, ranging from the mechanisms of their formation and destruction, to applications in synchronization, in creating stable aperiodic motion, in their robustness to noise and so on. Some of the main results have been summarised in a couple of review articles on SNAs and their applications.
In most natural systems, the transmission of signals takes place at finite velocity. This makes time delay coupling important in a proper model of such situations. Several studies have explored the manner in which this type of coupling affects complex phenomena (such as synchronization for example) in nonlinear dynamical systems. Time delay increases the dimensionality of the system and often also analytically intractable. Our recent work on time-delay coupled systems address specific features such as amplitude or oscillation death, changes in the modes of synchronization and in the phase-flip transition. Some of the main work has been summarized in two reviews.
We have also been looking at how systems are coupled to one another, when the coupling terms are not of the simple, diffusive form, or when “unlike” variables are coupled to each other. This latter situation often occurs in experimental systems. We have termed this the conjugate coupling, when dissimilar varables are coupled. Examples range from laser systems to electrochemical cells, to ecological models of predator-prey dynamics…
Biochemical systems are commonly modelled by differential equations or simulated by stochastic algorithms—the macroscopic and microscopic descriptions respectively. The macroscopic dynamics can be justified only when the participating molecule numbers are high enough to be replaced by concentrations. For small systems, intrinsic fluctuations comes into play som owing to the random creation and decay of individual molecules: a microscopic description of such disordered molecular motion is physically more correct.
Synchrony or concerted dynamical behaviour is observed in a wide variety of natural systems. Such behaviour is also often robust, namely, systems with large stochastic fluctuations, possessing a range of internal time–scales are nevertheless capable of exhibiting sustained correlated dynamics over long times. In one of our recent works, we examined mechanisms by which two (or more) stochastic systems can be microscopically coupled so as to result in the phase synchronization of their dynamical variables. Study of model systems, that produce sustained oscillations shows that a suitable coupling of different networks can lead to such correlated behaviour, and that both in–phase and anti–phase synchronization can occur when the coupling is time delayed.
We have two main interests in this area: How is rhythmic (or oscillatory) behaviour created, and what affects sycnhrony in neuron models. We have studied a number of automaton models to examine the major features – in terms of topology and individual “neuron” dynamics – that contribute to the creation of sustained rhythms. We have also studied a number of (more realistic) dynamical models of neurons in order to see how the coupling induces synchrony, and also how noise can either assist or deter synchronous dynamics. We are also interested in devising new methods for deciphering inherent dynamics in neuronal signals (from, say, EEG data or from local field potential measurements) through an Empirical Mode Decomposition (EMD) analysis.
Our interests in Computational Biology fall in two classes: the analysis of genomic sequence data using bioinformatics methods, and the applications of dynamical systems theory and modeling to problems of biological interest.
We have analysed a variety of genomic sequences using various tools from mathematics/statistics and signal processing to identify features of biological interest such as genes, transposable elements, lateral gene transfer events, noncoding genes and so on. The tools and techniques used range from Fourier and Wavelet transforms to the use of Markov models, Shannon entropy and Support Vector Machines.
In cellular and subcellular processes, the dynamics is greatly affected by the small numbers of participating entities. This makes fluctuations very drastic, and a source of intrinsic noise. Thus the modeling of such phenomena proceeds via the Master equation, and approximations such as the Langevin equation. In the past several years we have been trying to understand the nature of synchrony in such intrinsically stochastic systems and the consequences for regulation at the cellular level.