Ordinary Differential Equations MATH 119A
Midterm Material

Extra practice problems are suggested below. These probelms are very similar to those assigned for homework.

Chapter 2 - 1D odes: u'=f(u)

Graphical analysis:
- Find fixed points and assess there stability.
- Draw phase portrait (phase line) showing fixed points and their stability as well as direction and magnitude of the flow.
- Plot estimates of solutions u vs t (including curvature of u(t)) for different initial conditions.

Analytically find fixed points and use linear stability analysis to find their stability. Know the limitation of linear stability analysis.

Know why oscillations are impossible for 1D odes on the real line (i.e. not flow on a circle).

Homework problems
Examples 2.2.1, 2.4.2, 2.4.3
Problems 2.2.1-9, 2.4.1-7


Chapter 3 - 1D odes: Bifurcations u'=f(u;r)

Know how to identify and classify bifurcations both graphically and analtyically
(saddle-node bifurcations, transcritical bifurcations, and subcritical and supercritical pitchfork bifurcations).

Know how to compute bifurctation diagrams.

Know how to relate bifurcation diagrams to phase portraits.

Know how to compute normal forms
(i.e. Taylor series expansion of equation around fixed point/bifurcation point, keeping only the lowest terms).

Understand ideas of imperfect bifurcations (including two-parameter bifurcation diagrams).

Homework problems
Examples 3.1.1-2, 3.2.1-2, 3.4.1, section 3.6
Problems 3.1.1, 3.1.4, 3.2.1, 3.2.3, 3.4.3-4, 3.6.2-4, 3.7.3


Chapter 4 - 1D odes: Flow on a circle

Know how the ideas of Chapters 2 and 3 carry over to flow on a circle (i.e. periodic domains).

Homework problems
Section 4.3
Problems 4.3.5


Chapter 5 - 2D odes: Linear systems x'=ax+by, y'=cx+dy

Know how to classify the fixed point (0,0) based on eigenvlaues
(spirals, nodes, saddles, star nodes, degenerate nodes, centers ... also lines of fixed points).

Draw phase plane:
- plot vector field/direction field (plotting nullclines helps),
- plot trajectories,
- calculate eigenvector and know their geometric interpretations.

Write the general solution of system of ODEs in terms of eigenvalues and eigenvectors.

Know basic/intuitive definitions of stability (attracting vs Lyapunov stability).

Homework problems
Examples 5.1.2, 5.2.1, 5.2.3
Problems 5.2.3-10.


Chapter 6 - 2D odes: Non-linear systems x'=f(x,y), y'=g(x,y)

Know how to find fixed points and classify them using linear stability analysis.

Draw phase plane: plot nullclines, vector field/direction field and trajectories

Know the relationship between x vs t, y vs t and trajectories in phase plane.

Know the concept of basins of attraction.

Homework problems
section 6.4 (rabbit-sheep problem)
Problems 6.3.1-6, 6.4.2-3.