In: Advanced Math
This problem involves a periodic system: while t can be any real number, the dependent
variable θ(t) ∈ [0, 2π). Consult chapter 4 of Strogatz. Consider the equation
θ ̇ = μ + sin θ.
(a) Draw phase portraits for different values of μ, find the
bifurcation values of μ, and
describe the fixed points and their stability in each regime.
(b) Now consider specifically the case where μ is just slightly less than 1: what is true about the fixed point(s) you found above now? The equation in this case is an example of an “excitable system,” which means it has a single attracting rest state, but a sufficiently large stimulus can cause the system to make a large excursion before returning to its rest state. Think of the initial conditions as different stimuli. Explain why this is an excitable system.
(c) Write a program to solve the system using the backward Euler method. Plot various trajectories θ(t) for different initial conditions on the same plot. On a second graph, plot V (t) = cos[θ(t)] for each of the same initial conditions. This is a very simple model for the response function of a neuron to a stimulus (see the Wikipedia page on the “Kuramoto model.” )