Computer Algebra Recipes

2.1. PHASE-PLANE ANALYSIS 57
ations observed in the lynx and snowshoe hare data curves are not as smooth
and regular as in our idealized mathematical model.
PROBLEMS:
Problem 2-1: Iron core inductor
Consider the simple circuit shown in Figure 2.6, consisting of a charged capacitor
C connected to a coil of N turns wrapped around an iron core. The current i
C
i
iron core
inductor
Figure 2.6: Iron core inductor circuit.
versus °ux © relation for the iron core inductor has the form i = N ©=L0+A © 3 ,
where L0 is the self-inductance of the coil, © is the °ux threading through one
turn of the coil, and A>0.
(a) Using Kirchho®'s voltage rule, show that the governing ODE is given by
Ä©+® ©+¯ © 3 =0;
where ® and ¯ are left for you to identify.
(b) Reexpress the ODE in a dimensionless form with ® and ¯ scaled out.
(c) Analytically show that the origin of the phase plane is a vortex. Con¯rm
with a phase-plane ( _ © versus ©) portrait containing a representative orbit.
(d) Usethesceneoptiontoplot©(t).
Problem 2-2: Competing armies
The armies of two warring countries are modeled by the following equations:
_
C1 = ®C1 ¡ ¯C1 C2;
_
C2 =(® +1)C2 ¡ °¯C1 C2;
with ® and ¯ both positive and °>1. Here C1 and C2 are the numbers of
individuals in the armies of countries 1 and 2.
(a) Discuss the model equations and how the model could be improved.
(b) Analytically locate and identify all the stationary points.
(c) Taking ® = 5, ° = 1:15, and ¯ = 1=2500, make a tangent ¯eld plot
that includes all stationary points and some representative trajectories.
Discuss possible outcomes on the basis of this plot.
(d) Using appropriate scene options, create plots of C1(t) andC2(t).