Dynamical Systems M.S. Comprehensive Exam Outline

I.

Differential Equations

A.

Existence and uniqueness (no proof)

B.

Basic bifurcations in 1-D and 2-D flows

i.

Saddle-node

ii.

Pitchfork(subcritical and supercritical)

iii.

Transcritical bifurcation

iv.

Hopf bifurcation (subcritical and supercritical)

v.

Hysteresis

C.

Linear stability analysis for equilibria.

D.

Flows on the circle.

E.

Poincare-Bendixson Theorem, conservative systems in the plane, pendulum.

II.

Maps

A.

1-D maps

i.

Quadratic family and the logistic map

ii.

Sensitive dependence on initial conditions

iii.

Tent map

iv.

Conjugacy

v.

One-sided shift map

vi.

Sharkovskii Theorem

B.

2-D maps

i.

Markov partitions

ii.

Smale's horseshoe

iii.

Conjugacy to a 2-sided shift

iv.

Transverse homoclinic point

C.

Linearized stability for maps

References:

S. Strogatz: Nonlinear Dynamics and Chaos, Addison-Wesley 1997.

K. Alligood, T. Sauer, J. Yorke: Chaos: An introduction to Dynamical Systems, Springer 1996.

J. Hale, H. Kocak: Dynamics and Bifurcations, Springer 1991.