Applications of Systems of Equations- Consider the nonlinear autonomous system
x' = y + ax^3 + axy^2
(14)y' = -x + ay^3 + ayx^2.
433a. Verify that the origin is a critical point of system (14).b. Write the associated linear system at (0, 0), calculate the eigenvalues of
the linear system and verify that the origin is a neutrally stable center
of the linear system.c. Define r^2 (t) = x^2 (t) + y^2 (t).
(i) For a < 0, prove dr/dt < 0 provided (x, y) -/= (0, 0). Conclude,
therefore, that the origin is an asymptotically stable critical point of the
nonlinear system (14).
(ii) For a > 0, prove dr / dt > 0 provided (x, y) -/= (0, 0). Hence, conclude
that the origin is an unstable critical point of the nonlinear system (14).10.3 Modified Richardson's Arms Race Models
Let x(t) ;:::: 0 and y(t) ;:::: 0 represent the amounts spent per year on arms by
two nations. According to Richardson's arms race model, the expenditures for
arms per year satisfy the following li near autonomous system of differential
equations(1)dx- = Ay- Cx+r
dt
dy
dt =Bx - Dy +swhere A, B, C, and D are nonnegative, real constants and r and s are real
constants. The term Ay represents the rate of increase in yearly expenditures
for arms by the first nation due to its fear of t h e second nation. The term
-Cx represents the rate of decrease in yearly expenditures for arms by the first
nation due to the resistance of its people to increased spending for arms. The
constant term r represents the underlying "grievance" which t h e first nation
feels toward the second nation provided r > 0 or the underlying "feeling of
goodwill" provided r < 0. The terms in the second equation in system (1)
can be interpreted in an analogous manner.