MathJax TeX Test Page My current research interests are:
Border-splitting bifurcation and its application to anti-lock braking systems

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Complex attractors are (numerically) evidenced in even simplest piecewise-smooth differential equations, where two linear vector fields switch between one another according to a certain law. Consider, for example, a couple of vector fields (F1,F2) and a couple of 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 NSF project CMMI-1436856 the aforementioned cycles and more complex attractors are obtained through a bifurcation from an equilibrium, 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 work on limit cycles in ABSs utilized perturbation theory (joint project with Tassilo Kuepper and Thomas Koppen). The preprint is available here.

My current work investigates border splitting bifurcation in Anti-Lock Braking System with two control modes: 1) build valve increases brake pressure, 2) dump valve decreases brake pressure. Here is the diagram of the respective ABS along with the limit cycle that the wheel slip and the brake torque exhibit. The build valve activates when the wheel slip gets smaller then minimal allowed value (dashed red line) and the damp valve activates when the wheel slip exceeds the maximal allowed value (dashed blue line).

abs

Viewing the distance between the two dashed line as a small parameter, I obtained the limit cycle through a suitable border-splitting bifurcation, see this preprint.

Graduate project available: Unfolding the fold-fold singularity in 3D (click)


Applications of perturbation theory in robot locomotion

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.

biped

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. The NSF project CMMI-1436856 aims to classify all walking gaits that the bipedal robot is capable to realize.

Graduate project available: Design of walking gaits in the simplest biped model via perturbation theory (click)


Applications of perturbation theory in neuroscience

Research Plan

Grazing bifurcations and occurence of chattering

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Nonsmooth dynamical systems may posses solutions that hit the switching manifold infinite number of times during a finite time interval (also known as chatter or Zeno phenomenon). A trajectory that undergoes such a phenomenon is colored in red at the figure.

Chattering reduces quality when cutting (Dombovari et al) or drilling (Ema et al). On the othe hand, chattering is what provides finite-time stability in robot locomotion (Lamperski et al) and can be utilized very advantageously. In either case the consequences of the occurence of chattering are important.

A comprehensive (Lyapunov-like) stability theory is available to study stability of Zeno solutions, but little is known about how these solutions emerge. Studying Zeno solutions at the very point of their birth would allow to gather important dynamical properties of Zeno phenomenon. However, to date, we don't even have a rigorous theory as for how Zeno solutions occur in a periodically driven mass-spring impact oscillator. Each solution (red curve at the figure) of such an oscillator jumps upwards instantaneously (white lines at the figure) upon approaching the obstacle (cylinder at the figure). There are two fundamental observations that make studying the emergence of Zeno solutions possible at least for the impact oscillator:

The only way for Zeno solutions to occur is through a grazing bifurcation (i.e. through a tangent collision of a stable orbit with the cylinder).
The points of collision with the switching manifold fill in invariant manifolds (·····) bounded by a so-called discontinuity arc (- - - -) (Chillingworth), see the figure. The dynamics on these invariant manifolds is investigated in Budd & Dux and Nordmark & Piiroinen.

At the same time, the simplest problem about transitions from a non-Zeno periodic solution to a Zeno one still lacks complete understading in the current theory of nonsmooth dynamical systems, see Wagg and Davis & Virgin for interesting results in this direction.

My collaborators on grazing bifurcations and chattering are Kryzhevich (St-Petersburg University) and Lamb (Imperial College London).

