Ordinary Differential Equations MATH 119A
Material for Final Exam

Extra practice problems to supplement the homework problems are suggested below. Items 1-4 are from Mid-Term Material.

1. 1D ODEs:
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. Analytical methods: Find fixed points and use linear stability analysis to find their stability. Know the limitation of linear stability analysis. Examples 2.2.1, 2.4.2, 2.4.3, Problems 2.2.1-9, 2.4.1-7

2. Bifurcations in 1D ODEs:
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. Understand ideas of imperfect bifurcations (including two-parameter bifurcation diagrams). 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

3. 2D Linear systems.
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. Examples 5.1.2, 5.2.1,3, Problems 5.2.3-10.

4. Phase plane analysis:
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. Be able to explain why trajectories can’t cross. Problems 6.3.1-6, 6.4.2-3.

5. Conservative systems: especially of the form x''=F(x) (Section 6.5)
Know how to calculate the "Energy function", i.e. the conserved quantity; know how to show that "Energy" is conserved along trajectories (solutions of the ODE) and that trajectories are along level sets (contours) of the Energy function; know what types of fixed points are permissible and be able to explain why. Examples 6.5.1,2,3 Problems 6.5.4,5,13

6. Index Theory: (Section 6.8)
Know how to calculate the index of a closed curve and of a point; Know the basic properties of indices (including thm 6.8.1,2) and use them in simple applications. Example 6.8.5, Problem 6.8.8

7. Lyapunov functions: (Section 7.2)
Know the ideas behind Lyapunov functions and how to use them to show asymptotic stability (or instability). Know how to identify a gradient system and how calculate Lyapunov functions for gradient systems. Problems 7.2.6a,b (show that V are Lyapunov Fn), 7.2.10, Example 7.2.3

8. Dulac’s Criteon: (Section 7.2)
Know the Dulac’s Criteon and how to apply it to rule out the existence of limit cycles. Examples 7.2.4,5

9. Poincare-Bendixson Thm: (Section 7.3)
Know the Poincare-Bendixson Thm and how to apply it to prove the existence of limit cycles. Problem 7.3.3 Examples 7.3.1,2,3.

10. Polar coordinates:
Know to how analyze 2-D ODEs in polar coordinates. Problems 7.1.2, 7.3.5.

11. Bifurcations for 2-D ODEs: (Section 8.1,2)
Know how to find Hopf bifurcations and saddle-node bifurcations. Examples 8.1.1, 8.3.1,2