1550078481-Ordinary_Differential_Equations__Roberts_

436 Ordinary Differential Equations

At (2, 3) the associated linear system is

x' = fx(2, 3)(x - 2) + fy(2, 3)(y- 3) = -8(x - 2) + 3(y - 3)

y' = 9x(2, 3)(x - 2) + gy(2, 3)(y - 3) = 8(x - 2) - 6(y - 3).

The coefficient matrix of this linear system

A= (-8 3)

8 -6

has eigenvalues -2 and -12. Since both eigenvalues are negative, the critical
point (2, 3) is an asymptotically stable node of both the associated linear
system and the nonlinear system (6).

!Comments on Computer Software! Various computer software packages
include algorithms which numerically solve the system initial value problem

Y~ = fi(t,y1,Y2, ... ,yn); Y1(c) = d1

(7)

Y~ = fn(t, Yi, Y2, · · ·, Yn); Yn(c) = dn

on the interval [a, b] for c E [a , b] and for 2::; n::; N for some given maximum
integer value N. The software accompanying this text contains a program
named SOLVESYS which numerically solves the IVP (7) for a maximum value

of N = 6. Complete instructions for using SOLVESYS appear in Appendix A.

After the numerical solution has been calculated you may elect (i) to print
solution components, (ii) to graph any subset of the components in any order
on a rectangle R where a ::; t ::; b and YMIN ::; y ::; YMAX and you select
the values for YMIN and YMAX, or (iii) to produce a phase-plane portrait
of Yi versus Yj for any two distinct components Yi and Yj on any rectangle in
YjYi-space.

EXAMPLE 10 Computer Solution of a Nonlinear Modified

Richardson's Arms Race Model

Solve the system initial value problem

dx

dt = 3y - 2x^2 - 1 = f(x, y); x(O) = 2

(8)

dy

dt = 8x - y^2 - 7 = g(x, y); y(O) = O

on the interval [O, 2.5]. Display a graph of x(t) and y(t). Produce a phase-
plane graph of y(t) versus x (t) on the rectangle

R = { ( x, y) I 0 ::; x ::; 5 and 0 ::; y ::; 5}.