A selection of current research topics

My research focuses on analysis and design of dynamical systems, described by nonlinear differential equations or some other mechanism. In the end I hope to obtain a better understanding of the behaviour of some dynamic processes appearing in physics, social sciences, biology or some particular engineering processes.

What follows is a selection of research topics that have tempted me over the years; a substantial part of this research has been carried out with co-workers and students. Each topic is supplied with a short annotation.

As a general remark, most of my research starts from a simple idea, which in the end turns out to be false, but has a robustness property leading to an adapted (or twisted) version--hopefully interesting and containing some truth. There is a long list of topics and ideas which turned out to be ill-fated and which I'd rather not make public.

I have always been interested in results supported by mathematical proof, leaving some classes of systems out of reach from an analytical point of view. Approximations retaining the fundamental properties of the original remain an option for an analytical study. Numerical simulations may be helpful and are playing an increasing role in my ongoing research, but results supported by mathematical proof remain a priority.

Dynamical systems approach to optimization problems

We associate a dynamical system to a large class of optimisation problems such that the (unique) solution to the optimisation problem corresponds to a function of the (unique) equilibrium state of the dynamical system; the equilibrium should preferably be globally asymptotically stable. A simulation of the dynamical system (starting at an arbitrary initial condition) generates a trajectory converging to the equilibrium and therefore to the solution of the optimisation problem. The idea is certainly not new: the search for the minimum of an unconstrained convex real valued function can be implemented as the convergence to the equilibrium point of an associated gradient system.

We have shown that for a strictly convex separable cost function and linear constraints, the procedure leading to the minimization of the cost can be implemented as a dynamical system. The dynamical system arises directly from an interpretion of the Lagrange necessary conditions for optimality and not from the associated dual problem; a natural Liapunov function accompanies the dynamical system. Conditions for global convergence to the unique equilibrium are derived from a study of this Liapunov function. It turns out that the Liapunov function reveals itself as the (negative) of the Lagrange dual function.

The approach also accomodates cost functions which are not strictly convex but linear, and therefore embraces linear programming problems. An interesting feature is that the corresponding dynamical system has discontinuous dynamics. Any numerical simulation algorithm provides an estimate of the optimal solution. The approach has a conceptually attractive quality but calls for more research regarding specifically adapted simulation procedures, and speed of convergence properties.

An interesting feature is observed when the coefficient matrix of the linear constraints is sparse. A distributed version of a dynamical implementation may be possible: the evolution of the state components depends on a restricted set of "preferred" state components, not on the whole state. This may be important in practical applications. E.g. when considering the minimal cost flow problem, the corresponding dynamical system is a compartmental system described by nodes, each carrying a level of commodity (state variable) and arcs allowing for commodity flows between nodes. The dynamical system monitors the commodity level in the nodes. In a distributed implementation, the flow through an arc is determined by the state variables associated with its start and end node. In practical situations the state variables may correspond to physical quantities (e.g. the water level in a water distribution system or the queue length of the buffer of packages waiting to be processed by a router in a computer network). The functions determining the flow are independent of the inflow or outflow at the nodes and therefore the dynamical system implementation is robust with respect to the external inflow or outflow. The distributed nature of the optimising strategy often makes for a convenient practical implementation.

For details and extensions, see this paper.

Interconnection of simple nonlinear systems with saturated interaction

The subject belongs to the field nowadays referred to as complex dynamics and focuses on the dynamics of nonlinear systems composed of interconnections of elementary nonlinear systems. The general idea is that each elementary system (agent) is influenced by its neigbours' behaviour (agents to which it is interconnected) which may result in something structured or pleasing. Self-organisation is one way to describe this process.

This research originated from attempts to obtain a better understanding of the dynamics of the clustering process in Kuramoto systems.

While previous work in the scientific literature focuses on the formation of a single cluster, we are interested in the formation of several clusters, arising from the interplay between each agent's own dynamics and its interaction with neighbouring agents. The phenomenon may be observed in different fields: Think of groups of people gathered through a mutual interest, pricing policy of different distributors in an economic market or clustering of oscillators in brain cells.

The challenge is to model and analyse this phenomenon in a mathematical way: we propose a model with saturating interaction between agents. The model is amenable to mathematical analysis while capturing the main features of the dynamic clustering process. We have shown that the model exhibits a long term behaviour where agents find themselves organised into several groups or clusters.

We have completely characterized the cluster structure (i.e. the number of clusters and their composition) by means of a set of inequalities in the parameters of the model and have identified the intensity of the attraction as a key parameter governing the transition between different cluster structures. The results have been established under the assumption of an all-to-all connection structure (all agents are neighbours of one another). The relation with network of Kuramoto oscillators is discussed. The results have then been extended to general interconnection structures and saturation interactions with applications to swarming, opinion formation dynamics and the dynamics of the water level in interconnected water basins.

While this work on the clustering model follows from previous work on the Kuramoto model, on its turn it inspired us to start our work on optimisation and its relation to dynamical systems.

