Qualifying Exams in Applied Mathematics

Applied mathematics exam
The applied mathematics qualifying exam covers topics in dynamical systems, partial differential equations, and applied linear algebra.
  1. 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.
  2. 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).
  3. 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.
  4. 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.
  5. Elements of applied linear algebra: eigenvalues, Rayleigh quotients, the Jordan normal form, singular value decomposition, Gram–Schmidt orthogonalization, convergence of finite difference schemes.
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
  • Friedberg, Insel, Linear algebra
  • Roman, Advanced Linear algebra
  • Bleeker, Csordas, Basic partial differential equations
  • Evans, Partial differential equations
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