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To illustrate Theorem 9.7.2 consider the system (4). A routine calculation shows that
Fx(x, y) + Gy(x, y) = 2 — 4(x2 + /) = 2(1 — 2r2), (16)
where, as usual, r2 = x2 + y2. Hence Fx + Gy is positive for 0 < r < 1 /x/2, so there is no closed trajectory in this circular disk. Of course, we showed in Example 1 that there is no closed trajectory in the larger region r < 1. This illustrates that the information given by Theorem 9.7.2 may not be the best possible result. Again referring to Eq. (16), note that Fx + Gy < 0 for r > 1/x/2. However, the theorem is not applicable in this case because this annular region is not simply connected. Indeed, as shown in Example 1, it does contain a limit cycle.
The following theorem gives conditions that guarantee the existence of a closed trajectory.
Theorem 9.7.3 (Poincare-Bendixson6 Theorem). Let the functions F and G have continuous first
partial derivatives in a domain D of the xy-plane. Let D1 be a bounded subdomain
in D, and let R be the region that consists of D1 plus its boundary (all points of R are in D). Suppose that R contains no critical point of the system (15). If there
exists a constant t0 such that x = 0(t), y = ty(t) is a solution of the system (15)
that exists and stays in R for all t > t<), then either x = $(t), y = $(t) is a periodic
6Ivar Otto Bendixson (1861-1935) was a Swedish mathematician. This theorem appeared in a paper published by him in Acta Mathematica in 1901.
Chapter 9. Nonlinear Differential Equations and Stability
solution (closed trajectory), or x = 0(t), y = ty(t) spirals toward a closed trajectory as t ^ to. In either case, the system (15) has a periodic solution in R.
Note that if R does contain a closed trajectory, then necessarily, by Theorem 9.7.1, this trajectory must enclose a critical point. However, this critical point cannot be in R. Thus R cannot be simply connected; it must have a hole.
As an application of the Poincare-Bendixson theorem, consider again the system (4). Since the origin is a critical point, it must be excluded. For instance, we can consider the region R defined by 0.5 < r < 2. Next, we must show that there is a solution whose trajectory stays in R for all t greater than or equal to some tQ. This follows immediately from Eq. (8). For r = 0.5, dr/dt > 0, so r increases, while for r = 2, dr/dt < 0, so r decreases. Thus any trajectory that crosses the boundary of R is entering R. Consequently, any solution of Eqs. (4) that starts in R at t = t0 cannot leave but must stay in R for t > tQ. Of course, other numbers could be used instead of
0.5 and 2; all that is important is that r = 1 is included.
One should not infer from this discussion of the preceding theorems that it is easy to determine whether a given nonlinear autonomous system has periodic solutions or not; often it is not a simple matter at all. Theorems 9.7.1 and 9.7.2 are frequently inconclusive, while for Theorem 9.7.3 it is often difficult to determine a region R and a solution that always remains within it.
We close this section with another example of a nonlinear system that has a limit cycle.
The van der Pol (1889-1959) equation
u" — x(1 — u2)u' + u = 0, (17)
where x is a positive constant, describes the current u in a triode oscillator. Discuss the solutions of this equation.
If i = 0, Eq. (17) reduces to iW + u = 0, whose solutions are sine or cosine waves of period 2n. For x > 0 the second term on the left side of Eq. (17) must also be considered. This is the resistance term, proportional to u', with a coefficient — x(1 — u2) that depends on u. For large u this term is positive and acts as usual to reduce the amplitude of the response. However, for small u the resistance term is negative and so causes the response to grow. This suggests that perhaps there is a solution of intermediate size that other solutions approach as t increases.
To analyze Eq. (17) more carefully we write it as a second order system by introducing the variables x = u, y = u'. Then it follows that
x = y, y = -x + x(1 — x2) y. (18)
The only critical point of the system (18) is the origin. Near the origin the corresponding linear system is
;) '=(—? i) <19>
whose eigenvalues are (jx ± s//x2 — 4)/2. Thus the origin is an unstable spiral point for 0 < X < 2 and an unstable node for x > 2. In all cases, a solution that starts near the origin grows as t increases.
9.7 Periodic Solutions and Limit Cycles
With regard to periodic solutions Theorems 9.7.1 and 9.7.2 provide only partial information. From Theorem 9.7.1 we conclude that if there are closed trajectories, they must enclose the origin. Next, we calculate Fx (x, y) + Gy(x, y), with the result that
Fx(x, y) + Gy(x, y) = fz(1 — x2). (20)
Then, it follows from Theorem 9.7.2 that closed trajectories, if there are any, are not contained in the strip |x | < 1 where Fx + Gy > 0.
The application of the Poincare-Bendixson theorem to this problem is not nearly as simple as for the preceding example. If we introduce polar coordinates, we find that the equation for the radial variable r is