M374M
Take-Home Exam #2
1) (11pt) Consider
(a) Sketch the nullclines, and the direction of orbits in all regions separated by the nullclines.
(b) Find the equilibria and classify their type and stability using the Jacobian.
2) (13pt) In dimensionless form, a model for a thermo-chemical reaction is given below, where u ≥ 0 is the concen- tration of the reactant, q ≥ 0 is the temperature, and k > 0 is a parameter.
(a) Find the single equilibrium of the system.
(b) Determine the type and stability of the equilibrium for each k > 0.
(c) Find a trapping region R and a value k# and show that periodic orbits exist in R for k > k# . (The boundary of R may contain arcs of solution curves; any such arcs should be specified.)
3) (13pt) Consider εx3 + 3εx2 + 2x + 1 = 0, where 0 < ε ≪ 1 is a parameter.
Find a two-term perturbation approximation of each of the three roots x(ε).
4) (13pt) In dimensionless form, a model for the vertical motion of a projectile in a non-constant gravity field is given below, where u is the height above ground level, and 0 ≤ ε ≪ 1 is a parameter. (The case ε = 0 is a constant gravity model, whereas ε > 0 is Newton’s inverse-square law.)
(a) Find a perturbation approximation of u(t, ε) up to and including terms of order O(ε).
(b) Plot u versus t ∈ [0, 2.5] for the cases ε = 0 and ε = 0.1 and numerically determine the total fiight time for each case. Compared to the ε = 0 case, what is the percent change in the fiight time when ε = 0.1?