Bifurcations and destabilising feedback loops in multiple-scale switched control systems

Dynamical systems which are characterized by switchings between a number of distinct differentiable vector fields, with the switching law that depends on the value of some state variable, are common in applications. For example, some mathematical models and investgations of the character of neuromuscular control during tasks such as quiet standing or target tracking use switched control systems. Moreover, it is often the case that there are present different time scales in a given system, which imply that a dynamical system has a slow-fast structure. That is, there is a subset of dynamical states which have a characteristic time scale much slower than the time scale of some other subset of dynamical variables. In neuroscience, spiking and bursting of neurons provide an example – the build-up of voltage potential, leading to a release of a spike in neurons, has a much smaller time-scale than the time-scale of a spike. Thus to model spiking and bursting in neurons, slow-fast dynamical systems with so-called resets are used. The dynamics of such model systems are far from understood since there is no centre manifold theory which allows for a reduction of switched dynamical  systems as effectively as for smooth systems.

The aim of our current research is to investigate bifurcations in multiple-scale switched control systems in view of understanding links between the onset of destabilizing feedback loops, sudden changes between attractors and changes in their structure, due to, for example, bifurcations involving so-called Canard cycles and resets. An example of such a transition is depicted in the figure.

In the figure, we depict a locally superstable limit cycle attractor with two-spikes, existing in a planar slow-fast neuron model with resets, where w is the slow and v the vast variable. Through the so-called maximal Canard cycle and a number of bifurcations, this limit cycle, under small parameter variation, acquires an additional spike (dashed line) and thus largely increases its amplitude in the w direction. Understanding such dynamical scenarios (Discontinuity Induced Bifurcations) will allows us to understand multiple-scale switched control systems.

The applications of the developed theory may be diverse, ranging from the aforementioned neurobiological applications through automatics and robotics to climate models, where the understanding of possible onsets of positive feedback loops leading to a dramatic climate change is of vital importance.

This research is conducted in collaboration with Associate Research Professor Mathieu Desroches from Inria Sophia Antipolis – Méditerranée Research Centre (Nice, France) and Research Professor Serafim Rodrigues from the Basque Center for Applied Mathematics (Bilbao, Spain).

By Piotr S. Kowalczyk, Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology