For Spring 2018 our group seminar will take place each Friday from 2:00pm to 3:00pm in FO 3.616 (unless stated otherwise)

 Speaker Title/Abstract/Affiliation Date Time Room Slides Carlos García-Azpeitia Choreographies in the n-vortex problem Abstract: We study the equations of motion of n vortices of equal circulation in the plane, in a disk and on a sphere. The n vortices form a polygonal equilibrium in a rotating frame of reference. We investigate the bifurcation and numerical continuation of Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, correspond to choreographies of the n vortices. We include numerical results for all cases, and for various values of n. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México May 04, 2018 1:00 - 2:00PM FO 2.610F Tomoki Ohsawa Geometric Control of a Spherical Rolling Robot Abstract: I will talk about a geometric approach to controlling a spherical rolling robot like the Sphero (a programmable spherical toy robot that can be controlled by a cell-phone app). Specifically, we formulate the rolling sphere controlled by internal wheels as a control system on a principal bundle. We analyze its controllability using the connection naturally defined by the system as well as its curvature. We will also address an optimal control problem of the system and its relationship with an integrable Hamiltonian system. Department of Mathematical Sciences, The University of Texas at Dallas April 20, 2018 2:00 - 3:00PM FO 3.616 Nathan Williams Dynamical Algebraic Combinatorics Abstract: Cluster algebras provide a unified framework to explain certain phenomena arising in discrete dynamical systems---for example, both the integrality of the Somos-4 sequence and the periodicity of the pentagonal recurrence can be explained by appropriate cluster algebras. I will present a few recently-described related periodicity phenomena arising from root systems, and indicate some directions of generalization. Department of Mathematical Sciences, The University of Texas at Dallas March 02, 2018 2:00 - 3:00PM FO 3.616 Maxim Arnold Some properties of the string construction Abstract: In the theory of mathematical billiards, caustics correspond to the invariant curves splitting the phase space of the billiard transformation. The correspondence between the billiard table and its caustics resembles the correspondence between wave-fronts during the evolution. String construction helps to construct a table having particular caustic. I will discuss some properties of this construction. The talk is based on joint work with M. Bialy. Department of Mathematical Sciences, The University of Texas at Dallas February 23, 2018 2:00 - 3:00PM FO 3.616 Peter R. Wolenski Colloquium talk A Survey of Fully Convex Control problems Abstract: A Fully Convex Control (FCC) problem has the appearance of the classical calculus of variations Bolza problem $$\min \int_0^T L(x(t),\dot x(t))dt+l(x(0),x(T)),$$ where the minimization is over $x(\cdot)$ belonging to some class of arcs. The distinguishing features of FCC are that the data $L(\cdot, \cdot)$ and $l(\cdot,\cdot)$ (i) may take on the value $+\infty$ and (ii) are convex functions. Allowance of (i) provides great flexibility incorporating constraints so that most standard control problems come under its purlieu. However, broad generality is restrained by (ii), which although quite special, nonetheless includes the classical linear quadratic regulator and many of its generalizations. Moreover, the speciality of (ii) opens up the possibility of using convex dual formulations. Rockafellar developed FCC in the 1960-70's that evolved into more gen- eral nonsmooth control and variational analysis. We shall first survey these early results. Then we shall present new developments with state constraints and involve impulsive arcs. Department of Mathematics, Louisiana State University February 09, 2018 2:00 - 3:00PM FO 3.616 Wieslaw Krawcewicz Equivariant Brouwer Degree and its Applications (continuation of the talk of January 26) Department of Mathematical Sciences, The University of Texas at Dallas February 02, 2018 2:00 - 3:00PM FO 3.616 Wieslaw Krawcewicz Equivariant Brouwer Degree and its Applications Abstract: Since its introduction to mathematics, about 110 years ago, the topological degree was widely used as a tool to show the existence and multiplicity solutions in various types of differential equations. In my talk I would introduce the equivariant version of the Brouwer degree, provide its connection to the original Brouwer degree and list its most important properties and computational algorithms. In addition, we will discuss various problems related to the existence of periodic solutions in second order systems (discrete and continuous) with only minimal symmetry properties. Department of Mathematical Sciences, The University of Texas at Dallas January 26, 2018 2:00 - 3:00PM FO 3.616 Carlos García-Azpeitia Nonlinear Vibrations in the Fullerene Molecule C60 Abstract: The fullerene molecule is composed of 60 carbon atoms at the vertices of a truncated icosahedron. Kroto presents the definitive evidence of the existence of this molecule in 1986 and was awarded the Nobel price in chemistry for its discovery. The model of the fullerene molecule, consisting of 180 equations, is built on the framework of classical mechanics, where the force field comprise bond forces between carbon atoms connected by edges, and bond bending and torsion forces. The considered system of equations is equivariant under the action of the group I×O(3)×O(2), where I denotes the full icosahedral group. We prove the existence of global families of periodic solutions (nonlinear vibrational modes of oscillation) from the equilibrium configuration. We use the gradient equivariant degree to obtain the topological classification of all spatio-temporal symmetries. We find that the solutions are standing and rotating waves that propagate along the molecule with icosahedral, tetrahedral, pentagonal and triangular symmetries. This is a joint work with Wieslaw Krawcewicz, Manuel Tejada-Wriedt, Haopin Wu. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México January 19, 2018 2:00 - 3:00PM FO 3.616

