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7a. The amplitude ratio is (cK/g)/(^ac K/a) = a^J~c /gV"I .
7b. From Eq (2) a = .5, a = 1, g = .25 and c = .75, so the ratio is .5V .75 /.25^1 = 2V .75 = = 1.732.
7c. A rough measurement of the amplitudes is (6.1 - 1)/2 = 2.55 and (3.8 -. 9)/2 = 1.45 and thus the ratio is approximately 1.76. In this case the linear approximation is a good predictor.
11. The presence of a trapping company actually would require a modification of the equations, either by altering the coefficients or by including nonhomogeneous terms on the right sides of the D.E. The effects of indiscreminate trapping could decrease the populations of both rabbits and fox significantly or decrease the fox population which could possibly lead to a large increase in the rabbit population. Over the long run it makes sense for a trapping company to operate in such a way that a consistent supply of pelts is available and to disturb the predator-prey system as little as possible. Thus, the company should trap fox only when their population is increasing, trap rabbits only when their population is increasing, trap rabbits and fox only during the time when both their populations are increasing, and trap neither during the time both their populations are decreasing. In this way the trapping company can have a
moderating effect on the population fluctuations, keeping
the trajectory close to the center.
13. The critical points of the system are the solutions of the algebraic equations x(a - ox - ay) = 0, and y(-c + gx) = 0. the critical points are x = 0, y = 0;
x = a/o, y = 0; and x = c/g, y = a/a - co/ag = oA/a
where A = a/o - c/g > 0.
To study the critical point (0,0) we discard the nonlinear terms in the system of D.E. to obtain the corresponding linear system dx/dt = ax, dy/dt = -cy. The characteristic equation is r2 - (a+c)r - ac = 0 so ri = a, r2 = -c. Thus the critical point (0,0) is an unstable saddle point.
To study the critical point (a/o,0) we let x = (a/o) + u, y = 0 + v and substitute in the D.E. to obtain the almost linear system du/dt = -au - (aa/o)v - ou2 - auv, dv/dt = gAv + guv. The corresponding linear system is du/dt = -au - (aa/o)v, dv/dt = gAv. The characteristic equation is r2 + (a - gA)r - agA = 0 so r1 = -a, r2 = gA.
Thus the critical point (a/o,0) is an unstable saddle point.
To study the critical point (c/g, oA/a) we let x = (c/g) + u, y = (oA/a) + v and substitute in the D.E. to obtain the almost linear system
= -(co/g)u - (ac/g)v - ou2 - auv dt
= (oAg/a)u + guv dt
The corresponding linear system is du/dt = -(co/g)u - (ac/g)v, dv/dt = (oAg/a)u. The characteristic equation is r2 + (co/g)r + coA = 0, so r1,r2 = [-(co/g) ħ V (co/g) 2 - 4coA ] /2. Thus, depending on the sign of the discriminant we have that (c/g, oA/a) is either an asymptotically stable spiral point or an asymptotically stable node. Thus for nonzero initial data (x,y) (c/g, oA/a) as t .
Section 9.6, Page 519
1. Assuming that V(x,y) = ax2 + cy2 we find Vx(x,y) = 2ax,
Vy = 2cy and thus Eq.(7) yields V(x,y) = 2ax(-x3 + xy2) +
2cy(-2x2y - y3) = -[2ax4 + 2(2c-a)x2y2 + 2cy4]. If we choose a and c to be any positive real numbers with
2c > a, then V is a negative definite. Also, V is positive definite by Theorem 9.6.4. Thus by Theorem 9.6.1 the origin is an asymptotically stable critical point.
3. Assuming the same form for V(x,y) as in Problem 1, we
V(x,y) = 2ax(-x3 + 2y3) + 2cy(-2xy2) = -2ax4 + 4(a-c)xy3.
If we choose a = c > 0, then V(x,y) = -2ax4 < 0 in any
neighborhood containing the origin and thus V is negative semidefinite and V is positive definite. Theorem 9.6.1 then concludes that the origin is a stable critical point. Note that the origin may still be asymptotically stable, however, the V(x,y) used here is not sufficient to prove that.
6a. The correct system is dx/dt = y and dy/dt = -g(x). Since g(0) = 0, we conclude that (0,0) is a critical point.
From the given conditions, the graph of g must be positive for 0 < x < k and negative for -k < x < 0. Thus
if 0 < x < k then xg(s)ds > 0,
f x f 0
if -k < x < 0 then g(s)ds = - g(s)ds > 0.
Since V(0,0) = 0 it follows that V(x,y) = y2/2 + xg(s)ds
is positive definite for -k < x < k, - < y < . Next,
we have V(x,y) = Vx + Vy = g(x)y + y[-g(x)] = 0.
Since V(x,y) is never positive, we may conclude that it is negative semidefinite and hence by Theorem 9.6.1 (0,0) is at least a stable critical point.
V is positive definite by Theorem 9.6.4. Since Vx(x,y) = 2x, Vy(x,y) = 2y, we obtain
V(x,y) = 2xy - 2y2 - 2ysinx = 2y[-y + (x - sinx)]. If
x < 0, then V(x,y) < 0 for all y > 0. If x > 0, choose y
so that 0 < y < x - sinx. Then V(x,y) > 0. Hence V is not a Liapunov function.
Since V(0,0) = 0, 1 - cosx > 0 for 0 < |x| < 2p and y2 >
0 for y n 0, it follows that V(x,y) is positive definite
in a neighborhood of the origin. Next Vx(x,y) = sinx, Vy(x,y) = y, so
V(x,y) = (sinx)(y) + y(-y - sinx) = -y2. Hence V is negative semidefinite and (0,0) is a stable critical point by Theorem 9.6.1.