# Student Dynamics Seminar SPRING 2017

### About

The Student Dynamics Seminar exists for graduate students studying dynamical systems to give talks in a low pressure environment - no one with a PhD is allowed to attend. Talks can be about your own research, a paper you feel is important for people to understand, or something fundamental you feel was left out in a course, or anything you think is mathematically interesting involving dynamical systems.**It's a great place to practice your quals talk!**

### Meetings

The seminar meets**Thursdays from 4:00 - 4:50 pm in MCS B19**with tea served at 3:45 in the same room.

### Talks

Feb 16 2017**Speaker:**Parker Kuklinski**Title:**Spectral Analysis of Finite Quantum Walks**Abstract:**Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one dimensional quantum walk partially reflects information, as observed by computing absorption probabilities. In this presentation, I will shift from the local phenomena of absorption probabilities to the global behavior of finite quantum walks. These systems may be represented by block tri-diagonal matrices, so we use iterative methods to handle their respective spectra. The eigenvalues of the finite absorbing quantum walk accumulate on two sectors of the unit circle and the corresponding top eigenvectors can only be estimated. These results extend to three-state Grover walks and to quantum walks in two dimensions. In addition, these finite absorbing walks exhibit structure at smaller time scales in the form of modal phenomena, which I will computationally explore. This presentation will feature visualizations from QWSIM software.

**SPECIAL TIME: 3:15 - 4:00 pm & PLACE MCS 148**

**Speaker:**Alex Gelastopoulos**Title:**What's behind the Hodgkin-Huxley equations**Abstract:**The talk will be a back and forth between biology and mathematics. The goal is to convince you that you chose the least interesting of the two fields.

**Speaker:**Roland Welter**Title:**An invariant manifold theorem for C1 semigroups on Banach spaces**Abstract:**Calculating the invariant manifolds of a dynamical system is a general technique by which the dimension of the system can often be greatly reduced, significantly simplifying the problem. While the invariant manifold theorems for ODE's are well established, for PDE's the right set of sufficiently weak assumptions which still lead to useful conclusions is by no means clear. In this talk I'll present the invariant manifold theorem of Chen, Hale, and Tan, which provides a general context in which one can recover the main features of the theorems from ODE's.

**Speaker:**Patrick Cummings**Title:**NLS Approximation using Analytic Functions and Modified Energy Functionals**Abstract:**We consider two physical models that can be approximated by wave packets using the nonlinear Schr\"odinger equation. We first consider the Klein-Gordon-Zakharov system and prove an approximation result using analytic norms. We then consider a model equation that captures important properties of the water wave equation and discuss a new proof that both simplifies and strengthens previous results of Wayne and Schneider. Rather than using analytic norms, we construct a modified energy functional so that the approximation holds for the full interval of existence of the approximate NLS solution rather than just a subinterval (as is seen in the analytic case). Furthermore, the proof avoids problems associated with inverting the normal form transform by working with a modified energy functional motivated by Craig and Hunter et. al.

**Speaker:**Eric Cooper**Title:**Selection of dominant quasi-stationary states in 2d Navier-Stokes on the symmetric and asymmetric torus**Abstract:**Recent numerical studies have revealed certain families of functions play a crucial role in the long time behavior of 2D Navier-Stokes with periodic boundary conditions. These functions, called bar and dipole states, exist as quasi-stationary solutions and attract trajectories of all other initial conditions exponentially fast. Using geometric singular perturbation theory, this talk will focus on the separation in decay rates among certain lower and higher Fourier modes. The disparity in these decay rates for small values of the vorticity parameter leads to the evolution to these quasi-stationary states. Certain numerical results of the stochastically forced equations will also be discussed.

**SPECIAL TIME: 3:15 - 4:00 pm & PLACE SOC B65**

**Speaker:**Yujia Zhou**Title:**Slow M-current helps an inhibitory cell phase-lock to faster gamma pulses**Abstract:**Theta (4-10 Hz) and gamma (30-90 Hz) rhythms in the brain are commonly associated with memory and learning. The precision of co-firing between neurons and incoming inputs is critical in these cognitive functions. We consider an inhibitory neuron model with M-current under forcing from gamma pulses and a sinusoidal current in theta frequency. The M-current has a long time constant (~100 ms) and it is shown to have resonance with theta frequencies. However, we have found that in our model, the presence of a slow M-current helps the cell phase-lock to a faster gamma input. This phenomenon cannot be observed if the sinusoidal current is faster than the theta band. We characterize the dynamical mechanisms underlying the role of M-current in enabling a cell to entrain to gamma frequency input.

**Speaker:**Anthea Cheung**Title:**Patterns in the wake of traveling waves in the FitzHugh-Nagumo system**Abstract:**The Fitzhugh-Nagumo (FHN) equations are a system of reaction-diffusion equations first developed to model action potentials in the giant squid axon, and is now often used to model nerve conduction. Motivated by irregular patterns in patients undergoing epileptic seizures, we are interested in finding traveling wave solutions to the FHN system with irregular wakes. First we will discuss how pulses damped oscillatory tails can arise in the system, and why chaotic solutions could occur near such pulses. We will then discuss other reaction-diffusion systems with chaotic wakes and how those results may help us find parameter regimes with similar behavior in the FHN.

**Speaker:**Doris Wu**Title:**Analysis of Approximate Slow Invariant Manifold Method for Reactive Flow**Abstract:**The Approximate Slow Invariant Manifold (ASIM) method is a useful method for addressing model reduction in systems of reactive flow equations. It exploits separations of time scales between slow and fast species, and it generalizes the Intrinsic Low-Dimensional Manifold method, which was developed for model reduction in the context of reaction kinetics, to systems with diffusive and active transport. We will introduce the ASIM method, present the mathematical analysis of the ASIM method in the context of systems of reaction-diffusion equations, and explicitly determine the accuracy of the ASIM method. Our analysis includes precise statements of the errors at second order, and we find that these are proportional to the slow components of the reaction-diffusion equation, as well as to the curvature of the critical manifold.