 Speaker Title/Abstract/Affiliation Date Time Room Slides Mahsa Lotfi Recent Developments in Compressed Sensing Abstract:Compressed sensing refers to the recovery of high-dimensional but low-complexity objects from a limited number of measurements. Examples are the recovery of high-dimensional vectors with just a few nonzero components (but at unknown locations), and the "completion" of high-dimensional but low-rank matrices from measuring just a few components. In this talk I will focus on the vector recovery problem, and survey both well-established techniques as well as some of my doctoral work. Department of System Engineering, University of Texas at Dallas Dec 1, 2017 2:00 - 3:00PM SLC 1.202 Thanksgiving holiday Nov 23, 2017 Svetlozar Rachev Colloquium talk Option Pricing with Greed and Fear Factor: The Rational Finance Approach Abstract:We explain main concepts of Prospect Theory and Cumulative Prospect Theory within the framework of the rational dynamic asset pricing theory. We derived option pricing formulas when the asset returns are altered with a generalized Prospect Theory value function or a modified Prelec’s probability weighting function. We study the behavioral finance notion of “greed and fear” from the point of view of the rational dynamic asset pricing theory and derived the corresponding option pricing formulas in case of asset returns following continuous diffusion or discrete binomial trees. Department of Mathematics & Statistics, Texas Tech University Nov 17, 2017 2:00 - 3:00PM FN 2.102 Alireza Mohammadi Virtual Constraints Framework for Robotic Locomotion Control: Lessons and Directions for Cyberphysical Systems Abstract: Bioinspired robots, including autonomous bipeds, are prime examples of complex cyber-physical systems (CPS) with integrated computational and physical capabilities that can interact with humans in numerous modalities. Highly nonlinear and hybrid dynamics, nontrivial specifications involving nonlinear constraints on the state variables and control inputs, and safety critical- demands due to interaction with humans, are among a few properties that make the design and verification of these cyberphysical systems challenging. In this talk, we present the framework of virtual constraints and some of the advanced tools from Lagrangian and geometric mechanics, reduction of dynamical systems, computer algebra, pattern recognition, and extremum-seeking control that have been developed and/or tailored to address some of the problems in this framework. We show how some of these tools can be employed for effectively addressing CPS control challenges such as the curse of dimensionality, the tight safety-critical demands, and the need for controls hierarchical structure, through two robotic systems: snake robots and powered prostheses. Short Bio: Alireza Mohammadi received the M.Sc. and PhD degrees in Electrical and Computer Engineering from the University of Alberta, Edmonton, AB, Canada and the University of Toronto, Toronto, ON, Canada in 2011 and 2016, respectively. He received his B.Sc. degree in Electrical Engineering from Sharif University of Technology, Iran in 2009. He joined the Locomotor Control Systems Lab, the University of Texas at Dallas, Richardson, TX, USA as a Postdoctoral Research Associate in Nov. 2016. His research interests include cyberphysical systems, bioinspired robotics, wearable robots, nonlinear control, and hybrid systems. Locomotor Control Systems Laboratory, Department of Bioengineering, University of Texas at Dallas Nov 3, 2017 2:00 - 3:00PM SLC 1.202 Augusto Visintin Colloquium talk Compactness and Structural Stability of Nonlinear Flows Abstract: Parabolic equations are usually regarded as not representable by a minimization principle; however results of Brezis and Ekeland [1], Nayroles [3] and Fitzpatrick [4] provide evidence of the contrary. This can also be extended to the parabolic flow of pseudo-monotone operators. Compactness and stability of the dependence of the solution on the data and the potential can be proved on the basis of De Giorgi's notion of Gamma-convergence [2]. Here by compactness we mean that any sequence of potentials has a Gamma-convergent subsequence. Stability means that, if the data converge and the potentials Gamma-converge, then the sequence of solutions converges to a solution, up to extracting a subsequence, [5]. This was already known for stationary problems, and is here extended to evolutionary problems via a novel notion of evolutionary Gamma-convergence. These results can be extended in several directions, including doubly-nonlinear parabolic inclusions. This research is surveyed in [6]. References [1] H. Brezis, I. Ekeland: Un principe variationnel associ e a certaines equations paraboliques. I. Le cas ind ependant du temps, II. Le cas d ependant du temps. C. R. Acad. Sci. Paris S er. A-B 282 (1976) 971{974, and ibid. 1197{1198 [2] E. De Giorgi, T. Franzoni: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975) 842{850 [3] B. Nayroles: Deux th eor emes de minimum pour certains syst emes dissipatifs. C. R. Acad. Sci. Paris S er. A-B 282 (1976) A1035{A1038 [4] S. Fitzpatrick: Representing monotone operators by convex functions. Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), 59{65, Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, 1988 [5] A. Visintin: Variational formulation and structural stability of monotone equations. Calc. Var. Partial Di erential Equations 47 (2013), 273{317 [6] A. Visintin: On Fitzpatrick's theory and stability of flows. Rend. Lincei Mat. Appl. 27 (2016) 1{30 Department of Mathematics, University of Trento, Italy Oct 20, 2017 11:00 - 11:50AM FN 2.102 Swaroop Darbha String Stability of Automatic Vehicle Following Systems Abstract: Automatic Vehicle Following Systems (AVFS) have been investigated for the past seventy years and have recently been deployed in passenger vehicles in the form of Adaptive Cruise Control (ACC) Systems. AVFS couple the motion of vehicles by feedback and consequently, errors in spacing and velocity can propagate in a collection of vehicles employing AVFS. Amplification of errors in spacing can cause accidents. The topic of string stability in automatic vehicles is concerned with propagation of spacing errors in a collection of vehicles and is the central focus of my talk. In this talk, I will relate how vehicle models, information flow among vehicles and spacing policies play an important role in the propagation of errors by considering various types of information flow graphs. In particular, I will argue that symmetric information flow graphs are not suited for collections of vehicles desiring to maintain a rigid formation. Short Bio: Swaroop Darbha received his Bachelor of Technology from the Indian Institute of Technology - Madras in 1989, M. S. and Ph. D. degrees from the University of California in 1992 and 1994 respectively. He was a post-doctoral researcher at the California PATH program from 1995 to 1996. He has been on the faculty of Mechanical Engineering at Texas A&M University since 1997, where he is currently a professor. His current research interests lie in the development of vehicle control and diagnostic systems for Autonomous Ground Vehicles, development of planning, control and resource allocation algorithms for a collection of Unmanned Aerial Vehicles. Department of Mechanical Engineering, Texas A&M University Oct 12, 2017 4:00 - 5:00PM FN 2.102 Oleg Makarenkov Stabilization of quasistatic evolution of elastoplastic systems subject to periodic loading Abstract: This paper develops an analytic framework to design both stress and stretching/compressing T-periodic loadings which make the quasi-static evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function t → (e(t),p(t)), where ei(t) and pi(t) are the elastic and plastic deformations of spring i, defined on [t0,∞) by the initial condition (e(t0),p(t0)). