We cover the Hartman-Grobman theorem, which says that the local phase portrait of a nonlinear system near a hyperbolic fixed point is topologically equivalent (i.e., it maps homeomorphically) to the phase portrait of the linearized system, and the stability of the fixed point is preserved in the linear system. Therefore, we can analyze 2D nonlinear systems in the phase plane and apply what we already know from linear ODEs to classify the fixed points.

We will spend some time with the Rabbits-Sheep model from Strogatz 6.4: x′ = x*(3 - x - 2*y), y′ = y*(2 - x - y). We will study all its fixed points via linearization and finally plot the phase portrait. New ideas illustrated include basins of attraction, basin boundaries, separatrices, and we will examine some characteristic trajectories using XPP.

We will also spend a good deal of time with the Morris-Lecar model, and study how the stable equilibrium loses stability as applied current is incrementally increased. New concepts include limit cycles, which we will see emerge at a Hopf bifurcation. XPP will be used to compute the eigenvalues of the linearized system and we will examine in detail how the eigenvalues are complex conjugates below the Hopf point, to complex and positive beyond it, being pure imaginary at the Hopf point.

We will intuitively analyze how and why the stable resting membrane potential destabilizes in terms of magnitude and kinetics of the calcium and potassium currents, which relates directly to the linearized system and eigenvalues.