Here is a list of graduate students who are willing to tutor
privately for the Spring 2024 term.

Moriah Aberle – maberle@hawaii.edu
Arturo Jaime – ajaime@hawaii.edu
Dennis Le – led6@hawaii.edu
Rico Vicente – rvicente@hawaii.edu
Kestrel Strom – kstrom2@hawaii.edu

Please contact them directly to make arrangements such as cost,
meeting time and place, etc. The Mathematics Department is not
responsible for these arrangements.

The Robert and Doris Pulley Award in Mathematics from the ARCS Foundation (Achievement Rewards for College Scientists, a national non-profit organization that boosts U.S. competitiveness by raising funds to support promising U.S. citizens in doctoral programs in the sciences, engineering, and medicine) has been awarded to the following students. Advisers in parenthesis.

The applied mathematics qualifying exam covers topics in dynamical systems, partial differential equations, and applied linear algebra.

Material

Basic dynamical systems concepts: definition of a dynamical system (continuous and discrete), equilibrium states, ω,α-limit sets, invariant sets, stability of equilibrium states and periodic solutions, population dynamics models; linear systems, stable, unstable, center spaces; non-linear systems and existence/uniqueness of solutions; linearization, topological equivalence/conjugacy, center manifold theory (applications: species competition models, SIR models, predator-prey models); some global nonlinear techniques (nullclines, Lyapunov function, applications: nonlinear pendulum, SIR models); limit cycles. Poincaré–Bendixson theory in $\mathbb R^2$ (applications: Van der Pol oscillator, predator-prey models with limit cycle, oscillating chemical reactions); stability of periodic solutions, Poincaré map.

Bifurcation theory: family of systems, structural stability, definition of a bifurcation; Peixoto’s theorem, Morse–Smale systems; examples of one-parameter bifurcations of equilibrium states (application: laser phenomenon); genericity, transverse intersections, versal unfoldings (deformations) and codimension of a bifurcation (application: spruce budworm model (codimension-2 bifurcation)); the Hopf bifurcation (applications: oscillating chemical reactions, FitzHugh–Nagumo model); center manifold theory (for bifurcations); global bifurcations (homoclinic, heteroclinic).

Introduction to chaos: examples of chaotic behavior (discrete logistic model, Duffing oscillator, Lorenz system, Henon map, Horseshoe map, symbolic dynamics), sensitivity to initial condition; more on logistic model (period doubling, Feigenbaum constant, dense periodic orbits and Sharkovskii’s theorem); strange attractors; Lyapunov exponents.

Elements of partial differential equations: first order linear and quasilinear PDEs and the method of characteristics, second order linear PDEs and their classification, the Sturm–Liouville problem, Green’s functions and fundamental solutions, the Fourier transform, equilibrium solutions of time-dependent PDEs.

Elements of applied linear algebra: eigenvalues, Rayleigh quotients, the Jordan normal form, singular value decomposition, Gram–Schmidt orthogonalization, convergence of finite difference schemes.

Textbooks

The following textbooks are recommended:

Perko, Differential equations and dynamical systems

Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering

Hirsch, Smale, and Devaney, Differential equations, dynamical systems, and an introduction to chaos