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron C(t) in a vector space E of dimension d, it becomes natural to expect (based on a result by Krejci) that the solution t → (e(t),p(t)) always converges to a T-periodic function. The achievement of this paper is in spotting a class of sweeping processes and closed-form estimates on eligible loadings where the Krejci's limit doesn't depend on the initial condition (e(t0),p(t0)) and so all the trajectories approach the same T-periodic solution. The proposed class of sweeping processes is the one for which the normals of any d different facets of the moving polyhedron C(t) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. We discover that the normals of any d different facets of the moving polyhedron C(t) are linearly independent, if the number of stretching/compressing constraints is 2 less the number of nodes of the given network of springs and when the magnitude of the stress loading is sufficiently large (but admissible). In other words, we offer an analogue of the high-gain control method for elastoplastic systems, which can be used to design the properties of rheological models of materials (e.g. in creating smart materials). This is a joint work with Ivan Gudoshnikov Preprint: https://arxiv.org/abs/1708.03084 Department of Mathematical Sciences, The University of Texas at Dallas Oct 06, 2017 2:00 - 3:00PM SLC 1.202 Zalman Balanov Good deformations of multiple singularities: examples and applications Abstract: If a polynomial map defined in a finite-dimensional complex vector space admits only a finite number of roots and all of these roots are simple, then possible values of the Jacobian determinants at the roots are governed by the so-called Euler-Jacobi formula (in short, EJF). To extend the EJF to settings including multiple roots is not an east task, in general (essentially, there are no effective formulas allowing one to compute a residue at a multiple point in the multi-dimensional case). In a parallel way, the local behavior of a differential system near a hyperbolic equilibrium is equivalent to the behavior of the linearized system at the equilibrium (Grobman-Hartman Theorem). At the same time, if the equilibrium in question is not hyperbolic (for example, the Jacobian of the linearization vanishes), then describing a phase portrait near the equilibrium is a problem of formidable complexity. The standard theoretical method allowing one (at least, theoretically) to attack both problems is known under the wording resolution of singularity. Practical implementation of this method is based on deformations of a given map decomposing a compound singularity into simple ones. In concrete settings, to construct such a deformation is an art rather than a science. In my talk, I will describe some good deformations” and support them by two applications: (a) EJF including simple and double points, and (b) classification of phase portraits of differential systems around the so-called nilpotent equilibria. This talk is based on recent results obtained by Y. Krasnov (Bar Ilan University, Israel) and myself. Department of Mathematical Sciences, The University of Texas at Dallas Sep 22, 2017 1:00 - 2:00PM SLC 1.202 Carlos García-Azpeitia Global bifurcation of vortex and dipole solutions in Bose-Einstein condensates Abstract: The Gross-Pitaevskii equation models the dynamics of a Bose-Einstein condensate (BEC) with a symmetric harmonic trap. Periodic solutions of this equation play an important role in the understanding of the long term behavior of its solutions. We present the existence of several global branches of solutions among which there are vortex solutions and dipole solutions. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México Sep 15, 2017 1:00 - 2:00PM SLC 1.202 Zhaosheng Feng Colloquium talk Symmetry Analysis to the KdV-Burgers-Kuramoto equation Abstract: In this talk, we consider the KdV-Burgers-Kuramoto equation, a partial differential equation that occupies a prominent position in describing some physical processes in motion of turbulence and other unstable process systems. We convert the problem into an equivalent 3D-dimensional system and analyze its local dynamical behaviors. By means of the Lie symmetry reduction method and the Preller-Singer procedure, we show that there exist nontrivial bounded wave solutions under certain parametric conditions. Numerical simulations of wave phenomena are illustrated, which provide us rich dynamical information and are in agreement with our theoretical analysis. Department of Mathematics, University of Texas-Rio Grande Valley Sep 8, 2017 3:00 - 4:00PM SLC 1.202

 Speaker Title/Abstract/Affiliation Date Time Room Slides Dmitry Turaev Mathematics Distinguished Colloquium "Richness of Hamiltonian chaos"   poster Abstract: Since the time of Newton, physicists have described natural processes through differential equations. Nowadays it is commonly acknowledged that solutions to differential equations are often chaotic: a small variation in initial conditions leads to a large and unpredictable change in the behavior of the system. One should, therefore, try to describe this behavior statistically. However, we will show that a system with chaotic behavior often exhibits a variety of statistical patterns of such diversity that any attempt to comprehensively describe it will fail. We will argue that this ultimate richness is the main characteristic feature of chaos in practically any given system and show how this fact could help explain otherwise unexplained physical phenomena, like the low temperature flicker-noise in metals. Department of Mathematics, Imperial College London July 28, 2017 12:00 - 1:00PM FO 2.404

