For students who have taken Math 1b, AM/Math 21a, Math 21b, there was 6-7 week of differential equations background (11 classes in Math 1b + 9 classes in 21b).  See my prior post for the Math 1b diff eq content that is relevant to Dynamical Systems.

Student diff eq background from Math 21b:

• Linear first order systems
  • Interpret a linear system written in vector/matrix notation.
  • Use an eigenbasis and eigenvalues for the matrix of a linear system to construct a general solution.
  • Sketch a phase portrait for a 2d system.
  • Match a 2d system to its phase portrait.
  • Work with the matrix exponential.
  • Define “asymptotically stable” and “equilibrium point”.
  • Use eigenvalues of a linear system (with fixed point at the origin) to determine whether the origin is asymptotically stable.
  • Relate the eigenvalues, determinant, and trace of the matrix to the phase portrait about the origin for a linear system (with fixed point at the origin).
• Nonlinear first order systems
  • Sketch nullclines, identify equilbrium points, and add vectors of the vector field to regions of the phase plane.
  • Use the Jacobian to identify the behavior of a system near an equilibrium point.
• Higher order linear constant coefficient homogeneous differential equations
  • Convert from a second order equation to a first order system.
  • Find the characteristic polynomial.
  • Construct a general solution by converting to a matrix equation.
  • Solve higher order homogenous constant coefficient linear differential equations.
  • For a nonhomogeneous equation with a sinusoidal forcing (not at the natural frequency), find the general solution.

Many of these topics overlap with Math 1b, but are presented in matrix form in 21b.