Mathematics 345
Homework Assignment 4
Due Thursday 11 February 2016
1. Consider the following differential equation derived from the Kepler problem in physics.
¨ =
− x −
2 + x −
3 .
(a) Write the equation in the form ¨ =
V ( x ).
− dV dx and determine the potential function
(b) Introduce a new variable ˙ = y . Rewrite the equation into a system of two
1st-order ODEs of the form
˙ = f ( x, y ) , y ˙ = g ( x, y ) .
(c) Show that the total energy
E =
1
2 y
2 + V ( x ) , where V ( x ) is the potential found in (a) is a conserved quantity. (Hint: Show that dE dt
= ∂E
∂x
˙ + ∂E
∂y y ˙ = 0.)
(d) Sketch the phase portrait of the system obtained in (b). (Clearly mark the nullclines, fixed point(s), vector field directions, and some representative phase trajectories).
(e) Show that the steady state found in (d), correspond to the local minimum of the potential function V ( x ). Therefore, it is consistent with predictions of Theorem
6.5.1 and with the phase portrait obtained in (d).
(f) Generate an ode file for the system found in (b) (using the ode files from Homework
#3 as a starting template). Use XPPAUT to generate a phase portrait, print a hard copy of it and attach it to this assignment.
2. Consider a dimensionless version of the Lotka-Volterra model that describes a predatorprey type interaction.
˙ = αx
− xy, y ˙ = xy
− y, where α > 0 is a parameter.
(a) Sketch all the nulllclines.
(b) Find all fixed points.
(c) Classify the fixed points and sketch a local phase portrait near each fixed point.
(d) Sketch the global phase portrait using the results obtained in (a)-(c).
(e) Show that E = x + y
− ln( x )
−
α ln( y ) is a conserved quantity.
(f) Based on Theorem 6.5.1, which one of the two steady states correspond to the local minimum of the conserved quantity E ?
(g) Generate an ode file for the system. Use XPPAUT to generate a phase portrait, print a hard copy of it and attach it to this assignment.