 Speaker Title/Abstract/Affiliation Date Time Room Slides Lakmi Niwanthi Stability of global solutions of Moreau sweeping processes Abstract: We develop a theory which allows making qualitative conclusions about the dynamics of both monotone and non-monotone Moreau sweeping processes. We first show that any sweeping processes with monotone right-hand-sides admits a globally exponentially stable solution defined on the entire real line. We prove that such a solution is almost periodic when the right-hand-side of the sweeping process is almost periodic in time. And then we describe the extent to which such a globally stable solution persists under non-monotone perturbations. Department of Mathematical Sciences, The University of Texas at Dallas December 9, 2016 3:00 - 4:00PM FO 2.610F Hooton Edwards Odd number type theorem for equivariant systems Abstract: Various odd number type theorems state necessary conditions for stabilizing an unstable periodic solution to a differential equation by Pyragas' delayed feedback control. We discuss an equivariant counterpart of these conditions for systems with a finite symmetry group. Department of Mathematical Sciences, The University of Texas at Dallas Nov 4, 2016 3:00 - 4:00PM JO 3.516 Hao-Pin Wu Department of Mathematical Sciences, The University of Texas at Dallas October 28, 2016 3:00 - 4:00PM JO 3.516 Dmitry Rachinskiy and Daniel Kim The Devil is in the Details: Spectrum of the Discrete Preisach Memory Model and Chebyshev Polynomials Abstract: We consider a matrix associated with a random walk on a graph that describes transitions between 2^N states of the discrete Preisach memory model. This matrix can be also associated with the last-in-first-out (LIFO) inventory management rule. An explicit solution for the spectrum is presented. Bohmer?s Devil staircase, which appears in the limit of large N, is related to golden ratio and the Rabbit number. Department of Mathematical Sciences, The University of Texas at Dallas (Dmtriy Rachinskiy) and University of North Texas (Daniel Kim) October 21, 2016 3:00 - 4:00PM JO 3.516

 Speaker Title/Abstract/Affiliation Date Time Room Slides Jianzhong Su Influence of Synaptic Coupling on Bursting Oscillatory Solutions of Neuron Models Abstract: Neurons often exhibit bursting oscillations, as a mechanism to modulate and set pace for other brain functionalities. These bursting oscillations are distinctly characterized by a silent phase of slowly evolving steady states and an active phase of rapid firing oscillations. These bursting neurons can be modeled by fast-slow systems consisting of several ordinary differential equations in two time scales. In a network of neurons, their collective oscillatory behavior may differ from individual neurons due to coupling and inputs from other neurons. We analyze the transition mechanisms between periodic and chaotic/random behavior in a coupled system of neurons, in a typical square wave neuron model. Using geometric and bifurcation analysis, we provide insight on how coupling can regularize chaotic trajectories, and study the transition behavior from a flow-induced Poincare map. Department of Mathematics, University of Texas at Arlington April 22, 2016 3:00 - 4:00PM FO 2.702 Kaveh Fathian Applications of Algebraic Geometry in Camera Pose Estimation Abstract: This talk consists of four parts. First, we show how the camera pose estimation is formulated as a mathematical problem. We then show how existing algorithms use the computational algebraic geometry tools to solve the problem. Next, we propose a novel formulation using quaternion representation of rotation that eschews the shortcomings of the existing methods. Lastly, we discuss some open problems and show how camera pose estimation and other engineering fields can benefit from possible solutions. Department of Electrical Engineering, University of Texas at Dallas Apr 15, 2016 3:00 - 4:00PM FO 2.702 Hao-Pin Wu Department of Mathematical Sciences, University of Texas at Dallas Apr 1, 2016 3:00 - 4:00PM FO 2.702 Thomas Carr An SIRS epidemic model with a rapidly decreasing probability for temporary immunity Abstract: We consider an SIRS model for disease dynamics that accounts for temporary immunity whereby recovered individuals return to the susceptible class. In particular, we allow for a general probability function of remaining immune for a given time after recovery such that the model is a system of integro-differential equations. We first show that by considering a rapidly decreasing probability function that the original model can be approximated by a system of delay-differential equations. Perturbation methods are then applied to the delay equations to determine how the amplitude of oscillations, which correspond to repeated epidemics, depends on the system parameters and, in particular, the zeroth, first, and second moments of the probability distribution. Department of Mathematics, Southern Methodist University Mar 25, 2016 3:00 - 4:00PM FO 2.702 Stefan Siegmund Bifurcations and continuous transitions of attractors in autonomous and non autonomous systems Abstract: Nonautonomous bifurcation theory is a research field which is still in its infancy. It studies the change of attractors of nonautonomous systems. We present a stability theorem ensuring the existence of nearby attractors of perturbed systems. They depend continuously on a parameter if and only if the attraction is uniform w.r.t. the parameter, i.e. the attractors are equiattracting. We apply these principles to explicit systems to clarify the meaning of continuous and abrupt transitions of attractors in contrast to bifurcations, i.e. splitting of minimal invariant subsets into others within the attractor. Several examples are treated, including a nonautonomous pitchfork bifurcation. Dresden University of Technology Mar 4, 2016 3:00pm-4:00pm FO 2.702 Dmitri Vainchtein Mixing by Resonances in Multi-scale Stokes Flows Abstract: Mixing in near-integrable flows with a clear separation of time or spacial scales is notoriously difficult to achieve. In many flows the intrinsic symmetries create invariant surfaces that act as barriers to chaotic advection and mixing. Thus, a key to efficient mixing is to add to the original (symmetric) flow a certain kind of perturbation that destroys those symmetries. In the present talk we discuss a quantitative long-time theory of mixing due to the presence of resonances in 3-D near-integrable Stokes flows. The resonance phenomena, such as scattering on resonance, capture into resonance, and separatrix crossings, involving different components of the original flow and the perturbation may destroy the invariant surfaces, paving a way to the large-scale mixing in a big fraction of the fluid flow. We explain the extend and the rate of mixing in terms of the evolution of the adiabatic invariants of the system. We show that when the leading phenomenon is scattering on resonances or separatrix crossings, the resulting mixing can be described in terms of a single 1-D diffusion-type equation, with parameters of the diffusion equation defined by the averaged statistics of a single passage through resonance or separatrix. Temple University Feb. 26, 2016 3:00pm-4:00pm FO 2.702 Carlos Garcia-Azpeitia Molecular chains interacting by Lennard-Jones and Coulomb forces Department of Mathematics, National Autonomous University of Mexico Jan 15, 2016 2:00 - 3:00PM FO 2.404

 Speaker Title/Abstract/Affiliation Date Time Room Slides Thomas Hagstrom Some Problems in the Numerical Analysis of Wave Equations in the Time Domain Abstract: Ecient time-domain solvers for wave propagation problems must include three crucial components: i. Radiation boundary conditions which provide arbitrary accuracy at small cost (spectral convergence, weak dependence on the simulation time and wavelength) ii. Robust high-resolution volume discretizations applicable in complex ge- ometry (i.e. on grids that can be generated eciently) - we believe that high-resolution methods enabling accurate simulations with minimal dofs-per-wavelength are necessary to solve dicult 3 + 1-dimensional problems with the possibility of error control. iii. Algorithms for directly propagating the solution to remote locations - avoid sampling the wave whenever possible. In this talk we will discuss recent developments in all three areas, with a focus on unresolved issues arising in complex and nonlinear models. Department of Mathematics, Southern Methodist University Dec 11, 2015 1pm-2pm FN 2.104 Benito Chen-Charpentier Some Applications of Delay Differential Equations to Biology Authors: Benito Chen-Charpentier, University of Texas at Arlington Ibrahim Diakite, University of Harvard, Medical School Abstract: Many physical and biological processes such as transmission, gestation, maturation and reproduction take time to complete. These times are usually incorporated in mathematical models as delays. The introduction of delays in differential equations changes their solution, may change the uniqueness of solutions and may change the stability of the equilibrium points. The delays can also change the bifurcation diagrams obtained as other parameters in the model are varied. In this presentation we study the interaction between delay and bifurcation parameters for a virus propagation mathematical model and show the bifurcation diagram changes. Other models with delays will also be presented. Department of Mathematics, University of Texas at Arlington Dec 4, 2015 1pm-2pm FN 2.104 Pavel Krejci Colloquium talk Kurzweil Integral and Discontinuous Hysteresis Institute of Mathematics, Czech Academy of Sciences, Prague Nov 20, 2015 3pm-4pm FN 2.102 Mikhail Kamenskiy Colloquium talk On M.A.Krasnoselskii and I.G. Malkin bifurcations Department of Mathematics, Voronezh State University, Russia Nov 20, 2015 2pm-3pm FN 2.102 Sergey Antonyan Colloquium talk Universal Free G-Spaces UNAM, Mexico Nov 13, 2015 1pm-2pm GR 3.302 Wieslaw Krawcewicz Equivariant Degrees and its Applications Abstract: 1. Equivariant degree with no free parameters and its applications to symmetry BVP for ODEs 2. Twisted equivariant degree with one free parameter and its applications to symmetric Hopf bifurcation (for ODEs) 3. Gradient equivariant and its applications to variational systems (e.g. Newtonian systems). Some elementary examples of symmetric systems will be discussed. Department of Mathematical Sciences, UT Dallas Nov 6, 2015 1pm-2pm FN 2.104 Irina Berezovik Building and analyzing a mathematical model for spreading of infectious deceases in multiple communities Department of Mathematical Sciences, UT Dallas Oct 30, 2015 1pm-2pm FN 2.104 Wieslaw Krawcewicz Equivariant Degree: basic ideas, axioms and applications to equations with symmetries Department of Mathematical Sciences, UT Dallas Oct 23, 2015 1pm-2pm FN 2.104 Nick Gans Real-Time Optimization for Visual Search Abstract: Deploying autonomous robots and vehicles to search for people or objects has numerous applications, including surveillance, security, human/machine interaction, search and rescue, environmental monitoring and scientific exploration. In some cases, a specific object may be desired. In others, only basic descriptions may be given, such as color. The search task could be as vague as ï¿½find the most interesting object in the room.ï¿½ I will discuss solving this task by casting it as an optimization problem. The first challenge is to describe the search task with an objective mathematical function that takes its maximum value when the desired object is in view. The second challenge is developing real-time optimization algorithms that move the robot to poses that systematically improve the objective function until it reaches its maximum. Bio: Dr. Gans is an Assistant Professor at The University of Texas at Dallas. His research interests include nonlinear and adaptive control, with focus on vision-based control and estimation, robotics and autonomous vehicles. Current research includes control of self-optimizing autonomous sensors to maximize sensor information, human-machine interfaces, vision-based control of autonomous ground and air vehicles. Dr. Gans has published over 75 peer-reviewed conference and journal papers, and he holds two patents in these areas. Dr. Gans earned his BS in electrical engineering from Case Western Reserve University in 1999, then his M.S. in electrical and computer engineering in 2002 and his Ph.D. in systems and entrepreneurial engineering from the University of Illinois Urbana-Champaign in 2005. Prior to joining UT Dallas, he worked as a postdoctoral researcher with the Mechanical and Aerospace Engineering Department at the University of Florida and as a postdoctoral associate with the National Research Council, where he conducted research on control of autonomous aircraft for the Air Force Research Laboratory Munitions Directorate and developed the Visualization Laboratory for simulation of vision-based control systems. Department of Electrical Engineering, UT Dallas Oct 16, 2015 1pm-2pm FN 2.104 Edward Hooton Sliding Hopf bifurcation in interval systems Department of Mathematical Sciences, UT Dallas Sep 25, 2015 1pm-2pm FN 2.104 Sergey Kryzhevich One-dimensional chaos in a system with dry friction Faculty of Mathematics and Mechanics, Saint-Petersburg State University, Russia Fulbright Fellow at UTD Mathematics Department, Instructor of Nonlinear Dynamics Sep 11, 2015 1pm-2pm FN 2.104

