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To determine an equation for Q we multiply the first of Eqs. (4) by y, the second by x, and subtract, obtaining
/Jx x& = J + y2. (9)
J dt dt J w
Upon calculating dx/ dt and dy/dt from Eqs. (6), we find that the left side of Eq. (9) is r2(dQ/dt), so Eq. (9) reduces to
= 1. (10)
dt V '
The system of equations (8), (10) for r and Q is equivalent to the original system (4). One solution of the system (8), (10) is
r = 1, Q = t + t0, (11)
where t0 is an arbitrary constant. As t increases, a point satisfying Eqs. (11) moves clockwise around the unit circle. Thus the autonomous system (4) has a periodic solution. Other solutions can be obtained by solving Eq. (8) by separation of variables; if r = 0 and r = 1, then
dr = dt. (12)
r(1 - r2)
Equation (12) can be solved by using partial fractions to rewrite the left side and then integrating. By performing these calculations we find that the solution of Eqs. (10) and (12) is
r = , 1 =, Q = -t +10, (13)
\/1 + c0e 2t
where c0 and t0 are arbitrary constants. The solution (13) also contains the solution
(11), which is obtained by setting c0 = 0 in the first of Eqs. (13).
The solution satisfying the initial conditions r = p, Q = a at t = 0is given by
r = , 1 t, Q = (t a). (14)
V1 + [(1/p2) 1]e2t
If p < 1, then r ^ 1 from the inside as t ^ to; if p > 1, then r ^ 1 from the outside as t ^ to. Thus in all cases the trajectories spiral toward the circle r = 1 as t ^ to. Several trajectories are shown in Figure 9.7.1.
Chapter 9. Nonlinear Differential Equations and Stability
FIGURE 9.7.1 Trajectories of the system (4); a limit cycle.
In this example the circle r = 1 not only corresponds to periodic solutions of the system (4), but, in addition, other nonclosed trajectories spiral toward it as t ^ to. In general, a closed trajectory in the phase plane such that other nonclosed trajectories spiral toward it, either from the inside or the outside, as t ^ to, is called a limit cycle. Thus the circle r = 1 is a limit cycle for the system (4). If all trajectories that start near a closed trajectory (both inside and outside) spiral toward the closed trajectory as t ^ to, then the limit cycle is asymptotically stable. Since the limiting trajectory is itself a periodic orbit rather than an equilibrium point, this type of stability is often called orbital stability. If the trajectories on one side spiral toward the closed trajectory, while those on the other side spiral away as t ^ to, then the limit cycle is said to be semistable. If the trajectories on both sides of the closed trajectory spiral away as t ^ to, then the closed trajectory is unstable. It is also possible to have closed trajectories that other trajectories neither approach nor depart from, for example, the periodic solutions of the predator-prey equations in Section 9.5. In this case the closed trajectory is stable.
In Example 1 the existence of an asymptotically stable limit cycle was established by solving the equations explicitly. Unfortunately, this is usually not possible, so it is worthwhile to know general theorems concerning the existence or nonexistence of limit cycles of nonlinear autonomous systems. In discussing these theorems it is convenient to rewrite the system (1) in the scalar form
dx/ dt = F(x, y), dy/dt = G(x, y). (15)
Theorem 9.7.1 Let the functions F and G have continuous first partial derivatives in a domain D of the xy-plane. A closed trajectory of the system (15) must necessarily enclose at least one critical (equilibrium) point. If it encloses only one critical point, the critical point cannot be a saddle point.
9.7 Periodic Solutions and Limit Cycles
Although we omit the proof of this theorem, it is easy to show examples of it. One is given by Example 1 and Figure 9.7.1 in which the closed trajectory encloses the critical point (0, 0), a spiral point. Another example is the system of predator-prey equations in Section 9.5; see Figure 9.5.2. Each closed trajectory surrounds the critical point (3, 2); in this case the critical point is a center.
Theorem 9.7.1 is also useful in a negative sense. If a given region contains no critical points, then there can be no closed trajectory lying entirely in the region. The same conclusion is true if the region contains only one critical point, and this point is a saddle point. For instance, in Example 2 of Section 9.4, an example of competing species, the only critical point in the interior of the first quadrant is the saddle point (0.5, 0.5). Therefore this system has no closed trajectory lying in the first quadrant.
A second result about the nonexistence of closed trajectories is given in the following theorem.
Theorem 9.7.2 Let the functions F and G have continuous first partial derivatives in a simply connected domain D of the xy-plane. If Fx + Gy has the same sign throughout D, then there is no closed trajectory of the system (15) lying entirely in D.
A simply connected two-dimensional domain is one with no holes. Theorem 9.7.2 is a straightforward consequence of Greens theorem in the plane; see Problem 13. Note that if Fx + Gy changes sign in the domain, then no conclusion can be drawn; there may or may not be closed trajectories in D.