Singularities and bifurcations

Roughly speaking, a bifurcation is a change in the qualitative (topological) structure of the solutions of a dynamical system as a parameter varies.

A bifurcation occurs, e.g., when the number of equilibria changes, which in turn occurs when the number of solutions of an equation changes. The study and classification of such equations can be accomplished using singularity theory, a theory that rests heavily upon a topological and differentiable notion of equivalence.

A neuromorphic electronic circuit

While the brain's inner workings are still mostly unknown, the behavior of a single neuron is relatively well understood from the perspective of dynamical systems theory. By applying singularity theory to a neuron's critical manifolds, we designed an electronic circuit that mimics such manifolds and can smoothly transition between the two main neuronal operating modes: tonic spiking and bursting [CF17].

The resulting circuit solely uses six transistors and passive elements. For comparison, the simplest available neuromorphic circuit capable of transitioning between tonic spiking and bursting uses fourteen MOSFET transistors. Another advantage of our approach is that the circuit parameters are constructively tuned by geometric inspection of its static input-output characteristic, which avoids laborious and non-constructive parameter fitting procedures.

Bifurcations occurring on a Stirling engine

The Stirling engine is an external combustion thermodynamic engine that operates by cyclic expansion and contraction of a working fluid (typically air). Due to their high efficiency and capability of operating at low temperatures and with any heat sources, Stirling engines have found many applications, ranging from electricity generators and cryocoolers to solar power generators and combined heat and power systems.

Considering that energy conversion in the engine can only occur through cyclic motions, we study the bifurcations that lead to the appearance of limit cycles [GC20]. The temperature of the hot piston and the mechanical phase are our bifurcation parameters.

A notion of equivalence for linear complementarity problems with application to nonsmooth bifurcations

Linear complementarity problems can model many systems of interest to control engineering. Some examples are mechanical systems with unilateral constraints and electronic circuits with switching devices.

In [CMF20, MCF24], we introduce a new notion of equivalence between linear complementarity problems that sets the basis for the translation of the powerful tools of smooth bifurcation theory to this class of models. Leveraging this notion of equivalence, we introduce new tools to analyze, classify, and design nonsmooth bifurcations in linear complementarity problems and their interconnection.