 Speaker Title/Abstract/Affiliation Date Time Room Host Wieslaw Krawcewicz Multiple Symmetric Solutions to Newtonian Systems with Symmetries: The Gradient Equivariant Degree Approach Finding $p$-periodic solutions to the second order Newtonian system of the type $$\tag{1}\label{eq:sys1} \ddot x=-\nabla f(x),$$ where $f:\mathbb{R}^n\to \mathbb{R}$ is a $C^2$-differentiable function, can be reduced to the problem of finding critical points of the functional $J: \mathscr H\to \mathbb{R}$, where $\mathscr H:=H^1(S_p^1;V)$, $V:=\mathbb{R}^n$ and $S_p^1:=\mathbb{R}/(p/2\pi) \mathbb{Z}$, is given by $$J(x):=\int_0^p \left(\frac 12 |\dot x(t)|^2-f(x(t)) \right) dt, \quad |x(t)|^2:=x(t)\bullet x(t).$$ It is well known that the functional $J$ is $S^1$-invariant (where $S^1$ acts on $\mathscr H$ by shifting the argument) (see for example [7,4] see also [8,9,10]), however the system \eqref{eq:sys1} is time reversible, which induces a natural $O(2)$-action on $\mathscr H$. In the case $V$ is an orthogonal $\Gamma$-representation (we assume that $\Gamma$ is a finite group acting on $V=\mathbb{R}^n$ by permuting vector-coordinates), the problem \eqref{eq:sys1} can be reduced to the $G$-equivariant equation, $G=\Gamma\times O(2)$, $$\nabla J(x)=0,\quad x\in \mathscr H,$$ and the gradient $G$-equivariant degree (developed by Geba in [5]) can be applied. In my talk, I will present our most recent results (obtained together with Mietek Dabkowski and Yanli Lu, cf. [6]) on the usage of the gradient Γ×O(2)$\mathrm{\Gamma }×O\left(2\right)$$\Gamma\times O(2)$-equivariant degree to classify according to their symmetries the orbits of non-constant p$p$$p$-periodic solutions to system $\text{(1)}$$\eqref{eq:sys1}$, including the properties of the Euler ring U(Γ×O(2))$U\left(\mathrm{\Gamma }×O\left(2\right)\right)$$U(\Gamma\times O(2))$ (see [11,3]), computations of the Γ×O(2)$\mathrm{\Gamma }×O\left(2\right)$$\Gamma\times O(2)$-basic degrees (see [2,1]), etc. References: [1] Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree. AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006. [2] Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed Point Theory App. 8 (2010), 1-74. [3] Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to O(2)×S1$O\left(2\right)×{S}^{1}$$O(2) \times S^1$-symmetric variational problems: Equivariant gradient degree approach.Israel Math. Conf. Proc., Conf. Nonlinear Analysis and Optimization, Haifa, Israel, June 18-24, 2008, AMS Contemporary Mathematics, 540 (2010), 45-84. (2008) [4] J. Fura, A. Ratajczak and S. Rybicki, Existence and continuation of periodic solutions of autonomous Newtonian systems, J. Diff. Eqns 218 (2005), 216-252. [5] K. Geba, Degree for gradient equivariant maps and equivariant Conley index, in Topological Nonlinear Analysis II (Frascati, 1995), Progr. Nonlinear Differential Equations App. 27, Birkh\"auser, Boston, 1997, 247-272. [6] M. Dabkowski, W. Krawcewicz and Y. Lv, Multiple Periodic Solutions for Symmetric Second Order Newtonian Systems with Even Potentials, Preprint 2015. [7] H. Ruan and S. Rybicki, Applications of equivariant degree for gradient maps to symmetric Newtonian systems, Nonlinear Anal. 68 (2008), 1479-1516. [8] S. Rybicki, A degree for S1${S}^{1}$$S^1$-equivariant orthogonal maps and its applications to bifurcation theory. Nonlinear Anal. 23 (1994), 83-102. [9] S. Rybicki, Applications of degree for S1${S}^{1}$$S^1$-equivariant gradient maps to variational nonlinear problems with S1${S}^{1}$$S^1$-symmetries. Topol. Methods Nonlinear Anal. 9 (1997), 383-417. [10] S. Rybicki, Degree for S1${S}^{1}$$S^1$-equivariant strongly indefinite functionals. Nonlinear Anal. 43 (2001), 1001-1017. [11] T. tom Dieck, Transformation Groups. Walter de Gruyter, 1987.   Department of Mathematical Sciences, University of Texas at Dallas Jan 23, 2015 3pm-4pm FO 2.404 Jameson Graber Mean field games: an introduction Mean field game theory has been making many advances in the past decade. Its many applications are found in economics and finance, networks and cybersecurity, and even biology. In this presentation we introduce the fundamentals of the theory, starting with an explanation of the two conceptual components--game theory and mean field theory--and then putting them together. We will see how this leads to interesting mathematical models composed of nonlinear partial differential equations, and we will discuss some of the technical tools used to analyze them. Finally, we will present some of latest results on mean field games and make some remarks on open problems.   Department of Management, University of Texas at Dallas Jan 30, 2015 3pm-4pm FO 2.404 Robert Gregg Energy Shaping Approaches from Robot Walking to Lower-Limb Orthotics Department of Bioengineering, University of Texas at Dallas Feb 6, 2015 3pm-4pm FO 2.404 Ivan Gudoshnikov On the stability of perturbed linear dynamical systems in ordered spaces Department of Mathematical Sciences, University of Texas at Dallas Feb 13, 2015 3pm-4pm FO 2.404 Guojun Gan Math Colloquium Department of Mathematics, University of Connecticut Feb 20, 2015 2pm- 3pm FO 2.404 Viswanath Ramakrishna rescheduled to April 10 due to inclement weather Feb 27, 2015 3pm-4pm FN 2.106 Hasan Poonawala Preserving Strong Connectivity in Communication Networks Subtitle: Use of the structure of the Perron eigenvector for reducible stochastic matrices in control applications Abstract: Co-operation between multiple mobile robots often requires that the communication network formed by these robots be strongly connected. As the robots move in order to achieve some task, the loss of communication links may lead to loss of strong connectivity. We model the directed communication network using a stochastic matrix whose entries depend on the states of the robots. When the network is strongly connected, the stochastic matrix is irreducible and its Perron eigenvector has strictly positive components. If the network loses strong connectivity due to motion of the robots, the stochastic matrix may become reducible and some of the components of its Perron eigenvector may vanish. In this talk, we will describe which components will vanish, and how this knowledge can be used to derive a control law which preserves the strong connectivity of the network, under suitable conditions. Department of Engineering & Computer Science, University of Texas at Dallas Mar 6, 2015 3pm-4pm FO 2.404 Anne Martin Improving Amputee Walking Gait using Ideas from Underactuated Robot Control Because current commercial prostheses do not completely replicate the function of the physiological foot and ankle, amputee walking gait is typically both less efficient and less stable than healthy human walking. Powered prostheses offer one potential solution, although current devices and control methods are unable to completely restore healthy gait. In addition, for both powered and passive prostheses, the effects of foot design on walking ability are largely unknown. Thus, the development of a modeling and control method that can predict both healthy and amputee walking could allow for improvements in the physical design of prostheses and in the control design of powered prostheses. Unfortunately, existing models are either too computationally expensive or too simplistic to be used for this purpose. This talk presents a model of intermediate complexity based on the Hybrid Zero Dynamics (HZD) control approach that was originally developed for point-foot, underactuated robots. By incorporating feet into the model and correctly choosing the objective function for gait optimizations, the HZD-based model can be extended to accurately predict the lower-limb kinematics and energy expenditure of healthy human walking across a wide range of speeds. To use the model to investigate human amputee gait, it has been further extended to allow asymmetrical gait. This talk will describe the model, discuss the technical details of its extension from the original robotic formulation, demonstrate its predictive capabilities for healthy human walking gait, and present ongoing work in predicting human amputee gait and in developing a control formulation for a powered prosthesis. Department of Mechanical Engineering, University of Texas at Dallas Mar 13, 2015 3pm-4pm FO 2.404 Spring break Mar 20, 2015 Martin Brokate Weak differentiability of scalar hysteresis operators Technische Universitaet, Munich Mar 25, 2015 3pm-4pm FO 2.208 Mohamed Khamsi On fixed Point Theory of Monotone Mappings Nonexpansive mappings are those maps which have Lipschitz constant equal to 1. The fixed point theory for such mappings is rich and varied. It finds many applications in nonlinear functional analysis. The existence of fixed points for nonexpansive mappings in Banach and metric spaces have been investigated since the early 1960s. Recently a new direction has been discovered dealing with the extension of the Banach Contraction Principle to metric spaces endowed with a partial order. The first attempt was successfully carried by Ran and Reurings. In particular, they showed how this extension is useful when dealing with some special matrix equations. Another similar approach was carried by Nieto et al. and used such arguments in solving some differential equations. Jachymski gave a more general unified version of these extensions by considering graphs instead of a partial order. In this talk, we will discuss the case of monotone nonexpansive mappings. Some of the results obtained are new and open the door to some new directions in metric fixed point theory. Department of Mathematical Sciences, University of Texas at El Paso Mar 27, 2015 3pm-4pm FO 2.404 Viswanath Ramakrishna The Controlled Invariance Problem Department of Mathematical Sciences, University of Texas at Dallas Apr 10, 2015 2pm-3pm FO 2.404 Masoud Yari Nonlinear analysis of a mathematical model of propagation and degradation of vegetation patterns in semi-arid ecosystems online presentation via WebEx The purpose of this talk is to present and analyze a macro-scale mathematical model of vegetation propagation/degradation in semi-arid ecosystems. By using a non-linear analysis, I will derive dynamic phase transition equations and introduce the phase transition parameters. These phase transition parameters play a fundamental role in the structural stability analysis of solutions and geometric objects in the phase space as well as understanding transitions to higher states. I will focus on different localized patterns and their transitions, following by a discussion on strategies in qualitative parameter approximation for such systems. Such an analysis is useful in determining uncertainty regions, finding tipping points, and revealing some other hidden dynamics. The idea behind this approach goes back to pioneer studies of morphogenesis in biological systems. The principle of local self-activation and lateral inhibition, in the study of self-assembly of patterns in morphogenesis, is well formulated in works of Gierer and Meinhardt. The same principle can model the dynamics in semi-arid ecosystems that exhibit localized patterns in vegetation due to limitation of water resources, leading to formation of visually captivating features in aerial images, such as spot, gap, and labyrinth patterns as well as roll patterns on gradual slopes. School of Engineering & Computing Sciences, Texas A&M University-Corpus Christi Apr 17, 2015 3pm-4pm FO 2.404 Zhaosheng Feng Degenerate Parabolic System and Its Approximate Solutions In this talk, we are concerned with approximate solutions to a degenerate parabolic system. We provide a connection between the Abel equation of the first kind, an ordinary differential equation that is cubic in the unknown function, and the degenerate parabolic system, a partial differential equation that is the dispersion model of biological populations with both density-dependent diffusion and nonlinear rate of growth. We present the integral forms of the Abel equation with the initial condition. By virtue of the integral forms and the Banach Contraction Mapping Principle we derive the asymptotic expansion of bounded solutions in the Banach space, and use the asymptotic formula to construct approximate solutions to the degenerate parabolic system. Department of Mathematics, University of Texas-Pan American Apr 24, 2015 3pm-4pm FO 2.404 Sophia Jang Dynamics of phytoplankton-zooplankton systems with toxin producing phytoplankton There are many toxin producing phytoplankton (TPP) species in natural systems that can have extremely harmful effects on the pop- ulations interacting with them. We rst review several systems of plankton interactions with TPP consideration in the literature. We then present our phytoplankton-zooplankton models with toxin pro- ducing phytoplankton to study the effects of TPP upon extinction and persistence of the populations. One model assumes spatial ho- mogeneity while the other model considers spatial heterogeneity by incorporating diffusion. It is concluded that TPP may promote sur- vival of phytoplankton population and may destabilize the interactions depending on parameter regimes. Numerical simulations indicate that passive diffusion of both populations can simplify the dynamics of the interactions and exhibit plankton patchiness. Department of Mathematics and Statistics Texas Tech University May 1, 2015 3pm-4pm FO 2.404 Oleg Makarenkov Bifurcation of stable limit cycles in switched systems Department of Mathematical Sciences, University of Texas at Dallas May 8, 2015 3pm-4pm FO 2.404

