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
\begin{equation}\tag{1}\label{eq:sys1}
\ddot x=\nabla f(x),
\end{equation}
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
$$
J(x):=\int_0^p \left(\frac 12 \dot x(t)^2f(x(t)) \right) dt, \quad x(t)^2:=x(t)\bullet x(t).
$$
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 vectorcoordinates), 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 $\Gamma\times O(2)$equivariant degree to classify according to their symmetries the orbits of nonconstant $p$periodic solutions to system \eqref{eq:sys1}, including the properties of the Euler ring $U(\Gamma\times O(2))$ (see [11,3]), computations of the $\Gamma\times O(2)$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), 174.
[3] Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to $O(2) \times S^1$symmetric variational
problems: Equivariant gradient degree approach.Israel Math. Conf. Proc., Conf. Nonlinear Analysis and Optimization, Haifa, Israel, June 1824, 2008, AMS Contemporary Mathematics, 540
(2010), 4584.
(2008)
[4] J. Fura, A. Ratajczak and S. Rybicki,
Existence and continuation of periodic solutions of
autonomous Newtonian systems, J. Diff. Eqns 218 (2005),
216252.
[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, 247272.
[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),
14791516.
[8] S. Rybicki, A degree for $S^1$equivariant orthogonal maps and its applications to bifurcation theory. Nonlinear Anal. 23 (1994), 83102.
[9] S. Rybicki, Applications of degree for $S^1$equivariant gradient maps to variational nonlinear problems with $S^1$symmetries. Topol. Methods Nonlinear Anal. 9 (1997), 383417.
[10] S. Rybicki, Degree for $S^1$equivariant strongly indefinite functionals. Nonlinear Anal. 43 (2001), 10011017.
[11] T. tom Dieck, Transformation Groups. Walter de Gruyter, 1987.
Department of Mathematical Sciences, University of Texas at Dallas

Jan 23, 2015 
3pm4pm 
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 componentsgame theory and mean field theoryand 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 
3pm4pm 
FO 2.404 

Robert Gregg 
Energy Shaping Approaches from Robot Walking to LowerLimb Orthotics Department of Bioengineering, University of Texas at Dallas 
Feb 6, 2015 
3pm4pm 
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 
3pm4pm 
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 
3pm4pm 
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:
Cooperation 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 
3pm4pm 
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 pointfoot, underactuated robots. By incorporating feet into the model and correctly choosing the objective function for gait optimizations, the HZDbased model can be extended to accurately predict the lowerlimb 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 
3pm4pm 
FO 2.404 


Spring break

Mar 20, 2015 



Martin Brokate 
Weak differentiability of scalar hysteresis operators
Technische Universitaet, Munich 
Mar 25, 2015 
3pm4pm 
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 
3pm4pm 
FO 2.404 

Viswanath Ramakrishna 
The Controlled Invariance Problem
Department of Mathematical Sciences, University of Texas at Dallas 
Apr 10, 2015 
2pm3pm 
FO 2.404 

Masoud Yari 
Nonlinear analysis of a mathematical model of
propagation and degradation of vegetation patterns in semiarid ecosystems
online presentation via WebEx
The purpose of this
talk is to present and analyze a macroscale mathematical model of
vegetation
propagation/degradation in semiarid ecosystems. By using a nonlinear
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 selfactivation and lateral
inhibition, in the study of selfassembly of patterns in morphogenesis, is
well
formulated in works of Gierer and Meinhardt. The same principle can model
the
dynamics in semiarid 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 UniversityCorpus Christi 
Apr 17, 2015 
3pm4pm 
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 densitydependent 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 TexasPan American 
Apr 24, 2015 
3pm4pm 
FO 2.404 

Sophia Jang 
Dynamics of phytoplanktonzooplankton 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 phytoplanktonzooplankton 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 
3pm4pm 
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 
3pm4pm 
FO 2.404 
