MathJax TeX Test Page My current research interests are in applications of the theory of dynamical systems to real-life models that involve sudden switches, discontinuities and jumps. Specifically, my research makes contributions in the following applied sciences.
 ⌂ Materials Science

Project 1: Stability of quasi-static evolution of elastoplastic materials under switching topology: Stable growth of self-healing materials
1. Building a simple Lattice Spring Model with massless nodes and changing topology of connections
2. Converting the LSM into a Moreau Sweeping Process with changing geometry of moving constraint C(t) (differential equation x'=0 constrained by C(t))
3. Studying stability and convergence of the dynamics
Project 2: Stability of dynamic evolution of elastoplastic materials (detailed research plan)
1. Building a simple Lattice Spring Model with masses in nodes
2. Converting the LSM into a Perturbed Sweeping Process (differential equation x'=f (x) with a moving constraint)
3. Studying routes to different types of dynamic behavior
Publications and preprints:
References:
Illustration: The red dot in the animation below corresponds to stresses of the springs of the network given. The network of springs is subject to periodic stretching/compressing. The animation shows that stretching/compressing of particular springs corresponds to sweeping of the red dot along particular facets of the polyhedron. The construction of the polyhedron and other details are explained in our preprint. The 5-spring model we simulated is taken from Rachinskiy arXiv:1611.07099. The animation is created by Gudoshnikov.

 ⌂ Neuroscience

Project 1: Length of clusters of spikes in Izhikevich's model of stellate cells
1. Studying the theory of Hopf bifurcation of sub-threshold oscillations
2. Studying the theory of grazing bifurcation of spiking oscillations
3. Computing the ratio of the area of attraction of a) and b) and compare this ration against the length of clusters of spikes coming from simulations with noise
Project 2: Occurrence of bursting oscillations from sub-threshold oscillations
1. Studying the theory of grazing bifurcations of limit cycles in mechanical oscillators
2. Extending a) to the case of multiple impacts per period
3. Applying the result to a normal form linear system with neuron-type resets
detailed research plan

Relevant publications:
References:

 ⌂ Climate modeling

Project 1: Fold-fold singularities of a piecewise-smooth model of glacial cycles
1. Studying the piecewise-smooth Budyko-Widiasih model of glacial cycles
2. Finding stable switched equilibria of fold-fold type
3. Identifying the class of fold-fold singularities that lead to the occurrence of limit cycles under varying parameters
Project 2: Limit cycles of a piecewise-smooth model of glacial cycles coming from hysteresis switching
1. Studying the piecewise-smooth Budyko-Widiasih model of glacial cycles
2. Extending to 3D the available result about limit cycles of piecewise-smooth systems with hysteresis switching
Work in progress and relevant publications:
References:

 ⌂ Power electronics

Project 1: Stabilization of switched power converters to a nonequilibrium point
1. Studying the stable convex combination method for switching between smooth systems (Bolzern-Spinelli)
2. Extending the method to switching between discontinuous systems
3. Implementing the method on the models of buck and boost power converters
Project 2: Stabilization of switched power converters to a limit cycle
1. Computing limit cycles of piecewise affine models of power converters in closed form
2. Combining a) with the theory of stability of limit cycles of switched systems (Astrom)
Preprints and relevant publications:
References:
 ⌂ Anti-lock braking systems

Consider two vector fields (F1,F2) and two switching manifolds (S1,S2). Each trajectory x(t) follows the vector field F1 until x(t) crosses S2, where the system switches to the vector field F2 that governs the trajectory until it reaches S1 where a switch back to F1 occurs. The existence of stable limit cycles in this kind of systems is known since the textbook by Barbashin, whose approach employs a Lyapunov-like technique. The blue-green trajectory on the rightmost figure is an example of such a cycle.

In my paper SIAM J. Appl. Dyn. Syst. 16 (2017) the aforementioned cycles are obtained in dimension 2 through a bifurcation from an fold-fold singularity, where a suitably defined parameter μ crosses its bifurcation value μ=0. In 3-dimensional systems, the value μ=0 corresponds to the well-known Teixeira singularity (as it is termed after the pioneering work by Teixeira) or U-singularity (how it is termed in the book by Filippov). The Teixeira singularity is shown at the leftmost figure (the red point on the intersection of blue and green dashed lines) along with a sample trajectory (that sticks to the switching manifold L and slides along L until it approaches a pseudo-equilibrium) .

My interest in studying attractors of switching systems is motivated by closed-loop control problems in Anti-Lock Braking Systems (ABSs) (Drakunov et al, Bruijn et al, Tanelli et al) and switching convertors (Tse et al, Gupta et al, Sreekumar et al, Schild et al), intermittent therapy modeling in medicine (Tanaka et al), grazing management in ecology (Meza et al). My earlier results Internat. J. Control 90 (2017) on limit cycles in ABSs used perturbation theory (joint project with Tassilo Kuepper and Thomas Koppen).

My current work investigates border splitting bifurcation in Anti-Lock Braking System in 3D.

Here is an interactive diagram of a sample ABS that shows how the switching of the valves links to the motion of the phase point along a limit cycle of the respective switched systdem. The animation is created by Gudoshnikov.

 ⌂ Robotic walking

A bipedal robot can walk down a shallow slope without any control, see this video. The simulation below reproduces the walking gait by viewing the biped as a coupled pendulum combined with a velocity jump applied when the swing leg touches the ground. The animation is created by Gudoshnikov.

Garcia, Chatterjee and Ruina discovered (paper, preprint) that the walking gait observed can be mathematically established through a bifurcation of a periodic solution in the switched pendulum model when the slope of the ground crosses zero. A rigorous proof of the discovery by Garcia, Chatterjee and Ruina has been recently delivered in my course in UTD Summer School on Nonsmooth Dynamical Systems.

My current goal is to classify all the walking gaits that the bipedal robot is capable to realize and to study scenarious of gait initiation/termination (detailed research plan).

 ⌂ Dynamics of Lorentz gas

A possible explanation of the loss of thermal equilibria in gases of hard balls (see Kinoshita et al) is over regularization of the ergodic Sinai's (i.e. dispersing) billiard. This regularization is accomplished by approximating the billiard by a Hamiltonian system with a steep potential (thus smoothing the instantaneous collisions) and the absence of thermal equilibrium in the gas corresponds to the dramatic phenomenon of the lack of ergodicity in the regularized billiard, see Turaev & Rom-Kedar for the theory and Kaplan et al for experiments.

The main assumption of paper by Turaev & Rom-Kedar is the presence of a closed orbit that hits one of the balls tangentially (grazes). My goal is to prove that the set of dispersing Sinai's billiards with periodic grazing orbits is dense in the set of all dispersing Sinai's billiards. It is well known that periodic points of the respective dynamical system are dense in the phase space (Bunimovich-Sinai-Katok-Strelcin). However, if one moves one of the discs (say, disk C) towards the closed orbit (e.g. towards the dashed red line - - - -), the trajectory may run outwards because the disk influences this trajectory through the contacts with the other parts of the trajectory. Roughly speaking, my goal is to prove that a suitable part of the trajectory (here dotted red line ·····) can be frozen to a necessary extent by considering the periodic orbits which are almost tangent (i.e. the neighborhood $\delta>0$ is very small) to one of the discs (here disc B).