 Speaker Title/Abstract/Affiliation Date Time Room Slides Host Dmitry Rachinskiy Combining plays: from Prandtl-Ishlinskii operator to network models Sep 5, 2014 3pm-4pm FO 2.404 Maxim Arnold Escaping orbits in switching Hamiltonian systems Sep 12, 2014 4pm- 5pm FO 2.404 Maxim Arnold Escaping orbits in switching Hamiltonian systems II Sep 19, 2014 3pm-3:40pm FO 2.404 Faculty meeting Sep 26, 2014 Zalman Balanov Complex structures in Algebra, Analysis, Topology and Differential Equations Oct 3, 2014 3pm-4pm FO 2.404 Yifei Lou An introduction to mathematical models in image processing with a focus on PDEs and optimization techniques Oct 10, 2014 3pm-4pm FO 2.404 Sergey Antonyan The Gromov-Hausdorff metric and hyperspaces The Gromov-Hausdorff distance dGH${d}_{GH}$$d_{GH}$ is a useful tool for studying topological properties of families of metric spaces. M. Gromov first introduced the notion of the Gromov-Hausdorff distance dGH${d}_{GH}$$d_{GH}$ in his ICM 1979 address in Helsinki on synthetic Riemannian geometry. Two years later dGH${d}_{GH}$$d_{GH}$ appeared in the book M. Gromov. Structures metriques pour les varietes riemanniennes. Vol. 1 of Textes Mathematiques [Mathematical Texts], CEDIC, Paris, 1981. Edited by J. Lafontaine and P. Pansu. For two compact metric spaces X$X$$X$ and Y$Y$$Y$ the number dGH(X,Y)${d}_{GH}\left(X,Y\right)$$d_{GH}(X,Y)$ is defined to be the infimum of all Hausdorff distances dH(i(X),j(Y))${d}_{H}\left(i\left(X\right),j\left(Y\right)\right)$$d_H(i(X),j(Y))$ for all metric spaces M$M$$M$ and all isometric embeddings i:X→M$i:X\to M$$i:X\to M$ and j:Y→M.$j:Y\to M.$$j:Y\to M.$ Clearly, the Gromov-Hausdorff distance between isometric spaces is zero; it is a metric on the family GH$GH$$GH$ of isometry classes of compact metric spaces. The metric space (GH,dGH)$\left(GH,{d}_{GH}\right)$$(GH,d_{GH})$ is called the Gromov-Hausdorff hyperspace. It is a challenging open problem to understand the topological structure of this metric space. This talk contributes towards this problem. We mainly are interested in the following subspaces of GH$GH$$GH$ denoted by GH(Rn),$GH\left({\mathbb{R}}^{n}\right),$$GH(\mathbb{R}^n),$ n≥1$n\ge 1$$n\ge 1$, and called the Gromov-Hausdorff hyperspace of Rn.${\mathbb{R}}^{n}.$$\mathbb{R}^n.$ Here GH(Rn)$GH\left({R}^{n}\right)$$GH(R^n)$ is the subspace of GH$GH$$GH$ consisting of the classes [E]∈GH$\left[E\right]\in GH$$[E]\in GH$ whose representative E$E$$E$ is a metric subspace of the Euclidean space Rn.${\mathbb{R}}^{n}.$$\mathbb{R}^n.$ One of the results in this talk asserts that GH$GH$$GH$ is homeomorphic to the orbit space 2Rn/E(n),${2}^{{R}^{n}}/E\left(n\right),$$2^{R^n}/ E(n),$ where 2Rn${2}^{{R}^{n}}$$2^{R^n}$ is the hyperspace of all nonempty compact subsets of Rn${\mathbb{R}}^{n}$$\mathbb{R}^n$ endowed with the Hausdorff metric and E(n)$E\left(n\right)$$E(n)$ is the isometry group of Rn.${\mathbb{R}}^{n}.$$\mathbb{R}^n.$ This is applied to prove that GH(Rn)$GH\left({\mathbb{R}}^{n}\right)$$GH(\mathbb{R}^n)$ is homeomorphic to the Hilbert cube with a removed point.   Department of Mathematics, National Autonomous University of Mexico Oct 17, 2014 3pm-4pm FO 2.404 Wieslaw Qingwen Hu A state-dependent model with unimodal feedback Oct 24, 2014 3pm-4pm FO 2.404 Vadim Azhmyakov Modern Directions in Optimization and Control of Switched Systems: New Challenges for Mathematicians Hybrid and switched systems are mathematical models of heterogeneous control processes consisting of a continuous part, a finite number of continuous controllers, and a discrete supervisor. These models can represent an extremely wide range of systems of practical interest and are accepted as realistic models, for instance, in industrial electronics, power systems engineering, maneuvering aircrafts, automotive control systems, controllable chemical processes and communication networks. The emergence of a discrete event systems modeling framework is providing a new perspective for some important modern control processes and also constitutes a new challenges for Applied Mathematicians. In our talk, we discuss some specific families of switched systems, and the corresponding dynamic optimization problems. The class of problems to be considered concerns models, where discrete transitions are being triggered by the continuous dynamics, and are accompanied by discontinuous changes in the continuous state variable (state jumps). The control objective is to minimize a cost functional, where the control parameters are not only usual control inputs but also the discrete events associated with the switched nature of the systems dynamics. We describe some newly elaborated implementable solution procedures, consider illustrative examples and point the main directions of the recent investigations in the area.   Faculty of Electronic and Biomedical Engineering, University of Antonio Nariï¿½o Neiva, Republic of Colombia Oct 31, 2014 3pm-4pm FO 2.404 Oleg Vladimir Dragovic Introduction to pseudo-integrable billiards Nov 7, 2014 3pm-4pm FO 2.404 Mieczyslaw Dabkowski Catalan States of Lattice Crossing Nov 14, 2014 3:15pm- 4:15pm FO 2.404 Byungik Kahng homepage Trichotomy of Singularities of 2-Dimensional Bounded Invertible Piecewise Isometric Dynamics It is known that the singularities of 2-dimensional bounded invertible piecewise isometric dynamical systems can be classified as, removable, shuffling and sliding singularities, based upon their geometrical traits; and that only the sliding singularity can generate Devaney-chaos, while the others remain innocuous. However, the afore-mentioned classification had been somewhat incomplete in that the clear distinction between the sliding and the shuffling singularities had been unavailable. The presented research resolves this difficulty and completes the trichotomy, by completely characterizing the nature of sliding singularity. We use the invertible dual rotation map (or the Goetz map), as the main tool. The above abstract may change. I am assuming that I can complete what I am working on by the end of October. If I do not, then I cannot claim that my research would, "complete the trichotomy." I may end up talking about the background only and leave the technical part for next talk several weeks or months later.   Department of Mathematics and Information Sciences, University of North Texas at Dallas Nov 21, 2014 3pm-4pm FO 2.404 Oleg Thanksgiving holiday Nov 28, 2014 Julian Newman homepage Math Colloquium Dec 5, 2014 3:00pm-4:00pm FO 2.404