Kuramoto systems

Oscillators are systems exhibiting periodic behaviour. After some period of time they return to their original state, having completed one cycle; the number of cycles per second is their natural frequency. Examples abound in engineering systems, physics, biology.

Consider a set of isolated oscillators, each with its own natural frequency. Interactions between oscillators will cause the oscillators to show behaviour different from their natural oscillatory behaviour. For instance, pacemaker cells in the heart (considered to be oscillators as voltage potentials with periodic behaviour) will through strong interaction cause simultaneous firing, resulting in a regular heartbeat.

The Kuramoto model (and its extensions) is a class of models where each oscillator is described by its phase and the interaction is modeled as phase coupling. (Other models have been proposed, e.g. where the coupling is impulsive). The coupling may lead to the formation of clusters, referred to as synchronization (phase differences remain constant) or entrainment (long term averaged phase differences remain constant). Kuramoto considered the limit of an infinite number of oscillators and investigated a solution for which a group of oscillators is moving at a common constant frequency and the other oscillators are moving incoherently, with long term average frequencies somewhere between their natural frequencies and the frequency of the synchronized group. This behavior was shown to only occur greater than a minimal value. For smaller values of the coupling strength there is no synchronization and oscillators with different natural frequencies have different long term average frequencies.

For the model with a finite number of oscillators, we were able to formulate an inequality in terms of the parameters of the model, constituting a sufficient condition for entrainment with respect to a given subset. The proof of this result implies persistence of the entrainment under sufficiently small perturbations of the initial conditions. The result remains non-trivial in the limit of an infinite number of oscillators. Oscillators with separate natural frequency have small (long term average) interactions with each other. This allows us to estimate the critical values for the coupling strength, defining the onset of entrainment of a given subset, by neglecting the interactions between oscillators from this set with oscillators not belonging to this set. We also noticed that for some configurations the entrainment may disappear with increasing coupling strength. When the coupling strength is increased further the entrainment will reappear. A similar phenomenon can be observed in arrays of Josephson junctions.

For the Kuramoto model with an infinite number of oscillators we consider a perturbation of the solution with one synchronized cluster as investigated by Kuramoto. The perturbation results in the emergence of extra entrained subsets, with a size of the order of the perturbation. The natural frequencies of the oscillators in the entrained subsets can be characterized to first order of the perturbation size by a set of equations in the system parameters. Similarly to the model with a finite number of oscillators, there exists a distribution of the natural frequencies for which entrainment of the smaller entrained subsets disappears with increasing coupling strength in a particular interval. Another phenomenon following from the analytical results consists of the occurrence of entrainment in intervals where the frequency density is too low to account for it. The entrainment is a consequence of resonances with other entrained subsets emerging as a result of high frequency densities.

A selection of past research topics

At present just a few of these topics have been introduced and/or discussed.

Reconstruction of dynamics through an output mapping

Dynamical systems from disparate fields in the physical, engineering, medical world or in human sciences may be accurately modeled by differential equations. The system is characterised by its state in the sense that the state at some point in time represents all the information needed to determine the state (and therefore characterise the system) in future times. Many times the state is not observable: only a reading or a measurement reflecting some aspect or property of the system behaviour (stae trajectory) is available. This reading is a signal as a function of time into the reals obtained from a mapping from the state space into the reals (time enters into the picture as the state changes in time).

It is clear that an interesting property of a dynamical system would be that different states lead to different readings: this is called observability in control theory and has been widely studied in the field. We have shown that if 2n+1 samples (n is the dimension of the state space) are taken from a reading, leading up to a time series, then observability is guaranteed; more is true: the proof technique guarantees that a mapping from the state space to the time series is an embedding. It is worth remarking that our result states that the embedding featuring in the Whitney embedding theorem may be realised through a time series derived from 2n+1 samples of one measurement signal. Of course there are some technical conditions needed for these results to be true, explaining the genericity of the properties involved. We have also shown that for observability to be true less samples will not suffice and this in a robust way.

The observability result also has direct bearing on more specific questions like: is an attractor of a trajectory embedded through the process just described? Stated otherwise: can an attractor be reconstructed from the time series obtained in the way described above? The answer is positive and is briefly discussed in the concluding remarks of the main paper.

Stabilization by feedback

Positive systems

The Pontryagin principle and a Lagrangian description of inductor-capacitor circuits

Pole assignment through periodic output feedback

Stability and Averaging

In this work we focused in an IEEE paper and an Automatica paper on the study of stability-issues of ordinary differential equations by means of a Liapunov approach where the time derivative of the Liapunov function along solutions of the system, may have positive and negative values. This approach enabled us to study exponential stability of fast time-varying systems, thereby also setting the stability study by means of averaging in a Liapunov context.

It is well known that the fast time-varying character of a differential equation is equivalent with a small velocity field on a different time scale. The results just mentioned then prompted us to explore stability issues by means of averaging where the small velocity vectorfield was emulated at the origin by means of an homogeneous velocity field (with degree larger than one). Generalisations followed based on the notion of partial averaging.

We also introduced a notion of practical stability which we discussed further on for time-varying differential equations by means of averaging.

Prof. Dirk AeyelsProf. Dirk Aeyels
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