Bifurcations in Moreau's sweeping processes

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Consider a convex set $V(t)\subset\mathbb{R}^n$ that contains a small ball inside. Depending on the motion of the set $V(t)$, the ball will just stay immovable (in case it is not hit by the ring), or otherwise it is swept towards the interior of the set. In the latter case the velocity of the ball has to point inwards to the ring in order not to leave. If u(t) is the coordinate of the ball, the problem is formulated as $$ -\dot u(t)\in N_{C(t)}(u(t))+f(t,u(t)), $$ where $N_{C(t)}(u(t))$ is the outward normal cone to set $C(t).$ The evolution of u(t) is known as a Moreau's sweeping process. When the multi-function $t\mapsto C(t)$ is of bounded variation (often the case in applications), the solution of Moreau sweeping process can be a function of bounded variation only and the Lebesgue differentiation $\dot u(t)$ needs to be replaced by a Radon-Nikodym derivative with respect to a normalized measure. Kunze & Marques can serve as an introduction to the subject, where several existence and stability theorems are introduced. My interest in Moreau's sweeping processes is due to their intimate connection to measure differential inclusions, which are capable to describe and explain such outstanding phenomena of nonsmooth mechanics as Painleve paradoxes (Stewart).

To date, there is no any theory about generic singularities and bifurcations in Moreau's sweeping processes or Measure differential inclusions. I currently work on developing a bifurcation theory for Lebesgue differentiable Moreau's sweeping processes with with time-periodic right-hand-sides aiming to identify the most typical periodic regimes. My first paper in this direction (co-authored with Kamenskiy) is available here. Lyapunov stability theory for Moreau's sweeping processes and measure differential inclusions is available in Kunze & Marques and Leine & Wouw respectively.

Density of Sinai's billiards with grazing orbits

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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).

Non-monotone population models

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The generalized Gause prey-predator model with time-dependent coefficients reads as \begin{equation}\label{gause} \begin{array}{l} \dot x=x a(t,x)-yb(t,x),\\ \dot y=y(c(t,x)-d(t)), \end{array} \end{equation} where $a(t,x)$ is the specific growth rate of the prey in the absence of any predators, $b(t,x)$ is the predator response function, $c(t,x)$ is the proportion as to how the presence of prey enhances the growth of predator, $d(t)$ is the rate of how the predator population declines in the absence of prey.

I study periodic regimes under some monotonicity assumptions for the coefficients which are weak enough to not create any partial order for the flow of the system. In particular, it is not straightforward to spot a reasonable trapping region in this kind of model, see Ding & Jiang, Ding et al, Wolkowicz & Zhao. I took a perturbation approach instead and introduced such a perturbation that creates a trapping region $R_\varepsilon,$ which however converges to infinity as the perturbation parameters approaches $\varepsilon=0$, see Makarenkov.

The current goal is to extend this perturbative approach to generalized Gause prey-predator models with integro-differential ingredients, such as delays, impulses, switch-like interactions, etc. See Gouze & Sari and the above mentioned papers for motivations.

Bifurcations in nowhere differentiable differential equations

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Ordinary differential equations that lack Lipschitz regularity may occur when the method of characteristics is applied to solve certain transport equations, e.g. those governing incompressible fluid flows, see Ambrosio. This agrees with the classical theory of Kolmogorov that states that the dependence of the velocity vector on the position is only Holder with the exponent 1/3, see Falkovich et al. Multiple solutions that share same initial condition are possible in differential equations that lack Lipschitz regularity, see Hartman, Zuazua, Pugh & Wu, Zak. The commonly used technique for studying the behaviour of this class of systems is stochastic approximation (Ball, E & Vanden-Eijnden, Eyink) which embeds solutions' funnels of non-Lipschitz ODEs into stochastic flows.

The typical dynamical behaviour was investigated for those non-Lipschitz ODEs which are ε-close to smooth ones. The work by Kloeden & Kozyakin suggests that every transition in a smooth ODE induces a respective inflated transition in a nearby non-Lipschitz ODE. A result of this type is obtained by Kamenskii et al for Melnikov-type transitions. I am interested to understand the typical transitions in nondifferentiable ODEs of given Holder regularity.

Another class of applications that motivates my interest in studying nondifferentiable continuous ODEs is the continuous control method by Coron. This method replaces the trivial discontinuous control by a continuous one, while still ensuaring a finite-time stabilization. Nondifferentiability of the respective control system at the target attractor is a necessary condition for the method to work.